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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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. $


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.


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.


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}.


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.


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.


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.


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.


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


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


Improvement of graph theory Wei`s inequality

Abstract: In this paper we give a generalization of a result of Wei.


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.


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.


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.


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.


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.


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


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.


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.


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.


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.


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 $i\geq 1$.


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.


Riesz transforms for Jacobi expansions

Abstract: We define and study Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions.


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.


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.


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.


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 $q^{\epsilon} \le N \le q^{1 -\epsilon}$.


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.


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.


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.


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.


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.


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).


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.


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.


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.


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.


On A-tensors in Riemannian geometry

Abstract: We present examples, both compact and non-compact complete, of lo- cally non-homogeneous proper A-manifolds.


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.


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.


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.


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).


Markov loops and renormalization

Abstract: We study Poissonian ensembles of Markov loops and the associated renormalized self-intersection local times.


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.


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.


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.


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.


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.


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.


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.


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.


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 $M_{21}=\mathbf{PSL}(3,4)$.


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.


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.


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.


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.


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.


Connect sum and transversely non simple knots

Abstract: We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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 $r$.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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 $\mathbb R^{2}.$


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.


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 $Y_i=\sum^d_{j=1}c_{ij}\partial_j$.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


q-Abel polynomials

Abstract: This note gives a simple approach to q-analogues of some results associated with Abel polynomials.


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.


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.


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 $\mathbb{C}(z)$.


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.


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.


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.


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.


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 $\bP(G)$.


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.


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.


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, $T^1(\text{SL}_2 \mathbb{Z}\backslash \mathbb{H})$.