Discussion: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$

Abstract: Discussion of "The Dantzig selector: Statistical estimation when $p$ is much larger than $n$" [math/0506081]


Abstract: Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]


Discussion: The Dantzig selector: statistical estimation when $p$ is much larger than $n$


Countable Choice and Compactness

Abstract: We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.


Effect of linear lumping on controllability and observability

Abstract: The effect of linear lumping, linear transformation to reduce the number of state variables on controllability and observability of linear differential equations has been studied. Controllability of the original system implies the controllability of the lumped system. Examples taken from reaction kinetics illustrate our results.


Discussion: A tale of three cousins: Lasso, L2Boosting and Dantzig

Abstract: Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' by Emmanuel Candes and Terence Tao [math/0506081]


Rejoinder: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$

Abstract: Rejoinder to ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]


Counting Schr\"odinger boundstates: semiclassics and beyond

Abstract: This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassical rules are violated. Some new results are presented, concerning operators on product manifolds and graphs.


Sharpness of some properties of Wiener amalgam and modulation spaces

Abstract: We prove sharp estimates for the dilation operator $f(x)\longmapsto f(\lambda x)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schr\"odinger propagator on modulation spaces.


Finite sets of $d$-planes in affine space

Abstract: Let $A$ be a subvariety of affine space $\mathbb{A}^n$ whose irreducible components are $d$-dimensional linear or affine subspaces of $\mathbb{A}^n$. Denote by $D(A)\subset\mathbb{N}^n$ the set of exponents of standard monomials of $A$. We show that the combinatorial object $D(A)$ reflects the geometry of $A$ in a very direct way. More precisely, we define a $d$-plane in $\mathbb{N}^n$ as being a set $\gamma+\oplus_{j\in J}\mathbb{N}e_{j}$, where $#J=d$ and $\gamma_{j}=0$ for all $j\in J$. We call the $d$-plane thus defined to be parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$. We show that the number of $d$-planes in $D(A)$ equals the number of components of $A$. This generalises a classical result, the finiteness algorithm, which holds in the case $d=0$. In addition to that, we determine the number of all $d$-planes in $D(A)$ parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$, for all $J$. Furthermore, we describe $D(A)$ in terms of the standard sets of the intersections $A\cap\{X_{1}=\lambda\}$, where $\lambda$ runs through $\mathbb{A}^1$.


A note on minimal finite quotients of mapping class groups

Abstract: We prove that the minimal nontrivial finite quotient group of the mapping class group M_g of a closed orientable surface of genus g is the symplectic group PSp(2g,Z_2), for g = 3 and 4 (this might remain true, however, for arbitrary genus g > 2). We discuss also some results for arbitrary genus g.


Symmetric jump processes: localization, heat kernels, and convergence

Abstract: We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the H\"older continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.


Documentation for the ratpoints program

Abstract: This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm.


Equiconvergence theorems for Sturm--Liouville operators with distribution potentials^ the rate of equiconvergence

Abstract: We consider a Sturm--Liouville operator $Ly=-y''+qy$ in $L_2[0,\pi]$ with Dirichlet boundary conditions. We assume, that the potential $q$ is complex valued and belongs to Sobolev space $W_2^\theta[0,\pi]$, $\theta\in(-1,-1/2$. This operators were successfully defined in papers of Savchuk A.M. and Shkalikov A.A. There were also shown, that theese operators have a discrete spectrum, which we denote by $\{\lambda_n\}$, and $\lim\lambda_n=+\infty$. All but finitely many of them are simple. The eigenfunctions form the Riesz basis in $L_2[0,\pi]$. We investigate a uniform on $[0,\pi]$ equiconvergence of series for this system and for trigonometric system $\{\sin(nt)\}_1^\infty$. We obtain not only a theorems of equiconvergence, but also estimate a rate of this equiconvergence.


Bari-Markus property for Riesz projections of Hill operators with singular potentials

Abstract: The Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,\pi],$ with $H^{-1}$ periodic potentials, considered with periodic, antiperiodic or Dirichlet boundary conditions, have discrete spectrum, and therefore, for sufficiently large $N,$ the Riesz projections $$ P_n = \frac{1}{2\pi i} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{z: |z-n^2|= n\} $$ are well defined. It is proved that $$\sum_{n>N} \|P_n - P_n^0\|^2_{HS} < \infty, $$ where $P_n^0$ are the Riesz projection of the free operator and $\|\cdot\|_{HS}$ is the Hilbert--Schmidt norm.


Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schr\"odinger Operators on Bounded Lipschitz Domains

Abstract: We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\"odinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$.


Polynomial representation for long knots

Abstract: We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.


Maximalite des varietes toriques de dimension 4

Abstract: A complex algebraic variety defined over the reals is maximal when the sum of its Betti numbers for Borel Moore homology with $\zz$ coefficients coincides with the sum of the Betti numbers of its real part. We will show in this paper that toric varieties of dimension 4 are maximal.


On some results of Cufaro Petroni about Student t-processes

Abstract: This paper deals with Student t-processes as studied in (Cufaro Petroni N 2007 J. Phys. A, Math. Theor. 40(10), 2227-2250). We prove and extend some conjectures expressed by Cufaro Petroni about the asymptotical behavior of a Student t-process and the expansion of its density. First, the explicit asymptotic behavior of any real positive convolution power of a Student t-density with any real positive degrees of freedom is given in the multivariate case; then the integer convolution power of a Student t-distribution with odd degrees of freedom is shown to be a convex combination of Student t-densities with odd degrees of freedom. At last, we show that this result does not extend to the case of non-integer convolution powers.


Anick's fibration and the odd primary homotopy exponent of spheres

Abstract: For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on the bottom cell. A much simpler construction of such a fibration for p>=3 was given by Gray and the author using new methods. In this paper the new methods are used to start over, first constructing Anick's fibration for p>=3, and then using it to obtain the exponent result for spheres.


Scattering theory for the Gross-Pitaevskii equation in three dimensions

Abstract: We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schr\"odinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.


Invariant subspaces of subgraded Lie algebras of compact operators

Abstract: We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the structure of such algebras. As an application, we prove a number of results on the existence of invariant subspaces for algebraic structures of compact operators. Along the way we obtain new criteria for the triangularizability of a Lie algebra of compact operators.


Fundamental groups of symmetric sextics

Abstract: We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with at least two type $\bold{E}_6$ singular points. As a simple application, we compute the fundamental groups of 125 other sextics, most of which are new.


Characters of prime degree

Abstract: Let $G$ be a finite nilpotent group, $\chi$ and $\psi$ be irreducible complex characters of $G$ of prime degree. Assume that $\chi(1)=p$. Then either the product $\chi\psi$ is a multiple of an irreducible character or $\chi\psi$ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.


Equivariant K-theory, groupoids and proper actions

Abstract: In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite equivariant CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.


On semi-infinite cohomology of finite dimensional graded algebras

Abstract: We describe a general setting for the definition of semi-infinite cohomology of finite dimensional algebras, and provide its categorical interpretation. We apply this interpretation to compute semi-infinite cohomology of some modules over the small group at a root of unity, generalizing an earlier result of S. Arkhipov (conjectured by B. Feigin).


Relative Cohen--Macaulayness of bigraded modules

Abstract: In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.


Les quartiers de la lune de Troie ou le bouclier d'Achille, essai sur le corps de classe

Abstract: This is a text in french about classical class field theory, readable with a minimum of prerequisites


Remarks on modular symbols for Maass wave forms

Abstract: In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions (pseudo--measures in the sense of [MaMar2]). After explaining the basic issues and analogies in the extended Introduction, I construct modular symbols in the sec. 1 and the related L\'evy--Mellin transforms in the sec. 2. The whole paper is an extended footnote to the Lewis--Zagier fundamental study [LZ2].


Bimonotone enumeration

Abstract: Solutions of a diophantine equation $f(a,b) = g(c,d)$, with $a,b,c,d$ in some finite range, can be efficiently enumerated by sorting the values of $f$ and $g$ in ascending order and searching for collisions. This article considers functions that are bimonotone in the sense that $f(a,b) \le f(a',b')$ whenever $a \le a'$ and $b \le b'$. A two-variable polynomial with non-negative coefficients is a typical example. The problem is to efficiently enumerate all pairs $(a,b)$ such that the values $f(a,b)$ appear in increasing order. We present an algorithm that is memory-efficient and highly parallelizable. In order to enumerate the first $n$ values of $f$, the algorithm only builds up a priority queue of length at most $\sqrt{2n}+1$. In terms of bit-complexity this ensures that the algorithm takes time $O(n \log^2 n)$ and requires memory $O(\sqrt{n} \log n)$, which considerably improves on the memory bound $\Theta(n \log n)$ provided by a naive approach, and extends the semimonotone enumeration algorithm previously considered by R.L. Ekl and D.J. Bernstein.


Recurrence and transience of a multi-excited random walk on a regular tree

Abstract: We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.


Branching Process approach for 2-SAT thresholds

Abstract: It is well known that, as $n$ tends to infinity, the probability of satisfiability for a random 2-SAT formula on $n$ variables, where each clause occurs independently with probability $\alpha/2n$, exhibits a sharp threshold at $\alpha=1$. We study a more general 2-SAT model in which each clause occurs independently but with probability $\alpha_i/2n$ where $i \in \{0,1,2\}$ is the number of positive literals in that clause. We generalize branching process arguments by Verhoeven(99) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.


Dimension quotients

Abstract: We present two approaches, one homological and the other simplicial, for the investigation of dimension quotients of groups. The theory is illustrated, in particular, with a conceptual discussion of the fourth and fifth dimension quotients.


Comparing Classes of Finite Structures

Abstract: We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This reducibility is calibrated by comparing several classes of structures. The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs. The class of finite graphs and the class of finite linear orders are maximal among all classes of finite structures. We also prove some general characterizations of reducibility to certain classes. Examples of large chains and antichains of classes are constructed.


Classification from a Computable Viewpoint

Abstract: Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work, showing that certain classes of mathematical structures admit classification, while others do not. In the present paper, we describe some recent work on classification in computable structure theory.


Index Sets of Computable Structures

Abstract: The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, $\mathbb{Q}$-vector spaces, Archimedean real closed ordered fields, reduced Abelian $p$-groups of length less than $\omega^{2}$, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be $m$-complete $\Pi_{n}^{0}$, $d$-$\Sigma_{n}^{0}$, or $\Sigma_{n}^{0}$, for various $n$. In each case, the calculation involves finding an \textquotedblleft optimal\textquotedblright% \ sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable $\Pi_{n}$, $d$-$\Sigma_{n}$, or $\Sigma_{n}$) yields a bound on the complexity of the index set. When we show $m$% -completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.


Structures in Familiar Classes Which Have Scott Rank $\omega_1^{CK}$

Abstract: There are familiar examples of computable structures having various computable Scott ranks. There are also familiar structures, such as the Harrison ordering, which have Scott rank $\omega_1^{CK}+1$. Makkai produced a structure of Scott rank $\omega_1^{CK}$, which can be made computable, and simplified so that it is just a tree. In the present paper, we show that there are further computable structures of Scott rank $\omega_1^{CK}$ in the following classes: undirected graphs, fields of any characteristic, and linear orderings. The new examples share with the Harrison ordering, and the tree just mentioned, a strong approximability property.


$L_{p,q}$-Cohomology of Warped Cylinders

Abstract: We extend some results by Gol'dshtein, Kuz'minov, and Shvedov about the $L_p$-cohomology of warped cylinders to $L_{p,q}$-cohomology for different $p$ and $q$. As an application, we establish some sufficient conditions for the nontriviality of the $L_{p,q}$-torsion of a surface of revolution in terms of some Hardy constants.


Factorial threefold hypersurfaces

Abstract: Let $X$ be a hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most isolated ordinary double points. We prove that $X$ is factorial in the case when $X$ has at most $(d-1)^{2}-1$ singular points.


A Generalized Backward Equation For One Dimensional Processes

Abstract: Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f, the backward Kolmogorov equation gives a condition for f(t,X) to be a local martingale. We generalize the backward equation in two main ways. First, it is extended to non-differentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability condition, and the martingale condition for f(t,X) is then expressed in terms of the marginal distributions, drift measure and jumps of X. The proof involves the stochastic calculus of Dirichlet processes and a time-reversal argument. These results are then applied to show that a continuous and strong Markov martingale is uniquely determined by its marginal distributions.


Higher Order Riesz Transforms for Laguerre Expansions

Abstract: In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings.


A connection whose curvature is the Lie bracket

Abstract: Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie bracket of vectorfields on the manifold. The motion of a ball rolling on an oriented surface is the parallel transport of a similar connection on the trivial SO(3)-bundle over the surface. If the surface is a plane or a sphere, then the curvature of the connection is a scalar multiple of the Lie bracket in the Lie algebra of SO(3).


On Sp_4 modularity of Picard--Fuchs differential equations for Calabi--Yau threefolds (with an appendix by Vicentiu Pasol)

Abstract: Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.


Cocompact imbeddings and structure of weakly convergent sequences

Abstract: Concentration compactness method is a powerful techniques for establishing existence of minimizers for inequalities and of critical points of functionals in general. The paper gives a functional-analytic formulation for the method in Banach space, generalizing the Hilbert space case elaborated in \cite{ccbook}. The key object is a dislocation space - a triple $(X,F,D)$, where $F$ is a convex functional that defines a norm on Banach space $X$, and $D$ is a group of isometries on $X$. Bounded sequences in dislocation spaces admit a decomposition into an asymptotic sum "profiles" $w^{(n)}\in X$ dislocated by actions of $D$, that is, a sum of the form $\sum_ng^{(n)}_kw^{(n)}$, $g^{(n)}_k\in D$, while the remainder term converges weakly under actions of any sequence $g_k\in D$ ({\em $D$-weak convergence}). This decomposition allows to extend the weak convergence argument from variational problems with compactness to problems where $X$ is {\em cocompactly} (relatively to the group $D$) imbedded into a Banach space $Y$, that is, when every sequence $D$-weakly convergent in $X$ is convergent in the norm of $Y$. We prove a general statement on existence of minimizers in cocompact imbeddings that applies, in particular to Sobolev imbeddings which lack compactness (unbounded domain, critical exponent) including the subelliptic Sobolev spaces and spaces over Riemannian manifolds.


Moduli of twisted orbifold sheaves

Abstract: We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces.


Knuth Relations for the Hyperoctahedral Groups

Abstract: C. Bonnaf{\'e}, M. Geck, L. Iancu, and T. Lam have conjectured a description of one-sided cells in unequal parameter Hecke algebras of type $B$ which is based on domino tableaux of arbitrary rank. In the integer case, this generalizes the work of D. Garfinkle whose methods we adapt to construct a family of operators which generate the conjectured combinatorial description.


The Canonical Model of a Singular Curve

Abstract: We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the blowup of C with respect to the canonical sheaf \omega. We also prove some new results: we determine just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal.


Type ${\rm III_1}$ factors generated by regular representations of infinite dimensional nilpotent group $B_0^{\mathbb N}$

Abstract: We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group $B_0^{\mathbb N}$. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type ${\rm I}_\infty$ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,14]). In the present article we determine the type of corresponding factors. Namely we prove that the von Neumann algebra generated by the regular representations of infinite-dimensional nilpotent group $B_0^{\mathbb N}$ is type ${\rm III}_1$ hyperfinite factor. The case of the nilpotent group $B_0^{\mathbb Z}$ of infinite in both directions matrices will be studied in [6].


Geometric Interpretation of Second Elliptic Integrable System

Abstract: In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space $G/G_0$ can be embedded into the twistor space of the corresponding symmetric space $G/H$. Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift $J$ taking values in $G/G_0 \hookrightarrow \Sigma(G/H)$. We begin the paper by an example: $G/H=\R^4$. We study also the structure of 4-symmetric bundles over Riemannian symmetric spaces.


Alternative proofs of linear response for piecewise expanding unimodal maps

Abstract: We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous paper, but require proving Holder continuity of the infinitesimal conjugacy (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order if f_t is smooth enough and stays in the topological class of f_0 and if the observable smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e., the formula for the derivative at 0) obtained in our previous paper and gives additional information on this formula.


The value of Repeated Games with an informed controller

Abstract: We consider the general model of zero-sum repeated games (or stochastic games with signals), and assume that one of the players is fully informed and controls the transitions of the state variable. We prove the existence of the uniform value, generalizing several results of the literature. A preliminary existence result is obtained for a certain class of stochastic games played with pure strategies.


Counting points of homogeneous varieties over finite fields

Abstract: Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer coefficients. Moreover, the shifted periodic polynomial function, where $q^n$ is formally replaced with $q^n + 1$, is shown to have non-negative coefficients.


Differential operators and Cherednik algebras

Abstract: We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result, [GS, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].


Actions of F_\infty whose II_1 factors and orbit equivalence relations have prescribed fundamental group

Abstract: We show that given any subgroup F of R_+ which is either countable or belongs to a certain "large" class of uncountable subgroups, there exist continuously many free ergodic probability measure preserving actions \sigma_i of the free group with infinitely many generators such that their associated group measure space II_1 factors M_i and orbit equivalence relations R_i have fundamental group equal to F and with M_i (respectively R_i) stably non-isomorphic. Moreover, these actions can be taken so that R_i has no outer automorphisms and any automorphism of M_i is unitary conjugate to an automorphism that acts trivially on $L^\infty(X_i) \subset M_i$.


On strongly $g(x)$-clean rings

Abstract: Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with $g(s)=0$, $u$ a unit of $R$, and $su=us$. The relation between strongly $g(x)$-clean rings and strongly clean rings is determined, some general properties of strongly $g(x)$-clean rings are given, and strongly $g(x)$-clean rings generated by units are discussed.


A Payne-Weinberger eigenvalue estimate for wedge domains on spheres

Abstract: A Faber-Krahn type argument gives a sharp lower estimate for the first Dirichlet eigenvalue for subdomains of wedge domains in spheres, generalizing the inequality in the plane, found by Payne and Weinberger. An application is an alternative proof to the finiteness of a Brownian motion capture time estimate.


On the zeta function of divisors for projective varieties with higher rank divisor class group

Abstract: Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).


The centers of Iwahori-Hecke algebras are filtered

Abstract: We show that the center of the Iwahori--Hecke algebra of the symmetric group $S_n$ carries a natural filtered algebra structure, and that the structure constants of the associated graded algebra are independent of $n$. A series of conjectures and open problems are also included.


Finitistic and Representation Dimensions

Abstract: We prove that the finitistic dimension conjecture, the Gorenstein Symmetry Conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for artin algebras which can be realized as endomorphism algebras of modules over representation-finite algebras. Note it is a question whether or not all artin algebras have such realizations. It was also shown that if every quasi-hereditary algebras has a left idealized extension which is a monomial algebra or an algebra whose representation dimension is not more than 3, then the finitistic dimension conjecture holds.


Zero loci of admissible normal functions with torsion singularities

Abstract: We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].


Maximal contact and normal crossings in resolution of singularities

Abstract: This paper has been withdrawn by the author


On monotonicity of F-blowup sequences

Abstract: For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the $e+1$-th blowup dominates the $e$-th, locally or globally. It is shown that the answer is affirmative (globally for any $e$) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowups. We also give a sufficient condition for local domination.


Notes on uniform exponential growth and Tits alternative

Abstract: These notes contain results concerning uniform exponential growth which were obtained in collaborations with E. Breuillard and A. Salehi-Golsefidy, mostly during 2005, improving Eskin-Mozes-Oh theorem \cite{EMO}, as well as a uniform uniform version of Tits alternative improving \cite{uti}.


Generalized Harish-Chandra descent and applications to Gelfand pairs

Abstract: In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over arbitrary local field F of characteristic zero. Our main tool is Luna slice theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pair (GL(n,C),GL(n,R)) is a Gelfand pair. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan. We use the techniques developed here in our subsequent work [AG3] where we prove an archimedean analog of the theorem on uniqueness of linear periods by H. Jacquet and S. Rallis.


An archimedean analog of Jacquet - Rallis theorem

Abstract: In this paper we prove that the symmetric pair $(GL_{n+k}(F),GL_n(F) \times GL_k(F))$ is a Gelfand pair for any local field F of characteristic 0. For non-archimedean F it has been proven in [JR]. We use techniques developed in [AG2] to generalize their proof to general local fields.


Dynamics of tuples of matrices

Abstract: In this article we answer a question raised by N. Feldman in \cite{Feldman} concerning the dynamics of tuples of operators on $\mathbb{R}^n$. In particular, we prove that for every positive integer $n\geq 2$ there exist $n$ tuples $(A_1, A_2, ..., A_n)$ of $n\times n$ matrices over $\mathbb{R}$ such that $(A_1, A_2, ..., A_n)$ is hypercyclic. We also establish related results for tuples of $2\times 2$ matrices over $\mathbb{R}$ or $\mathbb{C}$ being in Jordan form.


Ambitable topological groups

Abstract: A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.


Factors in random graphs

Abstract: Let $H$ be a fixed graph on $v$ vertices. For an $n$-vertex graph $G$ with $n$ divisible by $v$, an $H$-{\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. In this paper we consider the threshold $th_{H} (n)$ of the property that an Erd\H{o}s-R\'enyi random graph (on $n$ points) contains an $H$-factor. Our results determine $th_{H} (n)$ for all strictly balanced $H$. The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random $k$-uniform hypergraph, solving the well-known "Shamir's problem."


Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence

Abstract: Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom, respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second-order, $O(p^{-2/3})$, by the Tracy--Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.


The strength of the Weak Lefschetz Property

Abstract: We study a number of conditions on the Hilbert function of a level artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.


Separable d-permutations and guillotine partitions

Abstract: We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between $d$-dimensional permutations and guillotine partitions of a $2^{d-1}$-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain results on avoided patterns in permutations.


Cohomology of Flag Varieties and the Brylinski-Kostant Filtration

Abstract: Let G be a semisimple complex algebraic group with Borel subgroup B and let P be a parabolic subgroup of G. Let T*(G/P) denote the cotangent bundle of G/P. Ranee Brylinski discovered a connection between cohomology of G-equivariant line bundles on T*(G/B) and the so-called Brylinski-Kostant filtration, which describes the action of principal sl_2 triples on G-representations. In this paper we generalize these results to a larger class of sl_2 triples. Along the way we also obtain generalizations of results due to Broer on cohomology of G-equivariant bundles on T*(G/P) for various parabolics P.


Persistence of normally expanded submanifolds with boundary or corners

Abstract: We show that invariant submanifolds with boundary, and more generally with corners which are normally expanded by an endomorphism are persistent as $a$-regular stratifications. This result will be shown in class $C^s$, for $s\ge 1$. We present also a simple example of a submanifold with boundary which is normally expanded but non-persistent as a differentiable submanifold.


Influence of Speed Limit on Roadway Safety in Indiana

Abstract: The influence of speed limits on roadway safety is an extremely important social issue and is subject to an extensive debate in the State of Indiana and nationwide. With around 800-900 fatalities and thousands of injuries annually in Indiana, traffic accidents place an incredible social and economic burden on the state. Still, speed limits posted on highways and other roads are routinely exceeded as individual drivers try to balance safety and mobility (speed). This research explores the relationship between speed limits and roadway safety. Namely, the research focuses on the influence of the posted speed limit on the causation and severity of accidents. Data on individual accidents from the Indiana Electronic Vehicle Crash Record System is used in the research, and appropriate statistical models are estimated for causation and severity of different types of accidents on all road classes. The results of the modeling show that speed limits do not have a statistically significant adverse effect on unsafe-speed-related causation of accidents on all roads, but generally increase the severity of accidents on the majority of roads other than highways (the accident severity on highways is unaffected by speed limits). Our findings can perhaps save both lives and travel time by helping the Indiana Department of Transportation determine optimal speed limit policies in the state.


The Equations of Singular Loci of Ample Divisors on (Subvarieties of) Abelian Varieties

Abstract: In this paper we consider ideal sheaves associated to the singular loci of a divisor in a linear system $|L|$ of an ample line bundle on a complex abelian variety. We prove an effective result on their (continuous) global generation, after suitable twists by powers of $L$. Moreover we show that similar results hold for subvarieties of a complex abelian variety.


Local theta correspondence and the lifting of Duke, Imamoglu and Ikeda

Abstract: We use results on the local theta correspondence to prove that for large degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form is not a linear combination of theta series.


Periods for flat algebraic connections

Abstract: In previous work, we established a duality between the algebraic de Rham cohomology of a flat algebraic connection on a smooth quasi-projective surface over the complex numbers and the rapid decay homology of the dual connection relying on a conjecture by C. Sabbah, which has been proved recently by T. Mochizuki for algebraic connections in any dimension. In the present article, we verify that Mochizuki's results allow to generalize these duality results to arbitrary dimensions also.


Canonical metrics of commuting maps

Abstract: In the present work we establish the equality of the canonical metric of two commuting maps on an algebraic variety X. As a consequence the canonical height and measure associated to both maps are identical.


Changes of variables in modulation and Wiener amalgam spaces

Abstract: In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier Lebesgue spaces. Finally, counterparts of these results are discussed for spaces on the torus as well as for weighted spaces.


New techniques for pointed Hopf algebras

Abstract: We present techniques that allow to decide that the dimension of some pointed Hopf algebras associated with non-abelian groups is infinite. These results are consequences of arXiv:0803.2430v1. We illustrate each technique with applications.


On a Generalised Lehmer Problem for Arbitrary Powers

Abstract: We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generalise a recent result of Y. Yi and W. Zhang.


Obtaining the One-Holed Torus from Pants: Duality in an SL(3,C)-Character Variety

Abstract: The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. The coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map which turns the three-holed sphere into the one-holed torus induces a rank preserving Poisson map on C[X].


Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. I

Abstract: We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.


Homogenization of semi-linear PDEs with discontinuous effective coefficients

Abstract: We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in \`{C}esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "$L^p-$viscosity solution" introduced in \cite{CCKS}. We use BSDEs techniques to establish the existence of $L^p-$viscosity solution for the averaged PDE. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of $L^p$-viscosity solution.


Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease

Abstract: We analyze a finite volume scheme for nonlocal SIR model, which is a nonlocal reaction-diffusion system modeling an epidemic disease. We establish existence solutions to the finite volume scheme, and show that it converges to a weak solution. The convergence proof is based on deriving series of a priori estimates and using a general $L^p$ compactness criterion.


Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs

Abstract: Sensitivity indices when the inputs of a model are not independent are estimated by local polynomial techniques. Two original estimators based on local polynomial smoothers are proposed. Both have good theoretical properties which are exhibited and also illustrated through analytical examples. They are used to carry out a sensitivity analysis on a real case of a kinetic model with correlated parameters.


Limiting Carleman weights and anisotropic inverse problems

Abstract: In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.


Maxima of Dirichlet and triangular arrays of gamma variables

Abstract: Consider a rowwise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certain Dirichlet vectors of increasing dimension are also obtained using the gamma representation.


On the gap between representability and collapsibility

Abstract: A simplicial complex K is called d-representable if it is the nerve of a collection of convex sets in R^d; K is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d-1 that is contained in a unique maximal face; and K is d-Leray if every induced subcomplex of K has vanishing homology of dimension d and larger. It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d greater or equal to 2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results "d-representable" in the assumption can be replaced by "d-collapsible" or even "d-Leray". We investigate "dimension gaps" among these notions, and we construct, for all positive integers d, a 2d-Leray complex that is not (3d-1)-collapsible and a d-collapsible complex that is not (2d-2)-representable. In the proofs we obtain two results of independent interest: (i) The nerve of every finite family of sets, each of size at most d, is d-collapsible. (ii) If the nerve of a simplicial complex K is d-representable, then K embeds in R^d.


Integration with respect to local time and Ito's formula for smooth nondegenerate martingales

Abstract: We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in It\^o's s formula as an integral over space and time with respect to local time.


On the size of Nikodym sets in finite fields

Abstract: Let $\mathbb{F}_q$ denote a finite field of $q$ elements. Define a set $B\subset\mathbb{F}_q^n$ to be Nikodym if for each $x\in B^{c}$, there exists a line $L$ such that $L\cap B^c=\{x\}.$ The main purpose of this note is to show that the size of every Nikodym set is at least $C_n\cdot q^n$, where $C_n$ depends only on $n$.


Landau's necessary density conditions for LCA groups

Abstract: H. Landau's necessary density conditions for sampling and interpolation may be viewed as a general principle resting on a basic fact of Fourier analysis: The complex exponentials $e^{i kx}$ ($k$ in $\mathbb{Z}$) constitute an orthogonal basis for $L^2([-\pi,\pi])$. The present paper extends Landau's conditions to the setting of locally compact abelian (LCA) groups, relying in an analogous way on the basics of Fourier analysis. The technicalities--in either case of an operator theoretic nature--are however quite different. We will base our proofs on the comparison principle of J. Ramanathan and T. Steger.


Symplectic maps of complex domains into complex space forms

Abstract: Let $M\subset{\complex}^n$ be a complex domain of ${\complex}^n$ endowed with a rotation invariant \K form $\omega_{\Phi}= \frac{i}{2} \partial\bar\partial\Phi$. In this paper we describe sufficient conditions on the \K potential $\Phi$ for $(M, \omega_{\Phi})$ to admit a symplectic embedding (explicitely described in terms of $\Phi$) into a complex space form of the same dimension of $M$. In particular we also provide conditions on $\Phi$ for $(M, \omega_{\Phi})$ to admit global symplectic coordinates. As an application of our results we prove that each of the Ricci flat (but not flat) \K forms on ${\complex}^2$ constructed by LeBrun (Taub-NUT metric) admits explicitely computable global symplectic coordinates.


Riemannian geometry of Hartogs domains

Abstract: Let $D_F = \{(z_0, z) \in {\C}^{n} | |z_0|^2 < b, \|z\|^2 < F(|z_0|^2) \}$ be a strongly pseudoconvex Hartogs domain endowed with the \K metric $g_F$ associated to the \K form $\omega_F = -\frac{i}{2} \partial \bar{\partial} \log (F(|z_0|^2) - \|z\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of $D_F$ do not self-intersect, we find necessary and sufficient conditions on $F$ for $D_F$ to be geodesically complete and we prove that $D_F$ is locally irreducible as a Riemannian manifold. Finally, we compare the Bergman metric $g_B$ and the metric $g_F$ in a bounded Hartogs domain and we prove that if $g_B$ is a multiple of $g_F$, namely $g_B=\lambda g_F$, for some $\lambda\in \R^+$, then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space.


Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

Abstract: We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.


Symplectic duality between complex domains

Abstract: In this paper after extending the definition of symplectic duality (given by the first two authors in arXiv:math/0603141 for bounded symmetric domains) to arbitrary complex domains of ${\C}^n$ centered at the origin we generalize some of the results proved in arXiv:math/0603141 and arXiv:0707.2125 to those domains.


Lindelof type of generalization of separability in Banach spaces

Abstract: We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset \mathcal{F} of X^{\ast}, which separates X, there exists a countable separating subset \mathcal{F}_{0} of \mathcal{F}. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Markucevic-bases, Corson property and related geometric issues are discussed.


Hochschild homology and global dimension

Abstract: We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.


$N=1$ formal genus $0$ Gromov-Witten theories and Givental's formalism

Abstract: In [Gi3] Givental introduced and studied a space of formal genus zero Gromov-Witten theories $GW_0$, i.e. functions satisfying string and dilaton equations and topological recursion relations. A central role in the theory plays the geometry of certain Lagrangian cones and a twisted symplectic group of hidden symmetries. In this note we show that the Lagrangian cones description of the action of this group coincides with the genus zero part of Givental's quantum Hamiltonian formalism. As an application we identify explicitly the space of $N=1$ formal genus zero GW theories with lower-triangular twisted symplectic group modulo the string flow.


Classification of groups generated by 3-state automata over a 2-letter alphabet

Abstract: This article contains most of the known results on the classification of groups generated by 3-state automata over a 2-letter alphabet, extending the previous papers 0704.3876 and math/0612178.


Positive Realness of a Transfer Function Neither Implies Nor is Implied by the External Positivity of their Associate Realizations

Abstract: This letter discusses the differences between the properties of positive realness of transfer functions and external positivity in linear time-invariant dynamic systems. It is proved that each one of both properties does not imply to each other.


Lecture notes on duality and interpolation spaces

Abstract: Known or essentially known results about duals of interpolation spaces are presented, taking a point of view sometimes slightly different from the usual one. Particular emphasis is placed on Alberto Calderon's theorem describing the duals of his complex interpolation spaces [A_0,A_1]_\theta. The pace is slow, since these notes are intended for graduate students who have just begun to study interpolation spaces. This second version corrects some small misprints. It also draws attention to a convenient norming subspace of the dual of a complex interpolation space, and to the slight difference between the spaces \mathcal{G}(X_0,X_1) introduced by Calderon and by Stafney.


Definably complete and Baire structures and Pfaffian closure

Abstract: We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. However, unlike the o-minimal case, the structures considered form an elementary class. In this context we prove a version of Kuratowski-Ulam's Theorem and some restricted version of Sard's Lemma. We use the above results to prove the following version of Wilkie's Theorem of the Complement: given a definably complete Baire expansion K of an ordered field with a family of smooth functions, if there are uniform bounds on the number of definably connected components of quantifier free definable sets, then K is o-minimal. We further generalize the above result, along the line of Speissegger's theorem, and prove the o-minimality of the relative Pfaffian closure of an o-minimal structure inside a definably complete Baire structure.


The bigger Brauer group and twisted sheaves

Abstract: Given an algebraic stack with quasiaffine diagonal, we show that each G_m-gerbe comes from a central separable algebra. In other words, Taylor's bigger Brauer group equals the etale cohomology in degree two with coefficients in G_m. This gives new results also for schemes. We use the method of twisted sheaves explored by de Jong and Lieblich.


Whittaker Modules for Generalized Weyl Algebras

Abstract: We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl_2 and of Heisenberg Lie algebras, Smith's generalizations of U(sl_2), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(phi,t) are in bijection with the phi-stable left ideals of R. We determine the annihilator Ann_A(w) of the cyclic generator w of V. We also describe the annihilator ideal Ann_A(V) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U(sl_2).


The stable free rank of symmetry of products of spheres

Abstract: A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non simply connected spaces.


Harmonic maps from degenerating Riemann surfaces

Abstract: We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in $W^{1,2}$ and $C^{0}$ modulo bubbles of sequences of such maps.


On Morita theory for self-dual modules

Abstract: Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear form $b$ if and only if $V$ is self-dual. We show that whenever a Morita bimodule $M$ which induces an equivalence between two blocks $B(kG)$ and $B(kH)$ of group algebras $kG$ and $kH$ is self-dual then the correspondence preserves self-duality. Even more, if the bilinear form on $M$ is symmetric then for $p$ odd the correspondence preserves the geometric type of simple modules. In characteristic 2 this holds also true for projective modules.


On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels II

Abstract: We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part II, we will observe the location of the zeros of the above functions by numerical calculation.


Differentiating polynomials, and zeta(2)

Abstract: We study the derivatives of polynomials with equally spaced zeros and find connections to the values of the Riemann zeta-function at the positive even integers.


Algebras of higher operads as enriched categories

Abstract: We decribe the correspondence between normalised $\omega$-operads and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category. Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised $\omega$-operad, and categories enriched in globular sets for the induced lax monoidal structure.


On continuous fields of JB-algebras

Abstract: We introduce and study continuous fields of JB-algebras (which are real non-associate analogues of C*-algebras). In particular, we show that for the universal enveloping C*-algebra C*sub-u(B) for the JB-algebra B defined by a continuous field of JB-algebras A-sub-t, t belongs to T, on a locally compact space T, there exists a decomposition of C*-sub-u(B) into a continuous field of C*-algebras C*u(A-sub-t), t belongs to T, on the same space T, composed entirely of the universal enveloping C*-algebras of the corresponding JB-algebras from the aforementioned decomposition of the algebra B.


On Intrinsic Characterization of Real Locally C*- and Locally JB-Algebras

Abstract: In the present paper we obtain an intrinsic characterization of real locally C*-algebras (projective limits of projective families of real C*-algebras) among complete real lmc *-algebras, and of locally JB-algebras (projective limits of projective families of JB-algebras) among complete fine Jordan locally multiplicatively-convex topological algebras.


Braid group B_3 irreducibles, a DIY guide

Abstract: This note tells you how to construct a k(n)-dimensional family of (isomorphism classes of) irreducible representations of dimension n for the three string braid group B_3, where k(n) is an admissible function of your choosing; for example take k(n) = [ n/2 ] +1 as in arXiv:0803.2778 and arXiv:0803.2785.


Certain Properties of Pythagorean Triangles involving the interior diameter, and the exterior diameters

Abstract: There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is Pythagorean, the four diameters are integers. Consider a Pythagorean triangle with the property that one leglength is a perfect(or integer)square, and with one of the four diameters also a integer square.Of the eight resulting combinations, we prove that only six are possible or can occur. We then completely parametrically describe the six families; each corresponding to one of the six combinations.


M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

Abstract: In this paper, we combine results on extensions of operators with recent results on the relation between the M-function and the spectrum, to examine the spectral behaviour of boundary value problems. M-functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that can not be detected by the M-function.


Parabolic induction and restriction functors for rational Cherednik algebras

Abstract: We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon-Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category O over the rational Cherednik algebras of type A which do not contain an S_n-invariant vector, and confirm a conjecture of Okounkov and the first author on the number of such representations. In the second version we have added a result on the simplicity of the spherical Cherednik algebra of type A for -1<c<0, and a strengthened version of the main result of arXiv:math/0312474, as well as an appendix by the second author containing arXiv:0706.4308, on the reducibility of the polynomial representation of the trigonometric Cherednik algebra.


Characterizing Hilbert spaces using Fourier transform over the field of p-adic numbers

Abstract: We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier transform $F: L_2(Q_p,X)\to L_2(Q_p,X)$ in space of functions, which are square-integrable in Bochner sense and take value in $X$, is a bounded operator.


Affine partitions and affine Grassmannians

Abstract: We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type $A_{n}$, the affine colored partitions form another family of combinatorial objects in bijection with $n+1$-core partitions and $n$-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.


A categorification of quantum sl(2)

Abstract: We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our graphical calculus. Using iterated flag varieties, a representation of the 2-category is constructed for each positive integer N. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).


Free products, cyclic homology, and the Gauss-Manin connection

Abstract: We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex' of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce and study `free-product deformations' of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.


The Equivalence between Uniqueness and Continuous Dependence of Solution for BSDEs with Continuous Coefficient

Abstract: In this paper, we will prove that, if the coefficient $g=g(t,y,z)$ of a BSDE is assumed to be continuous and linear growth in $(y,z)$, then the uniqueness of solution and continuous dependence with respect to $g$ and the terminal value $\xi$ are equivalent.


Unramified extensions and geometric $\mathbb{Z}_p$-extensions of global function fields

Abstract: We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a construction of a geometric $\mathbb{Z}_p$-extension which has a certain property.


Integration with respect to fractional local times with Hurst index $H$ greater than 1/2

Abstract: Let ${\mathscr L}^H(x,t)=2H\int_0^t\delta(B^H_s-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$. In this paper, we use Young integration to study the integral of determinate functions $\int_{\mathbb R}f(x){\mathscr L}^H(dx,t)$. As an application, we investigate the {\it weighted quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\}(B^H_{t_{k+1}}-B^H_{t_{k}}), $$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$ [f(B^H),B^H]^{(W)}_t=-\int_{\mathbb R}f(x){\mathscr L}^H(dx,t), $$ provided $f$ is of bounded $p$-variation with $1\leq p<\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional It\^{o} formula for new classes of functions.


A simply connected surface of general type with p_g=0 and K^2=4

Abstract: As the sequel to [5, 7], we construct a simply connected minimal complex surface of general type with p_g = 0 and K^2 = 4 by using a rational blow-down surgery and Q-Gorenstein smoothing theory.


Categorification of integrable representations of quantum groups

Abstract: We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.


Deforming motivic theories I: Pure weight perfect Modules on divisorial schemes

Abstract: In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i:Y -> X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O_X-Modules supported on Y. The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh non-connected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting non-connected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes.


A new stochastic process to model Heart Rate series during exhaustive run and an estimator of its fractality parameter

Abstract: In order to interpret and explain the physiological signal behaviors, it can be interesting to find some constants among the fluctuations of these data during all the effort or during different stages of the race (which can be detected using a change points detection method). Several recent papers have proposed the long-range dependence (Hurst) parameter as such a constant. However, their results induce two main problems. Firstly, DFA method is usually applied for estimating this parameter. Clearly, such a method does not provide the most efficient estimator and moreover it is not at all robust even in the case of smooth trends. Secondly, this method often gives estimated Hurst parameters larger than 1, which is the larger possible value for long memory stationary processes. In this article we propose solutions for both these problems and we define a new model allowing such estimated parameters.


Variable selection for the multicategory SVM via adaptive sup-norm regularization

Abstract: The Support Vector Machine (SVM) is a popular classification paradigm in machine learning and has achieved great success in real applications. However, the standard SVM can not select variables automatically and therefore its solution typically utilizes all the input variables without discrimination. This makes it difficult to identify important predictor variables, which is often one of the primary goals in data analysis. In this paper, we propose two novel types of regularization in the context of the multicategory SVM (MSVM) for simultaneous classification and variable selection. The MSVM generally requires estimation of multiple discriminating functions and applies the argmax rule for prediction. For each individual variable, we propose to characterize its importance by the supnorm of its coefficient vector associated with different functions, and then minimize the MSVM hinge loss function subject to a penalty on the sum of supnorms. To further improve the supnorm penalty, we propose the adaptive regularization, which allows different weights imposed on different variables according to their relative importance. Both types of regularization automate variable selection in the process of building classifiers, and lead to sparse multi-classifiers with enhanced interpretability and improved accuracy, especially for high dimensional low sample size data. One big advantage of the supnorm penalty is its easy implementation via standard linear programming. Several simulated examples and one real gene data analysis demonstrate the outstanding performance of the adaptive supnorm penalty in various data settings.


Linearity defects of modules over commutative rings

Abstract: This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module.


Editorial: Statistics and "The lost tomb of Jesus"

Abstract: What makes a problem suitable for statistical analysis? Are historical and religious questions addressable using statistical calculations? Such issues have long been debated in the statistical community and statisticians and others have used historical information and texts to analyze such questions as the economics of slavery, the authorship of the Federalist Papers and the question of the existence of God. But what about historical and religious attributions associated with information gathered from archeological finds? In 1980, a construction crew working in the Jerusalem neighborhood of East Talpiot stumbled upon a crypt. Archaeologists from the Israel Antiquities Authority came to the scene and found 10 limestone burial boxes, known as ossuaries, in the crypt. Six of these had inscriptions. The remains found in the ossuaries were reburied, as required by Jewish religious tradition, and the ossuaries were catalogued and stored in a warehouse. The inscriptions on the ossuaries were catalogued and published by Rahmani (1994) and by Kloner (1996) but there reports did not receive widespread public attention. Fast forward to March 2007, when a television ``docudrama'' aired on The Discovery Channel entitled ``The Lost Tomb of Jesus'' touched off a public and religious controversy--one only need think about the title to see why there might be a controversy! The program, and a simultaneously published book [Jacobovici and Pellegrino (2007)], described the ``rediscovery'' of the East Talpiot archeological find and they presented interpretations of the ossuary inscriptions from a number of perspectives. Among these was a statistical calculation attributed to the statistician Andrey Feuerverger: ``that the odds that all six names would appear together in one tomb are 1 in 600, calculated conservatively--or possibly even as much as one in one million.''


The generality of the zero-one laws

Abstract: We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability.


Asymptotic stability of solitons for the Benjamin-Ono equation

Abstract: In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of Martel and Merle (arXiv:0706.1174v2). As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.


Cohomology of twisted tensor products

Abstract: It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. We give an explicit description of the $\Ext$-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a twisted tensor product. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra.


Principal 2-bundles and their gauge 2-groups

Abstract: In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.


Simple proofs for universal binary Hermitian lattices

Abstract: If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs.


In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies

Abstract: Batting average is one of the principle performance measures for an individual baseball player. It is natural to statistically model this as a binomial-variable proportion, with a given (observed) number of qualifying attempts (called ``at-bats''), an observed number of successes (``hits'') distributed according to the binomial distribution, and with a true (but unknown) value of $p_i$ that represents the player's latent ability. This is a common data structure in many statistical applications; and so the methodological study here has implications for such a range of applications. We look at batting records for each Major League player over the course of a single season (2005). The primary focus is on using only the batting records from an earlier part of the season (e.g., the first 3 months) in order to estimate the batter's latent ability, $p_i$, and consequently, also to predict their batting-average performance for the remainder of the season. Since we are using a season that has already concluded, we can then validate our estimation performance by comparing the estimated values to the actual values for the remainder of the season. The prediction methods to be investigated are motivated from empirical Bayes and hierarchical Bayes interpretations. A newly proposed nonparametric empirical Bayes procedure performs particularly well in the basic analysis of the full data set, though less well with analyses involving more homogeneous subsets of the data. In those more homogeneous situations better performance is obtained from appropriate versions of more familiar methods. In all situations the poorest performing choice is the na\"{{\i}}ve predictor which directly uses the current average to predict the future average.


Models of Z/p^2 Z over a d.v.r. of unequal characteristic

Abstract: Let R be a discrete valuation ring of unequal characteristic which contains a primitive p^2-th root of unity. If K is the fraction field of R, it is well known that (Z/p^2 Z)_K is isomorphic to \mu_{p^2,K}. We prove that any finite and flat R-group scheme of order p^2 isomorphic to (Z/p^2 Z)_K on the generic fiber (i.e. a model of (Z/p^2 Z)_K), is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models.


On bijections between 231-avoiding permutations and Dyck paths

Abstract: We construct a bijection between 231-avoiding permutations and Dyck paths that sends the sum of the major index and the inverse major index of a 231-avoiding permutation to the major index of the corresponding Dyck path. Furthermore, we relate this bijection to others and exhibit a bistatistic on 231-avoiding permutations which is related to the q,t-Catalan numbers.


An equivariant version of the monodromy zeta function

Abstract: We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the Grothendieck ring of finite G-sets tensored by the field of rational numbers. Main two ingredients of the definition are equivariant Lefschetz numbers and the lambda-structure on the Grothendieck ring of finite G-sets. We give an A'Campo type formula for the equivariant zeta function.


Radial Balanced metrics on the unit disk

Abstract: Let $\Phi$ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to the \K form $\omega =\frac{i}{2}\partial\bar\partial\Phi$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\complex}={\real}^2$), and the function $h(x)=e^{-\Phi (z)}$, $x=|z|^2$, extends to an entire analytic function on ${\real}$, then $g$ equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function $f(x)=1-x$.


Numerical Algorithms and Simulations for Reflected Backward Stochastic Differential Equations with two Continuous Barriers

Abstract: In this paper we study different algorithms for reflected backward stochastic differential equations (BSDE in short) with two continuous barriers basing on random work framework. We introduce different numerical algorithms by penalization method and reflected method. At last simulation results are also presented.


A new principle for choosing regularization parameter in certain inverse problems

Abstract: A new parameter choice rule for inverse problems is introduced. This parameter choice rule was developed for total variation regularization in electron tomography and might in general be useful for $L^1$ regularization of inverse problems with high levels of noise in the data.


On distributional properties of perpetuities

Abstract: We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.


Dirac-harmonic maps from degenerating spin surfaces I: the Neveu-Schwarz case

Abstract: We study Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy and show the so-called generalized energy identity in the case that the domain converges to a spin surface with only Neveu-Schwarz type nodes. We find condition that is both necessary and sufficient for the $W^{1,2} \times L^{4}$ modulo bubbles compactness of a sequence of such maps.


Dynamic Physical Systems: Energy Balances and Stability Issues

Abstract: This paper discusses some links properties of operators with the well- known physical concepts of hyperstability, passivity, energy dissipativeness and conservativeness with positive realness properties of the transfer functions in linear dynamic systems.


Effective models and extension of torsors over a discrete valuation ring of unequal characteristic

Abstract: Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\to X_K and let Y be the normalization of X_K in Y. If G=Z/p^n Z, n<3, under some hypothesis on X, we attach some invariants to Y_K \to X_K. If p>2, we determine, through these invariants, when Y\to X has a structure of torsor which extends that of Y_K\to X_K. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of G on Y.


Structure of the cuspidal rational torsion subgroup of J_1(p^n)

Abstract: In this article, we determine the structure of the $p$-primary subgroup of the cuspidal rational torsion subgroup of the Jacobian $J_1(p^n)$ of the modular curve $X_1(p^n)$ for a regular prime $p$.


On sums of primes and triangular numbers

Abstract: We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+T_x with x an integer and p a prime or zero; another conjecture asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a prime and x a positive integer.


Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains

Abstract: In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are similar to those of the p-Laplacian, the novelty resides in the fact that we consider the equations in bounded domains which only satisfy the exterior cone condition.


False discovery rate analysis of brain diffusion direction maps

Abstract: Diffusion tensor imaging (DTI) is a novel modality of magnetic resonance imaging that allows noninvasive mapping of the brain's white matter. A particular map derived from DTI measurements is a map of water principal diffusion directions, which are proxies for neural fiber directions. We consider a study in which diffusion direction maps were acquired for two groups of subjects. The objective of the analysis is to find regions of the brain in which the corresponding diffusion directions differ between the groups. This is attained by first computing a test statistic for the difference in direction at every brain location using a Watson model for directional data. Interesting locations are subsequently selected with control of the false discovery rate. More accurate modeling of the null distribution is obtained using an empirical null density based on the empirical distribution of the test statistics across the brain. Further, substantial improvements in power are achieved by local spatial averaging of the test statistic map. Although the focus is on one particular study and imaging technology, the proposed inference methods can be applied to other large scale simultaneous hypothesis testing problems with a continuous underlying spatial structure.


From cardinal spline wavelet bases to highly coherent dictionaries

Abstract: Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\em{dictionary}}, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterise the correlation of the dictionary elements by measuring their `coherence' and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation.


Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds

Abstract: Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure.


Transcience/recurrence for normally reflected Brownian motion in unbounded domains

Abstract: Let $D\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a Brownian motion in $D$ with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form $D=\{(x,z)\in R^{l+m}:|z|<H(|x|)\}$, where $d=l+m$ and $H$ is a sufficiently regular function. This class of domains includes various horn-shaped domains and generalized slab domains.


Conditional Haar measures on classical compact groups

Abstract: We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension $n$. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension $p$. The developed method leads to the following result: for this conditional measure, writing $Z_U^{(p)}$ for the first nonzero derivative of the characteristic polynomial at 1, \[\frac{Z_U^{(p)}}{p!}\stackrel{\mathrm{law}}{=}\prod_{\ell =1}^{n-p}(1-X_{\ell}),\] the $X_{\ell}$'s being explicit independent random variables. This implies a central limit theorem for $\log Z_U^{(p)}$ and asymptotics for the density of $Z_U^{(p)}$ near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu.


On regression adjustments in experiments with several treatments

Abstract: Regression adjustments are often made to experimental data. Since randomization does not justify the models, bias is likely; nor are the usual variance calculations to be trusted. Here, we evaluate regression adjustments using Neyman's nonparametric model. Previous results are generalized, and more intuitive proofs are given. A bias term is isolated, and conditions are given for unbiased estimation in finite samples.


Stability of Delayed Systems Modeled by Fractional Models

Abstract: The paper discusses linear fractional representations of parameter-dependent nonlinear systems with dynamics defined by real rational nonlinearities and a finite set of point delays. The global asymptotic stability is investigated via linear matrix inequalities.


Cohomology and generic cohomology of Specht modules for the symmetric group

Abstract: Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(\Sigma_d, S^\lambda). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht modules equating cohomology of S^{p\lambda} with S^{p^2\lambda}. This is the first theorem we know relating Specht modules S^\lambda and S^{p\lambda}. The second result equates cohomology of S^\lambda with S^{\lambda + p^a\mu} for large a.


Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

Abstract: Let $\alpha,\beta\in(0,1)$ and \[ K^{\alpha,\beta}:=\left\{a\in L^\infty(\T): \sum_{k=1}^\infty |\hat{a}(-k)|^2 k^{2\alpha}<\infty, \sum_{k=1}^\infty |\hat{a}(k)|^2 k^{2\beta}<\infty \right\}. \] Mark Krein proved in 1966 that $K^{1/2,1/2}$ forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended earlier results of Gabor Szeg\H{o} for scalar symbols and established the asymptotic trace formula \[ \operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) \quad\text{as}\ n\to\infty \] for finite Toeplitz matrices $T_n(a)$ with matrix symbols $a\in K^{1/2,1/2}_{N\times N}$. We show that if $\alpha+\beta\ge 1$ and $a\in K^{\alpha,\beta}_{N\times N}$, then the Szeg\H{o}-Widom asymptotic trace formula holds with $o(1)$ replaced by $o(n^{1-\alpha-\beta})$.


Quantum Bounded Symmetric Domains

Abstract: This is Leonid Vaksman's monograph "Quantum bounded symmetric domains" (in Russian), preceded with an English translation of the table of contents and (a part) of the introduction. Quantum bounded symmetric domains are interesting from several points of view. In particular, they provide interesting examples for noncommutative complex analysis (i.e., the theory of subalgebras of C^*-algebars) initiated by W. Arveson.


Triangle angles and sides in progression and the diophantine equation x^2+3y^2 =z^2

Abstract: The main result of this paper, is the complete parametric description of the family of triangles which have integer sidelengths and with one angle being sixty degrees.


Fold maps, framed immersions and smooth structures

Abstract: For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q-1)/2 cobordism groups of framed immersions. The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds.