Asymptotic Geometry in the product of Hadamard spaces with rank one isometries

Abstract: In this article we study asymptotic properties of certain discrete groups $\Gamma$ acting by isometries on a product $\XX=\XX_1\times \XX_2$ of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $\Gamma$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $\delta_\theta(\Gamma)$ of orbit points in $\XX$ with a prescribed so-called "slope" $\theta\in (0,\pi/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $\Psi_\Gamma$ to $\RR_{\ge 0}^2$ of $\delta_\theta(\Gamma)$ as a function of $\theta$ is upper semi-continuous and concave.


Contraction of a Generalized Metric Structure

Abstract: In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we introduce a new metric structure - the galactic spaces - which depends on an ordered field extension of R. Moreover, some natural transformations of the category of galactic spaces, the contractions, are of particular interest: they generalize usual homotheties, they have a ratio which may be an infinitesimal, they are able to modify the topology and they satisfy a nice composition rule. With the help of nonstandard extensions we can associate to any metric space an infinite family of galactic spaces; lastly, we study some limit properties of this family.


On the Dimension of Secant Varieties

Abstract: We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.


Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Abstract: We consider the stochastic heat equation with multiplicative noise $u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order $\alpha<d$, then the sufficient condition for the existence of the solution is $d \leq 2+\alpha$ (if $H>1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of $k$ independent $d$-dimensional Brownian motions.


The Maximum of the Maximum Rectilinear Crossing Numbers of d-regular Graphs of Order n

Abstract: We extend known results regarding the maximum rectilinear crossing number of the cycle graph (C_n) and the complete graph (K_n) to the class of general d-regular graphs R_{n,d}. We present the generalized star drawings of the d-regular graphs S_{n,d} of order n where n+d= 1 mod 2 and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of S_{n,d} for n = d = 0 mod 2 is introduced and we conjecture that this drawing maximizes the maximum rectilinear crossing numbers, too. We offer a simpler proof of two results initially proved by Furry and Kleitman as partial results in the direction of this conjecture.


Brief communication. An indefinite Sturm theory

Abstract: Sturm theory for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. Here we propose a Sturm oscillation theorem for indefinite systems of even order and with Dirichlet boundary conditions having strongly indefinite leading term.


The Lyapunov spectrum is not always concave

Abstract: We characterize one-dimensional compact repellers having nonconcave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra.


Trek separation for Gaussian graphical models

Abstract: Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar $d$-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.


A central limit theorem for random walk in random environment on marked Galton-Watson trees

Abstract: In this article we focus on a general model of random walk on random marked trees. We prove a recurrence criterion, analogue to the recurrence criterion proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In the critical case, we obtain a criterion for the positive/null recurrence. Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi (2007). We focus on the "diffusive" regime and improve their result in this case, by obtaining a functional Central Limit Theorem. Our result is also an extension of a result by Y. Peres and O. Zeitouni (2008), obtained in the setting of biased random walk in Galton-Watson trees.


Prediction with Restricted Resources and Finite Automata

Abstract: We obtain an index of the complexity of a random sequence by allowing the role of the measure in classical probability theory to be played by a function we call the generating mechanism. Typically, this generating mechanism will be a finite automata. We generate a set of biased sequences by applying a finite state automata with a specified number, $m$, of states to the set of all binary sequences. Thus we can index the complexity of our random sequence by the number of states of the automata. We detail optimal algorithms to predict sequences generated in this way.


n-Linear Algebra of type I and its applications

Abstract: With the advent of computers, one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure, namely, n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of n-vector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several related results which are analogues of the classical linear algebra theorems. In case of n-vector spaces, we can define several types of linear transformations. The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in finite element analysis of mechanical structures with uncertain parameters. Further, it is suggested that the concept of n-matrices can be used in real world problems which adopts fuzzy models like Fuzzy Cognitive Maps, Fuzzy Relational Equations and Bidirectional Associative Memories. The applications of these algebraic structures are given in the third chapter. The fourth chapter suggests problems to further a reader's understanding of the subject.


The reduced HOMFLY-PT homology for the Conway and the Kinoshita-Terasaka knots

Abstract: In this paper we compute the reduced HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots and show that they are isomorphic.


Generalized solutions for the Euler-Bernoulli model with distributional forces

Abstract: We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves discontinuous, and even distributional coefficients and right-hand side. The general problem is solved by application of functional analytic techniques to obtain estimates for the solutions to regularized problems. Finally, we prove coherence properties and provide a regularity analysis of the generalized solution.


Phylogenetic distances for neighbour dependent substitution processes

Abstract: We consider models of nucleotidic substitution processes where the rate of substitution at a given site depends on the state of its neighbours. For a wide class of such nonreversible models, we show how to compute consistent, mathematically exact, estimators of the time elapsed between aligned sequences, for an ancestral sequence and a present one, and also for two present sequences. In both cases, we provide asymptotic confidence intervals, valid for nucleotidic sequences of finite length. We compute explicit formulas for the estimators and for their confidence intervals in the simplest nontrivial case, the Jukes-Cantor model with CpG influence.


Canonical models of filtered $A_\infty$-algebras and Morse complexes

Abstract: The purpose of this paper is two-fold. First we explain the construction of the canonical model of filtered $A_\infty$-algebras given in the authors' book [FOOO]. The canonical model plays a crucial role in the study of Lagrangian Floer theory on toric manifolds in our recent papers, arXiv:0802.1703 and arXiv:0810.5654. Then using a variation of the arguments used in that construction, we define a natural filtered $A_\infty$-structure on the Morse complex of a Morse function and its $A_\infty$ homotopy to the $A_\infty$-algebras on a Lagrangian submanifold constructed in [FOOO]. The corresponding graphical moduli spaces `summing over trees' involve holomorphic discs connected by the gradient flow lines.


Multiparameter ergodic averages for two commuting actions of an amenable group

Abstract: We find limits of some multiple ergodic averages, generalizing a result of Bergelson to the setting of two commuting transformations and actions of amenable groups.


A Theorem on Analytic Strong Multiplicity One

Abstract: Let $K$ be an algebraic number field, and $\pi=\otimes\pi_{v}$ an irreducible, automorphic, cuspidal representation of $\GL_{m}(\mathbb{A}_{K})$ with analytic conductor $C(\pi)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $\pi$ can be decided completely by its local components $\pi_{v}$ with norm $N(v)<c\cdot C(\pi)^{2m+\varepsilon}.$


Essential Variables and Separable Sets in Universal Algebra

Abstract: The study of essential and strongly essential variables in functions defined on finite sets is a part of $k$-valued logic. We extend the main definitions from functions to terms. This allows us to apply concepts and results of Universal Algebra. On the basis of the concept of a separable set of variables in a term we introduce a new notion of complexity of terms, algebras and varieties and give examples.


\'Equations aux q-diff\'erences lin\'eaires: factorisation, r\'esolution et th\'eor\`emes d'indices

Abstract: We describe explicit algorithms for factoring q-difference operators and solving q-difference equations. These are well known results, presented in a "concrete" form. ----- Nous decrivons des algorithmes explicites pour la factorisation d'operateurs et la resolution d'equations aux q-differences. Il s'agit d'une presentation "concrete" de resultats bien connus.


On the normal holonomy representation of spacelike submanifolds in pseudo-Riemannian space forms

Abstract: In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichn\'erowicz property. In particular, we derive a classification of Lorentzian normal holonomy representations.


Topological Modular Forms of Level 3

Abstract: We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Gamma_0(3)).


Goldie Ranks of Skew Power Series Rings of Automorphic Type

Abstract: Let A be a semprime, right noetherian ring equipped with an automorphism alpha, and let B := A[[y; alpha]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.


Cubic ergodic averages for actions of amenable groups

Abstract: We unify and extend some previous results about cubic ergodic averages and sets of positive density in products of groups. This provides a joint generalization of earlier work of the author in the case of two commuting actions of an amenable group, and of Q. Chu's recent result on a a finite number of commuting transformations.


Codimension and pseudometric in co-Heyting algebras

Abstract: In this paper we introduce a notion of dimension and codimension for every element of a distributive bounded lattice $L$. These notions prove to have a good behavior when $L$ is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on $L$ which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of $L$ with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of $L$ if and only if it is compact or equivalently if every finite dimensional quotient of $L$ is finite. In this case we say that $L$ is precompact. If $L$ is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers $n,d$ of a term $t_{n,d}$ such that in every co-Heyting algebra generated by an $n$-tuple $a$, $t_{n,d}(a)$ is precisely the maximal element of codimension $d$.


On Bellissima's construction of the finitely generated free Heyting algebras, and beyond

Abstract: We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima's representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal sprectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. For example we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.


Generalized low-pass filters and multiresolution analyses

Abstract: We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.


Computing p-adic integrals using motivic integration

Abstract: We use the theory of motivic integration in order to give a geometric explanation of the behavior of some p-adic integrals.


Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

Abstract: The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems.


Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Abstract: Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.


Left eigenvalues of $2\times 2$ symplectic matrices

Abstract: We obtain a complete characterization of the $2\times 2$ symplectic matrices having an infinite number of left eigenvalues. Previously, we give a new proof of a result from Huang and So about the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al.


Index formulas on stratified manifolds

Abstract: Elliptic operators on stratified manifolds with any finite number of strata are considered. Under certain assumptions on the symbols of operators, we obtain index formulas, which express index as a sum of indices of elliptic operators on the strata.


Inegalite d'Ahlfors en dimension superieure

Abstract: We proove an Ahlfors' like inequality for the holomorphic curves of a complex compact Kobayashi hyperbolic manifold.


The Harish-Chandra isomorphism for Clifford algebras

Abstract: We study the analogue of the Harish-Chandra homomorphism where the universal enveloping algebra is replaced by the Clifford algebra, $Cl(g)$, of a semisimple Lie algebra $g$. Two main goals are achieved. First, we prove that there is a Harish-Chandra type isomorphism between the subalgebra of $g$-invariants in $Cl(g)$ and the Clifford algebra of the Cartan subalgebra of $g$. Second, the Cartan subalgebra is identified, via this isomorphism, with a graded space of the so-called primitive skew-symmetric invariants of $g$. This leads to a distinguished orthogonal basis of the Cartan subalgebra, which turns out to be induced from the Lie algebra Langlands dual to $g$ via the action of its principal three-dimensional subalgebra. This settles a conjecture of Kostant.


A formalism for the study of Natural Tensors Fields of type (0,2) on Manifolds and Fibrations

Abstract: In order to study tensor fields of type (0,2) on manifolds and fibrations we introduce the notion of s-spaces. With the help of these objects we generalized the concept of natural tensor without making use of the theory of natural operators and differential invariants.


Non-crossing linked partitions and multiplication of free random variables

Abstract: The material gives a new combinatorial proof of the multiplicative property of the S-transform. In particular, several properties of the coefficients of its inverse are connected to non-crossing linked partitions and planar trees.


Depth and Stanley depth of multigraded modules

Abstract: We study the behavior of depth and Stanley depth along short exact sequences of multigraded modules and under reduction modulo an element.


On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$

Abstract: In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed $N$ and for any $m\geq 2$. It is shown that when $m=2$ and $m=3$ the Euler-Maclaurin quadrature formula is optimal.


Harmonic functions for a class of integro-differential operators

Abstract: We consider the operator $\sL$ defined on $C^2(\bR^d)$ functions by \sL f(x)&=&{1/2}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial^2f(x)}{\partial x_i\partial x_j}+\sum_{i=1}^d b_i(x)\frac{\partial f(x)}{\partial x_i} &+&\int_{\bR^d\backslash\{0\}}[f(x+h)-f(x)-1_{(|h|\leq1)}h\cdot \grad f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on $n(x,h)$, we establish a Harnack inequality for functions that are nonnegative in $\bR^d$ and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on $n(x,h)$. A regularity theorem for those nonnegative harmonic functions is also proved


Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering

Abstract: We consider two classes of defocusing energy-supercritical nonlinear Schr\"odinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.


A limit approach to group homology

Abstract: In this paper, we consider for any free presentation $G = F/R$ of a group $G$ the coinvariance $H_{0}(G,R_{ab}^{\otimes n})$ of the $n$-th tensor power of the relation module $R_{ab}$ and show that the homology group $H_{2n}(G,{\mathbb Z})$ may be identified with the limit of the groups $H_{0}(G,R_{ab}^{\otimes n})$, where the limit is taken over the category of these presentations of $G$. We also consider the free Lie ring generated by the relation module $R_{ab}$, in order to relate the limit of the groups $\gamma_{n}R/[\gamma_{n}R,F]$ to the $n$-torsion subgroup of $H_{2n}(G,{\mathbb Z})$.


$L^2$-Betti numbers and non-unitarizable groups without free subgroups

Abstract: We show that there exist non-unitarizable groups without non-abelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with non-vanishing first $L^2$-Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of $L^2$-Betti numbers.


Geometrical description of smooth projective symmetric varieties with Picard number one

Abstract: In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of automorphisms. When this group, Aut(X), acts non-transitively on X, we describe a G-equivariant embedding of the variety X in a homogeneous variety (with respect to a larger group).


A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

Abstract: We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.


Some remarks on the size of tubular neighborhoods in contact topology and fillability

Abstract: The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N, and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nx{0} in the model space NxR^{2k}. In this article we make the observation that if (N,\xi_N) is a 3-dimensional overtwisted submanifold with trivial normal bundle in (M,\xi), and if its model neighborhood is sufficiently large, then (M,\xi) does not admit an exact symplectic filling.


The space of closed subgroups of $R^n$

Abstract: The Chabauty space of a topological group is the set of its closed subgroups, endowed with a natural topology. As soon as $n>2$, the Chabauty space of $R^n$ has a rather intricate topology and is not a manifold. By an investigation of its local structure, we fit it into a wider, but too wild, class of topological spaces (namely Goresky-MacPherson stratified spaces). Thanks to a localization theorem, this local study also leads to the main result of this article: the Chabauty space of $R^n$ is simply connected for all $n$. Last, we give an alternative proof of the Hubbard-Pourezza Theorem, which describes the Chabauty space of $R^2$.


Locally definable homotopy

Abstract: In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.


Vanishing Cycles in Holomorphic Foliations by Curves and Foliated Shells

Abstract: The purpose of this paper is the study of vanishing cycles in holomorphic foliations by complex curves on compact complex manifolds. The main result consists in showing that a vanishing cycle comes together with a much richer complex geometric object - we call this object a foliated shell.


Multiplicative formulas in Cohomology of $G/P$ and in quiver representations

Abstract: Consider a partial flag variety $X$ which is not a grassmaninan. Consider also its cohomology ring ${\rm H}^*(X,\ZZ)$ endowed with the base formed by the Poincar\'e dual classes of the Schubert varieties. In \cite{Richmond:recursion}, E. Richmond showed that some coefficient structure of the product in ${\rm H}^*(X,\ZZ)$ are products of two such coefficients for smaller flag varieties. Consider now a quiver without oriented cycle. If $\alpha$ and $\beta$ denote two dimension-vectors, $\alpha\circ\beta$ denotes the number of $\alpha$-dimensional subrepresentations of a general $\alpha+\beta$-dimensional representation. In \cite{DW:comb}, H. Derksen and J. Weyman expressed some numbers $\alpha\circ\beta$ as products of two smaller such numbers. The aim of this work is to prove two generalisations of the two above results by the same way.


Geodesic Webs of Hypersurfaces

Abstract: In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.


Factorization property of generalized s-selfdecomposable measures and class $L^f$ distributions$^1$

Abstract: The method of \emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \emph{factorization property}. These are the classes $\mathcal{U}^f_{\be}$ and $L^f$, respectively


Lowest Weights in Cohomology of Variations of Hodge Structure (II)

Abstract: Let $X$ be an irreducible complex analytic space with $j:U\into X$ an immersion of a smooth Zariski open subset, and let $\bV$ be a variation of Hodge structure of weight $n$ over $U$. Assume $X$ is compact K\"ahler. Then provided the local monodromy operators at infinity are quasi-unipotent, $IH^k(X, \bV)$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\bV)$ carries a mixed Hodge structure of weight $\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,\bV) \to H^k(U,\bV)$ is the lowest weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement $X-U$ is not a hypersurface.


Exotic Bialgebras from 9x9 Unitary Braid Matrices

Abstract: We present the exotic bialgebras that arise from a 9x9 unitary braid matrix.


Vorticity internal transition layers for the Navier-Stokes equations

Abstract: We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show -thanks to an asymptotic expansion- that there is a sharp but smooth variation of the fluid vorticity into a internal layer moving with the flow of the Euler equations; as long as this later exists and as t * nu is small, where nu is the viscosity coefficient.


Pairs of mutually annihilating operators

Abstract: Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)--> S^{-1}AS, S^{-1}BS).


On complex nilpotent Leibniz superalgebras of nilindex n+m

Abstract: We present the description up to isomorphism of Leibniz superalgebras with characteristic sequence $(n|m_1,...,m_k)$ and nilindex $n+m,$ where $m=m_1+ >...+m_k,$ $n$ and $m$ ($m\neq 0$) are dimensions of even and odd parts, respectively.


On the moduli space of quadruples of points in the boundary of complex hyperbolic space

Abstract: We consider the space $\mathcal M$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\ch{n},$ up to its holomorphic isometry group ${\rm PU}(n,1).$ One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $\mathcal M$. For $n=2$, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $\mathcal M $ for any dimension $n \geq 1$. The major innovation in our paper is the use of the Gram matrix instead of a standard position of points.


Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

Abstract: We consider an entire graph S of a continuous real function over (N-1)-dimensional Euclidean space with N larger than or equal to 3. Let D be a domain in N-dimensional Euclidean space with S as a boundary. Consider in D the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in D. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Amp\`ere-type equation.


Explicit Constructions of the non-Abelian $\mathbf{p^3}$-Extensions Over $\mathbf{\QQ}$

Abstract: Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(\mu_{p})}^{*}$. In this paper we let $k=\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\QQ$ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.


A locally finite tree that behaves like an infinite star

Abstract: We construct a tree T of maximal degree 3 with infinitely many leaves such that whenever finitely many of them are removed, the remaining tree is isomorphic to T. In this sense T resembles an infinite star.


A local branching heuristic for MINLPs

Abstract: Local branching is an improvement heuristic, developed within the context of branch-and-bound algorithms for MILPs, which has proved to be very effective in practice. For the binary case, it is based on defining a neighbourhood of the current incumbent solution by allowing only a few binary variables to flip their value, through the addition of a local branching constraint. The neighbourhood is then explored with a branch-and-bound solver. We propose a local branching scheme for (nonconvex) MINLPs which is based on iteratively solving MILPs and NLPs. Preliminary computational experiments show that this approach is able to improve the incumbent solution on the majority of the test instances, requiring only a short CPU time. Moreover, we provide algorithmic ideas for a primal heuristic whose purpose is to find a first feasible solution, based on the same scheme.


Dunkl Hyperbolic Equations

Abstract: We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.


The Virtually Cyclic Classifying Space of the Heisenberg Group

Abstract: We are interested in the relationship between the virtual cohomological dimension (or vcd) of a discrete group Gamma and the smallest possible dimension of a model for the classifying space of Gamma relative to its family of virtually cyclic subgroups. In this paper we construct a model for the virtually cyclic classifying space of the Heisenberg group. This model has dimension 3, which equals the vcd of the Heisenberg group. We also prove that there exists no model of dimension less than 3.


The length of chains in algebraic lattices

Abstract: We study how the existence in an algebraic lattice $L$ of a chain of a given type is reflected in the join-semilattice $K(L)$ of its compact elements. We show that for every chain $\alpha$ of size $\kappa$, there is a set $\B$ of at most $2^{\kappa}$ join-semilattices, each one having a least element such that an algebraic lattice $L$ contains no chain of order type $I(\alpha)$ if and only if the join-semilattice $K(L)$ of its compact elements contains no join-subsemilattice isomorphic to a member of $\B$. We show that among the join-subsemilattices of $[\omega]^{<\omega}$ belonging to $\B$, one is embeddable in all the others. We conjecture that if $\alpha$ is countable, there is a finite set $\B$.


Local Convergence of the Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds

Abstract: Local convergence analysis of the proximal point method for special class of nonconvex function on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.


Lifts of partial characters with respect to a chain of normal subgroups

Abstract: In this paper, we consider lifts of $\pi$-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we call an inductive pair, and every character induced from an inductive pair is a such a lift. With this condition, we will get a lower bound on the number of such lifts.


On the Banach-Mazur Type for Normed Spaces

Abstract: In order to measure qualitative properties we introduce a notion of a type for arbitrary normed spaces which measures the worst possible growth of partial sums of sequences weakly converging to zero. The ideas can be traced back to Banach and Mazur who used this type to compare the so-called linear dimension of classical Banach spaces. As an application we compare the linear dimension and investigate isomorphy of some classical Banach spaces.


Groups where all the irreducible characters are super-monomial

Abstract: Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\{p \}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group.


The Large Sieve and Galois Representations

Abstract: We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.


Some index formulae on the moduli space of stable parabolic vector bundles

Abstract: We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.


Asymptotic expansion of planar canard solutions near a non-generic turning point

Abstract: This paper deals with the asymptotic study of the so-called canard solutions, which arise in the study of real singularly perturbed ODEs. Starting near an attracting branch of the "slow curve", those solutions are crossing a turning point before following for a while a repelling branch of the "slow curve". Assuming that the turning point is degenerate (or non-generic), we apply a correspondence presented in a recent paper. This application needs the definition of a family of functions $\phi$ that is studied in a first part. Then, we use the correspondence is used to compute the asymptotic expansion in the powers of the small parameter for the canard solution.


First, second, and third change of rings theorems for the Gorenstein homological dimensions

Abstract: Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established, in this paper, for the Gorenstein projective dimension is a generalization of a result established over Noetherian rings and for finitely generated modules. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this paper for the Gorenstein projective dimension is a generalization of a $G$-dimension of a finitely generated module $M$ over a noetherian ring $R$.


The Ricci Flow for Nilmanifolds

Abstract: We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.


Steinberg lattice of the general linear group and its modular reduction

Abstract: We study the Steinberg lattice of the general linear group when reduced modulo a prime different from the defining characteristic.


Restricted limits of minimal affinizations

Abstract: We obtain character formulas of minimal affinizations of representations of quantum groups when the underlying simple Lie algebra is orthogonal and the support of the highest weight is contained in the first three nodes of the Dynkin diagram. We also give a framework for extending our techniques to a more general situation. In particular, for the orthogonal algebras and a highest weight supported in at most one spin node, we realize the restricted classical limit of the corresponding minimal affinizations as a quotient of a module given by generators and relations and, furthermore, show that it projects onto the submodule generated by the top weight space of the tensor product of appropriate restricted Kirillov-Reshetikhin modules. We also prove a conjecture of Chari and Pressley regarding the equivalence of certain minimal affinizations in type D4.


The number of elements in the mutation class of a quiver of type $D_n$

Abstract: We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\sum_{d|n} \phi(n/d)\binom{2d}{d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.


'Fair' Partitions of Polygons - an Introduction

Abstract: We address the question: Given a positive integer $N$, can any 2D convex polygonal region be partitioned into $N$ convex pieces such that all pieces have the same area and same perimeter? The answer to this question is easily `yes' for $N$=2. We prove the answer to be `yes' for $N$=4 and also discuss higher powers of 2.


How Euler would compute the Euler-Poincar\'e characteristic of a Lie superalgebra

Abstract: The Euler-Poincar\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to Euler, allows to do that, to a certain degree. The mathematics behind it is simple, we just glue the pieces of elementary homological algebra, first-year calculus and pedestrian combinatorics together, and present them in a (hopefully) coherent manner.


Average-Case Perturbations and Smooth Condition Numbers

Abstract: We define a new condition number adapted to directionally uniform perturbations. The definitions and theorems can be applied to a large class of problems. We show the relation with the classical condition number, and study some interesting examples.


Feature selection by Higher Criticism thresholding: optimal phase diagram

Abstract: We consider two-class linear classification in a high-dimensional, low-sample size setting. Only a small fraction of the features are useful, the useful features are unknown to us, and each useful feature contributes weakly to the classification decision -- this setting was called the rare/weak model (RW Model). We select features by thresholding feature $z$-scores. The threshold is set by {\it higher criticism} (HC). Let $\pee_i$ denote the $P$-value associated to the $i$-th $z$-score and $\pee_{(i)}$ denote the $i$-th order statistic of the collection of $P$-values. The HC threshold (HCT) is the order statistic of the $z$-score corresponding to index $i$ maximizing $(i/n - \pee_{(i)})/\sqrt{\pee_{(i)}(1-\pee_{(i)})}$. The ideal threshold optimizes the classification error. In \cite{PNAS} we showed that HCT was numerically close to the ideal threshold. We formalize an asymptotic framework for studying the RW model, considering a sequence of problems with increasingly many features and relatively fewer observations. We show that along this sequence, the limiting performance of ideal HCT is essentially just as good as the limiting performance of ideal thresholding. Our results describe two-dimensional {\it phase space}, a two-dimensional diagram with coordinates quantifying "rare" and "weak" in the RW model. Phase space can be partitioned into two regions -- one where ideal threshold classification is successful, and one where the features are so weak and so rare that it must fail. Surprisingly, the regions where ideal HCT succeeds and fails make the exact same partition of the phase diagram. Other threshold methods, such as FDR threshold selection, are successful in a substantially smaller region of the phase space than either HCT or Ideal thresholding.


Mixing time of exponential random graphs

Abstract: Exponential random graphs are used extensively in the sociology literature. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing, and more generally for understanding basic features of the network model itself. In practice sampling is typically carried out using Markov chain Monte Carlo, in particular either the Glauber dynamics or the Metropolis-Hasting procedure. In this paper we characterize the high and low temperature regimes of the exponential random graph model. We establish that in the high temperature regime the mixing time of the Glauber dynamics is $\Theta(n^2 \log n)$, where $n$ is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the Erd\H{o}s-R\'enyi random graph.


Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

Abstract: It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.


Solitary Wave Solutions for the Nonlinear Dirac Equations

Abstract: In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation \[ i \sum_{j=0}^{3} \ga^j \pd_j \psi - m\psi + F(\bar{\psi}\psi)\psi =0 \] where $ m >0$ and $ F(s) = |s|^{\theta}$ for $ 1\leq \theta < 2.$ More precisely we show that there exists $\e_0 > 0$ such that for $\omega \in(m - \e_0, m), $ there exists a solution $ \psi(t,x) = e^{-i\omega t}\phi_{\omega}(x), x_0 = t, x = (x_1, x_2, x_3),$ and the mapping from $ \omega $ to $ \phi_{\omega} $ is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schr\"{o}dinger equations.


Density of commensurators for uniform lattices of right-angled buildings

Abstract: Let G be the automorphism group of a regular right-angled building X. The "standard uniform lattice" \Gamma_0 in G is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of \Gamma_0 is dense in G. This result was also obtained by Haglund. For our proof, we develop carefully a technique of "unfoldings" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices \Gamma_n in G, each commensurable to \Gamma_0, and then apply the theory of group actions on complexes of groups to the sequence \Gamma_n. As further applications of unfoldings, we determine exactly when the group G is nondiscrete, and we prove that G acts strongly transitively on X.


Badly approximable affine forms and Schmidt games

Abstract: For any real number \t, the set of all real numbers x for which there exists a constant c(x) > 0 such that \inf_{p \in \ZZ} |\t q - x - p| \geq c(x)/|q| for all q in \ZZ {0} is an 1/8-winning set.


On sums and products in C[x]

Abstract: We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove an analogue of a theorem of Bourgain and Chang for the ring C[x].


A characterization of well-founded algebraic lattices

Abstract: We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice $L$ is well-founded if and only if $K(L)$, the join-semilattice of compact elements of $L$, is well-founded and contains neither $[\omega]^{<\omega}$, nor $\underline\Omega(\omega^*)$ as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice $L$ is well-founded if and only if $K(L)$ is well-founded and contains no infinite independent set. If $K(L)$ is a join-subsemilattice of $I_{<\omega}(Q)$, the set of finitely generated initial segments of a well-founded poset $Q$, then $L$ is well-founded if and only if $K(L)$ is well-quasi-ordered.


The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties

Abstract: Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a non-constant map to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can be often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial, or more generally semi-ample canonical bundle.


Homeomorphic Solutions to Reduced Beltrami Equations

Abstract: We study differential expressions related to linear families of quasiconformal mappings and give a simple and direct proof to a result due to Alessandrini and Nesi arXiv:0707.0727.


On almost Blow-analytic equivalence

Abstract: Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of blowings-up along smooth analytic centers. We apply the result to prove that Nash function germs that are analytically equivalent after analytic desingularizations are Nash equivalent after Nash desingularizations. Results are based on a precise Euclidean description of a sequence of blowings-up combined with N\'eron Desingularization.


An introduction to wonderful varieties with many examples of type F4

Abstract: We give an introduction to the theory of wonderful G-varieties, with many examples when G is simple of type F4. We present results and open problems about these varieties: on their classification, on their isotropy groups, on morphisms between them, and on their relations with the representation theory of G.


Combinatorial Formulas for Classical Lie Weight Systems on Arrow Diagrams

Abstract: In 2002 Haviv gave a way of assigning Lie tensors to directed trivalent graphs. Weight systems on oriented chord idagrams modulo 6T can then be constructed from such tensors. In this paper we give explicit combinatorial formulas of weight systems using Manin triples constrcted from classical Lie algebras. We then compose these oriented weight systems with the averaging map to get weight systems on unoriented chord diagrams and show that they are the same as the ones obtained by Bar-Natan in 1991. In the last section we carry out calculations on certain examples.


Axisymmetric Rotating Fluid Equations

Abstract: We investigate the equations of anisotropic axisymmetric incompressible viscous fluids in the exterior of a cylinder of $\R^3$, rotating around an inhomogeneous vector $B(t, r)$. We prove uniform local existence with respect to the Rossby number in suitable anisotropic Sobolev spaces. We also obtain the propagation of the isotropic Sobolev regularity.


On Invertibility of Sobolev Mappings

Abstract: We prove local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions.


Extension Phenomena for Holomorphic Geometric Structures

Abstract: The most commonly encountered types of complex analytic G-structures and Cartan geometries cannot have singularities of complex codimension 2 or more.


The Virtual Private Network Design Problem with Concave Costs (Oberwolfach abstract)

Abstract: The symmetric Virtual Private Network Design (VPND) problem is concerned with buying capacity on links (edges) in a communication network such that certain traffic demands can be met. We investigate a natural generalization of VPND where the cost per unit of capacity may decrease if a larger amount of capacity is reserved (economies of scale principle). The growth of the cost of capacity is modelled by a non-decreasing concave function $f$. We call the problem the concave symmetric Virtual Private Network Design (cVPND) problem. After showing that a generalization of the so-called Pyramidal Routing problem and hence also the cVPND have the so-called tree routing property, we study approximation algorithms for cVPND. For general $f$, using known results on the so-called Single Source Buy at Bulk problem by Grandoni and Italiano, we give a randomized 24.92-approximation algorithm.


Some algebraic properties of hypergraphs

Abstract: We consider Stanley--Reisner rings $k[x_1,...,x_n]/I(\mc{H})$ where $I(\mc{H})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize upon some known results about chordal graphs and study a weak form of shellability.


Dynamical Systems and Topological Surgery

Abstract: In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathematically by the above dynamical system.


Nonsmooth Hormander vector fields and their control balls

Abstract: We prove a ball-box theorem for nonsmooth Hormander vector fields of step s.


Minimal coexistence configurations for multispecies systems

Abstract: We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.


Lines on Fermat surfaces

Abstract: We prove that the Neron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well.


Dimension of elliptic harmonic measure of Snowspheres

Abstract: A metric space $\mathcal{S}$ is called a \defn{quasisphere} if there is a quasisymmetric homeomorphism $f\colon S^2\to \mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by $f$. It is shown that for certain self similar quasispheres $\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\mathcal{S}$.


Kinetic equations with Maxwell boundary conditions

Abstract: We prove global stability results of {\sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz\`es-Guiraud information in a crucial way.


Slow escaping points of meromorphic functions

Abstract: We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filling wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.


Evaluating the Impact of Missing Data Imputation through the use of the Random Forest Algorithm

Abstract: This paper presents an impact assessment for the imputation of missing data. The data set used is HIV Seroprevalence data from an antenatal clinic study survey performed in 2001. Data imputation is performed through five methods: Random Forests, Autoassociative Neural Networks with Genetic Algorithms, Autoassociative Neuro-Fuzzy configurations, and two Random Forest and Neural Network based hybrids. Results indicate that Random Forests are superior in imputing missing data in terms both of accuracy and of computation time, with accuracy increases of up to 32% on average for certain variables when compared with autoassociative networks. While the hybrid systems have significant promise, they are hindered by their Neural Network components. The imputed data is used to test for impact in three ways: through statistical analysis, HIV status classification and through probability prediction with Logistic Regression. Results indicate that these methods are fairly immune to imputed data, and that the impact is not highly significant, with linear correlations of 96% between HIV probability prediction and a set of two imputed variables using the logistic regression analysis.


Rooted trees and symmetric functions: Zhao's homomorphism and the commutative hexagon

Abstract: Recent work on perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf algebras to each other and to the Hopf algebras of symmetric functions, noncommutative symmetric functions, and quasi-symmetric functions.


Non existence of principal values of signed Riesz transforms of non integer dimension

Abstract: In this paper we prove that, given s> 0, if E is a subset of R^m with positive and bounded s-dimensional Hausdorff measure H^s and the principal values of the s-dimensional signed Riesz transform of H^s|E exist H^s-almost everywhere in E, then s is integer. Other more general variants of this result are also proven.


Categorical centers and Reshetikhin-Turaev invariants

Abstract: The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C).


Spectral norm of products of random and deterministic matrices

Abstract: We study the spectral norm of matrices M that can be factored as M=BA, where A is a random matrix with independent mean zero entries, and B is a fixed matrix. Under the (4+epsilon)-th moment assumption on the entries of A, we show that the spectral norm of such an m by n matrix M is bounded by \sqrt{m} + \sqrt{n}, which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of M. Rudelson and the author implies that the smallest singular value of a random m times n matrix with i.i.d. mean zero entries and bounded (4+epsilon)-th moment is bounded below by \sqrt{m} - \sqrt{n-1} with high probability.


On the Dec group of finite abelian Galois extensions over global fields

Abstract: If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras.


On the characterization of algebraically integrable plane foliations

Abstract: We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number, $r+1$, of points in the projective plane through which infinitely many algebraic leaves of the foliation go.


The double of a Hopf monad

Abstract: The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category Z-C of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We construct a Hopf monad Z_T on C, the centralizer of T, and a canonical distributive law of T over Z_T. By Beck's theory, this has two consequences. On one hand, D_T=Z_T T is a quasitriangular Hopf monad on C, called the double of T, and Z(T-C)= D_T-C as braided categories. As an illustration, we define the double D(A) of a Hopf algebra A in a braided autonomous category in such a way that the center of the category of A-modules is the braided category of D(A)-modules (generalizing the Drinfeld double). On the other hand, the canonical distributive law also lifts Z_T to a Hopf monad on T-C which gives the coend of T-C. Hence, for T=Z, an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C, which is useful in quantum topology.


Theorem on best Diophantine approximations for linear forms

Abstract: We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form.


Heat kernel bounds, ancient $\kappa$ solutions and the Poincar\'e conjecture

Abstract: We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $\kappa$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backward limits of these ancient solutions. The current paper together with \cite{Z:2} and a different proof of universal noncollapsing due to Chen and Zhu \cite{ChZ:1} lead to a simplified proof of the Poincar\'e conjecture without using reduced distance and reduced volume.


Uniqueness of contact Hamiltonians of topological strictly contact isotopies

Abstract: We prove that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians.


The example of a self-similar continuum which is not an attractor of any zipper

Abstract: The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.


Characterization of linear groups whose reduced C*-algebras are simple

Abstract: The reduced C*-algebra of a countable linear group G is shown to be simple if and only if G has no nontrivial normal amenable subgroups. Moreover, these conditions are shown to be equivalent to the uniqueness of tracial state on the aforementioned C*-algebra.


Pullback of varieties by finite maps

Abstract: We study the local geometry of the pullback of a variety via a finite holomorphic map. In particular, we are looking for properties of $V = F^{-1}(W)$ such that if $V$ has the property $A$, then $W$ must have the property $A$. We show that $A$ can be the property of normality or prefactoriality. We also show that $A$ can be the property of smoothness, under extra assumptions.


Isolated points in the space of left orderings of a group

Abstract: Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend known results to cover the case of uncountable groups. With minor technical restrictions on the group G, we also find that no dense left ordering is isolated in LO(G), and that the closure of the set of all dense left orderings of G yields a dense G-delta set within a Cantor set of left orderings in LO(G). Lastly, we show that certain conditions on a discrete left ordering of G can guarantee that it is not isolated in LO(G), and we illustrate these ideas using the Dehornoy ordering of the braid groups.


Universal deformation rings for the symmetric group S_4

Abstract: Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.


Fluctuation theory and exit systems for positive self-similar Markov processes

Abstract: For a positive self-similar Markov process, X, we construct a local time for the random set, $\Theta$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $\Theta$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder process linked to the L\'{e}vy process associated to X via Lamperti's transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying L\'{e}vy process oscillates.


Derived Mackey functors

Abstract: For a finite group $G$, the so-called $G$-Mackey functors form an abelian category $M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the derived category $D(M(G))$ would be similarly important as the "homological" counterpart of the $G$-equivariant stable homotopy category. It turns out that this is not so -- $D(M(G))$ is pathological in many respects. We propose and study a replacement for $D(M(G))$, a certain triangulated category $DM(G)$ of "derived Mackey functors" that contains $M(G)$ but is different from $D(M(G))$. We show that standard features of the $G$-equivariant stable homotopy category such as the fixed points functors of two types have exact analogs for the category $DM(G)$.


On Group bijections $\phi $ with $\phi(B)=A$ and $\forall a\in B, a\phi(a) \notin A$

Abstract: A {\em Wakeford pairing} from $S$ onto $T$ is a bijection $\phi : S \to T$ such that $x\phi(x)\notin T,$ for every $x\in S.$ The number of such pairings will be denoted by $\mu(S,T)$. Let $A$ and $ B$ be finite subsets of a group $G$ with $1\notin B$ and $|A|=|B|.$ Also assume that the order of every element of $B$ is $\ge |B|$. Extending results due to Losonczy and Eliahou-Lecouvey, we show that $\mu(B,A)\neq 0.$ Moreover we show that $\mu(B,A)\ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ unless there is $a\in A$ such that $|Aa^{-1}\cap B|=|B|-1$ or $Aa^{-1}$ is a progression. In particular, either $\mu(B,B) \ge \min \{\frac{||B|+1}{3},\frac{|B|(q-|B|-1)}{2q-|B|-4}\},$ or for some $a\in B,$ $Ba^{-1}$ is a progression.


The three-dimensional Finite Larmor Radius Approximation

Abstract: Following Fr\'enod and Sonnendr\"ucker, we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in a three-dimensional setting with the help of an averaging lemma.


The speed of a biased random walk on a percolation cluster at high density

Abstract: We study the speed of a biased random walk on a percolation cluster on $\Z^d$ in function of the percolation parameter $p$. We obtain a first order expansion of the speed at $p=1$ which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice.


Magmatic "Quantum-Like" Systems

Abstract: Quantum computation has suggested, among others, the consideration of "non-quantum" systems which in certain respects may behave "quantum-like". Here, what algebraically appears to be the most general possible known setup, namely, of {\it magmas} is used in order to construct "quantum-like" systems. The resulting magmatic composition of systems has as a well known particular case the tensor products.


Reduced genus-two Gromov-Witten Invariants for complex manifolds

Abstract: In this article, we construct the reduced genus-two Gromov-Witten invariants for certain almost K\"{a}hler manifold $(X, \omega, J)$ such that $J$ is integrable and satisfies some regularity conditions. In particular, the standard projective space $(\P^n, \omega_0, J_0)$ of dimension $n \le 7$ satisfies these conditions. This invariant counts the number of simple genus-two $J$-holomorphic curves that satisfy appropriate number of constraints.


On a class of linearizable planar geodesic webs

Abstract: We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb.


Copulas for Markovian dependence

Abstract: Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper demonstrates some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete co- and countermonotonicity and independence (Fr\'{e}chet copulas) are shown to imply quite a restricted type of Markov process and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable or, equivalently, conditionally i.i.d.


The diffeomorphism group of a Lie foliation

Abstract: We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus $T^n$, $n\geq 2$, namely codimension one foliations, flows, and the so-called non-quadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than $\pm \id$ and translations. The computation is an application of a general formula that we prove for the diffeomorphism group of any Lie foliation with dense leaves on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and G. Lachaud for codimension one foliations on $T^n$, $n\geq 2$, and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau's diffeological spaces.


A Bernoulli linked-twist map in the plane

Abstract: We prove that a Lebesgue measure-preserving linked-twist map defined in the plane is metrically isomorphic to a Bernoulli shift (and thus strongly mixing). This is the first such result for an explicitly defined linked-twist map on a manifold other than the two-torus. Our work builds on that of Wojtkowski who established an ergodic partition for this example using an invariant cone-field in the tangent space.


Note on on Dedekind type DC sums

Abstract: In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums.


On the comparison of the Dirichlet and Neumann counting functions

Abstract: The aim of the paper is to show that L. Friedlander's results on the relation between Dirichlet and Neumann counting functions (Arch. Ration. Mech. Anal. 116, 1991) remain valid in abstract setting.


Complementary Regions of Knot and Link Diagrams

Abstract: An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. This paper is an investigation into which sequences, either finite or infinite, are universal. We also consider how to minimize the number of odd-sided faces for projections of knots and links with n components.


Precise subelliptic estimates for a class of special domains

Abstract: For the $\bar\partial$-Neumann problem on a regular coordinate domain $\Omega\subset \C^{n+1}$, we prove $\epsilon$-subelliptic estimates for an index $\epsilon$ which is in some cases better than $\epsilon=\frac1{2m}$ ($m$ being the {\it multiplicity}) as it was previously proved by Catlin and Cho in \cite{CC08}. This also supplies a much simplified proof of the existing literature. Our approach is founded on the method by Catlin in \cite{C87} which consists in constructing a family of weights $\{\phi^\delta\}$ whose Levi form is bigger than $\delta^{-2\epsilon}$ on the $\delta$-strip around $\partial\Omega$.


On a C2-nonlinear subdivision scheme avoiding Gibbs oscillations

Abstract: This paper is devoted to the presentation and the analysis of a new nonlinear subdivision scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence, stability and order of approximation are analyzed. It is proved that this schemes converges towards limit functions of H\"older regularity index larger than 1.192. Numerical estimates provide an H\"older regularity index of 2.438. Up to our knowledge, this scheme is the first one that achieves simultaneously the control of the Gibbs phenomenon and regularity index larger than 1 for its limit functions.


A generalization of strongly Gorenstein projective modules

Abstract: This paper generalize the idea of the authors in J. Pure Appl. Algebra 210 (2007) 437--445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.


Applications of the Graph Minor Theorem to algebraic structures I

Abstract: We use the Graph Minor Theorem to characterize infinite sequences of finite subsets of factorial and commutative semigroups (here semigroups have a unity element), e.g. the multiplicative semigroup of a unique factorization domain.


Attractor Networks on Complex Flag Manifolds

Abstract: Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps are equivalent concepts and illustrate this with the example of linear flows on projective spaces. In these examples the fixed points are linearly ordered with respect to the Smale order which makes the attractor-repellor network overly simple. In this paper we provide a class of examples in which the attractor-repellor network and its lattice structure can be explicitly determined even though the Smale order is not total. They are associated with special flows on complex flag manifolds. In the process we show that the Smale order on the set of fixed points can be identified with the well-known Bruhat order. This could also be derived from results of Kazhdan and Lusztig, but we give a new proof using the Lambda-Lemma of Palis. For the convenience of the reader we also introduce the flag manifolds via elementary dynamical systems using only a minimum of Lie theory.


Doubling rational normal curves

Abstract: In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in \cite{fer}, we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers $ (2r,g,n) $ where $ r $ is the degree of the support, $ n \geq r $ is the dimension of the projective space containing the double curve, and $ g $ is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple $ (2r,g,n) $ is irreducible. Furthermore, we prove that the general double curve in the families associated to $ (2r,r+1,r) $ and $ (2r,1,2r-1) $ is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus $ g \leq -2 $ form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in $ \mathbb{P}^3 $ of arbitrary genus is a smooth point of the corresponding Hilbert scheme.


Point configurations that are asymmetric yet balanced

Abstract: A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.


On Poncelet's maps

Abstract: Given two ellipses, one surrounding the other one, Poncelet introduced a map $P$ from the exterior one to itself by using the tangent lines to the interior ellipse. This procedure can be extended to any two smooth, nested and convex ovals and we call this type of maps Poncelet's maps. We recall what he proved around 1814 in the dynamical systems language: In the two ellipses case and when the rotation number of P is rational there exists a natural number such that P^n is the identity, or in other words, the Poncelet's map is conjugated to a rational rotation. In this paper we study general Poncelet's maps and give several examples of algebraic ovals where the corresponding Poncelet's map has a rational rotation number and it is not conjugated to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical tools.


$L^p$-improving estimates for averages on polynomial curves

Abstract: In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a polynomial-like function from below using only the value of the function or its derivatives at some prescribed point. In this paper, it is shown that there is always a relatively large set of points (independent of the particular function to be integrated) for which such estimates are possible. Inequalities of this type are then applied to extend the results of Tao and Wright (JAMS, 2003) to obtain endpoint restricted weak-type estimates for averages over curves given by polynomials.


Finite groups have even more conjugacy classes

Abstract: In his paper "Finite groups have many conjugacy classes" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper.


On Taking Square Roots without Quadratic Nonresidues over Finite Fields

Abstract: We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.


Identities for the Riemann zeta function

Abstract: We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of $s$. The expansions also give a different approach to the analytic continuation of the Riemann zeta function.


Deterministic Primality Proving on Proth Numbers

Abstract: We present an algorithm to decide the primality of Proth numbers, N=2^e t+1, without assuming any unproven hypothesis. The expected running time and the worst case running time of the algorithm are O ((t log t + log N)log N) and O ((t log t + log N) log^2 N) bit operations, respectively.


Generalized (\kappa,\mu)-space forms

Abstract: Generalized (\kappa ,\mu)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case.


The second moment of Dirichlet twists of Hecke $L$-functions

Abstract: Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$.


Additive Bases in Abelian Groups

Abstract: Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of $G$; that is, for every $g\in G$ there exist $A_1\subset B_1, ..., A_k\subset B_k$ such that $g=\sum_{i=1}^k\sum_{a\in A_i} a$. This generalizes a result of Alon, Linial, and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where $B_1, ..., B_k$ are finite subsets of a vector space we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form $\sum_{i=1}^k \sum_{a\in A_i} a$, where $A_i$ vary over all subsets of $B_i$ for each $i=1, >..., k$. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings.


Property A and asymptotic dimension

Abstract: The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.] $asdim(X,d)\leq n$, [b.] For each $R,\epsilon > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_x\Delta A_y|}{| A_x\cap A_y|} < \epsilon$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$, [c.] For each $R > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_x\Delta A_y|}{| A_x\cap A_y|} < \frac{1}{n+1}$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$.


Poynting-Robertson Effect on the Lyapunov Stability of Equilibrium Points in the Generalized Photogravitational Chermnykh's Problem

Abstract: The Poynting-Robertson(P-R) effect on Lyapunov stability of equilibrium points is being discussed in the the generalized photogravitational Chermnykh's problem when bigger primary is a sours of radiation and smaller primary is an oblate spheriod. We derived the equations of motion, obtained the equilibrium points and examined the linear stability of the equilibrium points for various values of parameter which have been used in the present problem. We have examined the effect of gravitational potential from the belt. The positions of the equilibrium points are different from the position in classical case. We have seen that due to the P-R effect all the equilibrium points are unstable in Lyapunov sense.


Dunkl Operators and Canonical Invariants of Reflection Groups

Abstract: Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.


Central extensions of current algebras

Abstract: This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus algebras, and of some modular semisimple Lie algebras. The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest.


Nonparametric Estimation of Variance Function for Functional Data

Abstract: This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulaton studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses.


Stable isomorphism of dual operator spaces

Abstract: We prove that two dual operator spaces $X$ and $Y$ are stably isomorphic if and only if there exist completely isometric normal representations $\phi$ and $\psi$ of $X$ and $Y$, respectively, and ternary rings of operators $M_1, M_2$ such that $\phi (X)= [M_2^*\psi (Y)M_1]^{-w^*}$ and $\psi (Y)=[M_2\phi (X)M_1^*].$ We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. We provide examples motivated by CSL algebra theory.


Conformal Killing graphs with prescribed mean curvature

Abstract: We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.


Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory

Abstract: Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient. Additionally, as a byproduct of our discussion we give an example of a complete Kaehlerian non-projective algebraic surface, which may be of independent interest.


Examples of scalar-flat hypersurfaces in $\mathbb{R}^{n+1}$

Abstract: Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.


Higher order Schwarzian derivatives in interval dynamics

Abstract: We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up to any given order.


A lower bound on Seshadri constants of hyperplane bundles on threefolds

Abstract: We give the lower bound on Seshadri constants for the case of very ample line bundles on threefolds. We consider the situation when the Seshadri constant is strictly less than 2 and give a version of Bauer's theorem \cite[Theorem 2.1]{B1} for singular surfaces so we can prove the same result for smooth threefolds.


Generalization of the "Stark unit" for abelian L-functions with multiple zeros

Abstract: For an abelian extension of number fields we show that the Stark conjecture for all Artin L-functions with zero of order r is equivalent to existence of a special element in the rational span of the r-th exterior power of the Galois module of units of the bigger field.


Sparse graphs: metrics and random models

Abstract: Recently, Bollob\'as, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with $n$ vertices and $\Theta(n)$ edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function $\ka:[0,1]^2 \to [0,\infty)$. To understand these models, we should like to know when different kernels $\ka$ give rise to `similar' graphs, and, given a real-world network, how `similar' is it to a typical graph $G(n,\ka)$ derived from a given kernel $\ka$. The analogous questions for dense graphs, with $\Theta(n^2)$ edges, are answered by recent results of Borgs, Chayes, Lov\'asz, S\'os, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in $[0,1]$. Possible generalizations of these results to graphs with $o(n^2)$ but $\omega(n)$ edges are discussed in a companion paper [arXiv:0708.1919]; here we focus only on graphs with $\Theta(n)$ edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models, and vice versa.


On the complexity of Putinar's Positivstellensatz

Abstract: We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints.