Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

Abstract: We consider a complete nonsingular variety $X$ over $\bC$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes \Omega_X^j(\log D)\big) = 0$ for any nef line bundle $L$ on $X$ and all $i < j - c$, where $c$ is an explicit function of $(X,D,L)$. This implies e.g. the vanishing of $H^i(X, L \otimes \Omega_X^j)$ for $L$ ample and $i > j$, and gives back a vanishing theorem of Broer when $X$ is a flag variety.


Extreme lattices and vexillar designs

Abstract: We define a notion of vexillar design for the flag variety in the spirit of the spherical designs introduced by Delsarte, Goethals and Seidel. For a finite subgroup of the orthogonal group, we explain how conditions on the group have the orbits of any flag under the group action be a design and point out why the minima of a lattice in the sense of the general Hermite constant forming a 4-design implies being extreme. The reasoning proves useful to show the extremality of many new expected examples ($E_8$, $\La_{24}$, Barnes-Wall lattices, Thompson-Smith lattice for instance) that were out of reach until now.


Bieri-Neumann-Strebel-Renz invariants and homology jumping loci

Abstract: We investigate the relationship between the geometric Bieri-Neumann-Strebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as K\"ahler and quasi-K\"ahler manifolds.


Equivariant Lie-Rinehart cohomology

Abstract: In this paper, we study Lie-Rinehart cohomology for quotients of singularities by finite groups, and interpret these cohomology groups in terms of integrable connection on modules.


Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

Abstract: A Banach space operator $T\in B({\cal X})$ is polaroid if points $\lambda\in\iso\sigma\sigma(T)$ are poles of the resolvent of $T$. Let $\sigma_a(T)$, $\sigma_w(T)$, $\sigma_{aw}(T)$, $\sigma_{SF_+}(T)$ and $\sigma_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $\pi_0(M_C)=\{\lambda\in\iso\sigma(M_C) 0<\dim(M_C-\lambda)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$ and $B$ has SVEP at points $\mu\in\sigma_w(M_0)\setminus\sigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $\lambda\in\sigma_w(M_0)\setminus\sigma_{SF_+}(A)$ and $B^*$ has SVEP at points $\mu\in\sigma_w(M_0)\setminus\sigma_{SF_-}(B)$, then $\sigma(M_C)\setminus\sigma_w(M_C)=\pi_0(M_C)$. Here the hypothesis that $\lambda\in\pi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $\lambda\in\pi_0(A)$ are poles of the resolvent of $A$. For an operator $T\in B(\X)$, let $\pi_0^a(T)=\{\lambda:\lambda\in\iso\sigma_a(T), 0<\dim(T-\lambda)^{-1}(0)<\infty\}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $\pi_0^a(\M)$ and $B$ is polaroid on $\pi_0^a(B)$, then $\sigma_a(\M)\setminus\sigma_{aw}(\M)=\pi_0^a(\M)$.


Heat-kernel estimates for random walk among random conductances with heavy tail

Abstract: We study models of discrete-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$, with polynomial tail near 0 with exponent $\gamma>0$. We first prove for all $d\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $\gamma$.


Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields

Abstract: We construct a 'triangulated analogue' of coniveau spectral sequences: the motif of a variety over a countable field is 'decomposed' (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodsky's motives. To this end we construct a 'Gersten' weight structure for a certain triangulated category of 'comotives': the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology of its generic fibre; cohomology of function fields contain twisted cohomology of their residue fields (for all geometric valuations). We also develop further the general theory of weight structures for triangulated categories (independently from the 'motivic' part of the paper). Besides, we develop a certain theory of 'nice' pairings of triangulated categories; this subject seems to be new.


Brody curves omitting hyperplanes

Abstract: A Brody curve, a.k.a. normal curve, is a holomorphic map from the complex line to the complex projective space of dimension n, such that the family of its translations is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n=1.


Equipartition of energy for the wave equation associated to the Dunkl-Cherednik Laplacian

Abstract: This paper is concerned with energy properties of the wave equation associated to the Dunkl-Cherednik Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case.


Leaf superposition property for integer rectifiable currents

Abstract: We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.


Singular limits for the Riemann problem. General diffusion, relaxation, and boundary conditions

Abstract: We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations.


Stability of quantized time-delay nonlinear systems: A Lyapunov-Krasowskii-functional approach

Abstract: Lyapunov-Krasowskii functionals are used to design quantized control laws for nonlinear continuous-time systems in the presence of constant delays in the input. The quantized control law is implemented via hysteresis to prevent chattering. Under appropriate conditions, our analysis applies to stabilizable nonlinear systems for any value of the quantization density. The resulting quantized feedback is parametrized with respect to the quantization density. Moreover, the maximal allowable delay tolerated by the system is characterized as a function of the quantization density.


Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems

Abstract: We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks -- which are recognized as a source for non-uniqueness and instability. Our Haar-Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It extends the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controling the distance between two solutions.


On the NonKoszulity of (2p+1)-ary partially associative Operads

Abstract: We present a study of quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n=3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework. We prove that the operad associated to 3-ary partially associative algebra is not Koszul. Recall that, if n is even, the operad of n-ary partially associative algebras is Koszul.


Finite energy solutions to the isentropic Euler equations with geometric effects

Abstract: Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin r=0 is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higher-integrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values.


Curvature tensor under the complete non-compact Ricci Flow

Abstract: We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^n\times [0,T)$, the curvature tensor stays uniformly bounded on $M^n\times [0,T)$. Some other results are also presented.


Synchronization of discrete-time dynamical networks with time-varying couplings

Abstract: We study the local complete synchronization of discrete-time dynamical networks with time-varying couplings. Our conditions for the temporal variation of the couplings are rather general and include both variations in the network structure and in the reaction dynamics; the reactions could, for example, be driven by a random dynamical system. A basic tool is the concept of Hajnal diameter which we extend to infinite Jacobian matrix sequences. The Hajnal diameter can be used to verify synchronization and we show that it is equivalent to other quantities which have been extended to time-varying cases, such as the projection radius, projection Lyapunov exponents, and transverse Lyapunov exponents. Furthermore, these results are used to investigate the synchronization problem in coupled map networks with time-varying topologies and possibly directed and weighted edges. In this case, the Hajnal diameter of the infinite coupling matrices can be used to measure the synchronizability of the network process. As we show, the network is capable of synchronizing some chaotic map if and only if there exists an integer T>0 such that for any time interval of length T, there exists a vertex which can access other vertices by directed paths in that time interval.


Sequential multiple hypothesis testing in presence of control variables

Abstract: Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ at this stage can be used. The distribution of $Y$ depends on some unknown parameter $\theta$, and we consider the problem of testing multiple hypotheses $H_1: \theta=\theta_1$, $H_2: \theta=\theta_2, ...$, $H_k: \theta=\theta_k$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, another value $X_2$ for the control variable is chosen, and $Y_2$ as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses $H_1,...$, $H_k$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations $Y_1, Y_2,..., Y_n$ are independent, given controls $X_1,X_2,..., X_n$, $n=1,2,...$.


Characteristic foliation on a hypersurface of general type in a projective symplectic manifold

Abstract: Given a projective symplectic manifold $M$ and a non-singular hypersurface $X \subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an \'etale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families of curves.


Weak isomorphisms between Bernoulli shifts

Abstract: In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.


Commuting birth-and-death processes

Abstract: We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which the $m$ matrices that record the transition probabilities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute $n$ step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component being the closure of all processes whose nearest neighbor transition probabilities are positive. We exhibit an explicit monomial parametrization for this main component, and we explore the boundary components using primary decomposition.


$k$-distant crossings and nestings of matchings and partitions

Abstract: We define and consider k-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of k-distant crossings and nestings is symmetric. We also study the numbers of k-distant noncrossing matchings and partitions for small k, which are counted by well-known sequences, as well as the orthogonal polynomials related to k-distant noncrossing matchings and partitions. We extend Chen et al.'s r-crossings and enhanced r-crossings.


Square function and heat flow estimates on domains

Abstract: The first purpose of this note is to provide a proof of the usual square function estimate on Lp (?). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker version of the square function estimate, which is enough in most instances involving dispersive PDEs. Moreover, we obtain, by a relatively simple integration by parts, several useful Lp (?; H) bounds for the derivatives of the heat ?ow with values in a given Hilbert space H.


The Cauchy-Crofton formula and the Whitney arc property for definable sets

Abstract: We use the Cauchy-Crofton formula to show that every definable cell (bounded by a ball with rational radius) in an O-minimal expansion of a field extension of the real numbers satisfies the Whitney arc property.


Props in model categories and homotopy invariance of structures

Abstract: We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object X homotopy equivalent to an algebra A over a cofibrant prop P inherits a P-algebra structure so that X defines a model of A in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of A-infinity algebras.


On a connectedness theorem of Debarre

Abstract: Under a slightly stronger hypothesis, one improves a connectedness result of Debarre [D] for a product of two projective spaces in terms of the extension problem of formal-rational functions (see Theorems 1.3 and 1.4 of the introduction)


Some remarks about the second Leibniz cohomology group for Lie algebras

Abstract: We compare by a very elementary approach the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones. Examples are given of coupled cocycles. Some properties are deduced as to Leibniz deformations. We also consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional semisimple Lie algebras. Finally, a list of Kac-Moody types for indecomposable nilpotent Lie algebras of dimension $\leqslant 7$ is given.


Geometric Realizations of Hermitian curvature models

Abstract: We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehler condition at the point in question.


On $T_{2n-1}^\perp$ spaces

Abstract: This paper is devoted to the inequalities for mean values of functions from $T_{2n-1}^\perp$. The simple proof of the classical Jackson inequality in the case of the second modulus of continuity may be considered as the consequence of our estimates. The problems of the sharp constants in classical Stechkin's inequality are also discussed.


Nonparametric estimation of a trend based upon sampled continuous processes

Abstract: Let X be a second order random process indexed by a compact interval [0,T]. Assume that n independent realizations of X are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function mu = EX in the space C([0,T]) as n, p go to infinity and, using Gaussian process theory, we derive approximate simultaneous confidence bands for mu.


Mean field frozen percolation

Abstract: We define a modification of the Erdos-Renyi random graph process which can be regarded as the mean field frozen percolation process. We describe the behavior of the process using differential equations and investigate their solutions in order to show the self-organized critical and extremum properties of the critical frozen percolation model. We prove two limit theorems about the distribution of the size of the component of a typical frozen vertex.


On jet bundles and generalized Verma modules

Abstract: The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods and generalized Verma modules. We calculate the $P$-module of the dual jet bundle $J^l(L)^*$ and prove it equals the $l$'th piece of the canonical filtration for $H^0(X,L)^*$. We use the results obtained to prove the discriminant of any linear system on any grassmannian is irreducible.


Cone structure of $L^2$-Wasserstein spaces

Abstract: The purpose of this paper is to understand the geometric structure of the $L^2$-Wasserstein space $\pp$ over the Euclidean space.For this sake, we focus on its cone structure.One of our main results is that the $L^2$-Wasserstein space over a Polish space has a cone structure if and only if so does the underlying space.In particular, $\pp$ turns out to have a cone structure.It is also shown that $\pp$ splits $\R^d$ isometrically but not $\R^{d+1}$.


Note on the Fenchel transform in the Heisenberg group

Abstract: Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.


Volumes, Traces and Zeta Functions

Abstract: Let $Q(x)$ be a quadratic form over $\mathbb{R}^n$. The Epstein zeta function associated to $Q(x)$ is a well known function in number theory. We generalize the construction of the Epstein zeta function to a class of function $\phi(x)$ defined in $\mathbb{R}^n$ that we call $A-$homogeneous, where $A$ is a real aquare matrix of order $n$ having each eigenvalue in the left hal space $\Re\lambda>0$. Such a class includes all the homogeneous polynomials (positive outside the origin) and all the norms on $\mathbb{R}^n$ which are smooth outside the origin. As in the classical (i.e. quadratic) case we prove that such zeta functions are obtained from the Mellin transforms of theta function of Jacobi type associated to the $A-$homogeneous function $\phi(x)$. We prove that the zeta function associated to a $A-$homogeneous function $\phi(x)$ which is positive and smooth outside the origin is an entire meromorphic function having a unique simple pole at $s=\alpha$ the trace of the matrix $A$ with residue given by the product of the trace $\alpha$ and the Lebesgue volume of the unit ball associated to $\phi(x)$, that is the volume of the set $x\in\R^n$ satisfying $\phi(x)<1$. We also prove that the theta funtion associated to $\phi(x)$ has an asymptotic expansion near the origin. We find that the coefficients of such expansion depend on the values that the zeta function associated to $\phi(x)$ assumes at the negative integers.


Hardy inequalities for weighted Dirac operatos

Abstract: An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight $r^{-b}$ for functions in $\R^n$. The exact Hardy constant $c_b=c_b(n)$ is found and generalized minimizers are given. The constant $c_b$ vanishes on a countable set of $b$, which extends the known case $n=2$, $b=0$ which corresponds to the trivial Hardy inequality in $\R^2$. Analogous inequalities are proved in the case $c_b=0$ under constraints and, with error terms, for a bounded domain.


Symplectic implosion and non-reductive quotients

Abstract: The symplectic implosion construction of Guillemin, Jeffrey and Sjamaar associates to a Hamiltonian action of a compact group K on a symplectic manifold X its 'imploded cross section'. When X is a complex projective variety and K acts linearly on X, this construction is closely related to geometric invariant theory (GIT) for the action on X of a maximal unipotent subgroup U of the complexification G of K. The aim of this paper is to generalise symplectic implosion to give a symplectic construction for GIT-like quotients by unipotent radicals U of arbitrary parabolic subgroups P of the complex reductive group G acting linearly on the projective variety X.


Partial symmetry, reflection monoids and Coxeter groups

Abstract: This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.


Estimates for invariant metrics on $\Bbb C$-convex domains

Abstract: Geometric lower and upper estimates are obtained for invariant metrics on $\Bbb C$-convex domains containing no complex lines.


Tutte polynomial and G-parking functions

Abstract: Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we give the definition of the bridge vertex of a $G$-parking function and obtain an expression of the Tutte polynomial $T_G(x,y)$ of $G$ in terms of $G$-parking functions. We find the Tutte polynomial enumerates the $G$-parking function by the number of the bridge vertices.


Sparse recovery under matrix uncertainty

Abstract: We consider the model {eqnarray*}y=X\theta^*+\xi, Z=X+\Xi,{eqnarray*} where the random vector $y\in\mathbb{R}^n$ and the random $n\times p$ matrix $Z$ are observed, the $n\times p$ matrix $X$ is unknown, $\Xi$ is an $n\times p$ random noise matrix, $\xi\in\mathbb{R}^n$ is a noise independent of $\Xi$, and $\theta^*$ is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that $X$ is observed with additive error. For dimensions $p$ that can be much larger than the sample size $n$, we consider the estimation of sparse vectors $\theta^*$. Under matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of nonzero components of $\theta^*$), even if the noise level is very small. We suggest new estimators called matrix uncertainty selectors (or, shortly, the MU-selectors) which are close to $\theta^*$ in different norms and in the prediction risk if the restricted eigenvalue assumption on $X$ is satisfied. We also show that under somewhat stronger assumptions, these estimators recover correctly the sparsity pattern.


Refinements of Lattice paths with flaws

Abstract: The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\sum\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ valleys (resp. double descents). Some similar results are derived.


A Unification of Two Refinements of Euler's Partition Theorem

Abstract: We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.


An empirical central limit theorem in L^1 for stationary sequences

Abstract: In this paper, we derive asymptotic results for L^1-Wasserstein distance between the distribution function and the corresponding empirical distribution function of a stationary sequence. Next, we give some applications to dynamical systems and causal linear processes. To prove our main result, we give a Central Limit Theorem for ergodic stationary sequences of random variables with values in L^1. The conditions obtained are expressed in terms of projective-type conditions. The main tools are martingale approximations.


On a congruence only holding for primes II

Abstract: We present a primality criterium based on congruences for cyclotomic polynomials, and point out a way to generalize our result in order to obtain a family of similar criteria. No practical use is aimed however.


Formulas for the Connes-Moscovici Hopf algebra

Abstract: We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\mathcal{H}_{\tmop{CM}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\mathcal{H}^1_{\tmop{CM}}$ containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to the Fa\`a di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather than diffeomorphisms. On the other hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to a sub-Hopf algebra of the classical shuffle Hopf algebra which appears naturally in resummation theory, in the framework of formal and analytic conjugacy of vector fields. Using the very simple structure of the shuffle Hopf algebra, we derive once again explicit formulas for the coproduct and the antipode in $\mathcal{H}^1_{\tmop{CM}}$.


Remarques sur une conjecture de Lang

Abstract: The aim of this paper is to study a conjecture predicting a lower bound on the canonical height on abelian varieties, formulated by S. Lang and generalized by J. H. Silverman. We give here an asymptotic result on the height of Heegner points on the modular jacobian $J_{0}(N)$, and we derive non-trivial remarks about the conjecture.


Minoration de la hauteur de Neron-Tate sur les surfaces abeliennes

Abstract: This paper contains results concerning a conjecture made by Lang and Silverman predicting a lower bound for the canonical height on abelian varieties of dimension 2 over number fields. The method used here is a local height decomposition. We derive as corollaries uniform bounds on the number of torsion points on families of abelian surfaces and on the number of rational points on families of genus 2 curves.


Differentiability of the arithmetic volume function

Abstract: We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants.


Almost prime values of the order of elliptic curves over finite fields

Abstract: Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the reduced curve over $\F_p$. We prove in this paper that, under the GRH, there are at least $2.778 C_E^{\rm twin} x / (\log{x})^2$ primes $p$ such that $|E(\F_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng and Murty & Miri. This is also the first result where the dependence on the conjectural constant $C_E^{\rm twin}$ appearing in the twin prime conjecture for elliptic curves (also known as Koblitz's conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one used by previous authors. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant appearing in the upper bound for the number of primes $p$ such that $|E(\F_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.


Universality in Complex Wishart ensembles: The 1 cut case

Abstract: We studied universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues and the number of each of these eigenvalue goes to infinity in the asymptotic limit. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. In our previous work the case when the support consists of 2 intervals was studied. This paper complements our previous analysis and studied the case when the support consists of a single interval. By using Riemann-Hilbert analysis, we have shown that under proper rescaling of the eigenvalues, the limiting correlation kernel is given by the sine kernel and the Airy kernel in the bulk and the edge of the spectrum respectively. As a consequence, the behavior of the largest eigenvalue in this model is described by the Tracy-Widom distribution.


Vibration Spectra of the $m$-Tree Fractal

Abstract: We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases.


Intriguing sets of partial quadrangles

Abstract: The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in particular, the well-studied objects known as \textit{generalised quadrangles} are each partial quadrangles. An \textit{intriguing set} of a generalised quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalised quadrangles by Bamberg, Law and Penttila to partial quadrangles, which surprisingly gives insight into the structure of hemisystems and other intriguing sets of generalised quadrangles.


Dirac operators for coadjoint orbits of compact Lie groups

Abstract: The coadjoint orbits of compact Lie groups carry many K\"ahler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi-Civita connection of the Riemannian metric, and we use the complex structure to give a fairly explicit construction of a canonical Dirac operator for the Riemannian metric, in a way that avoids use of the Spin^c groups. Substantial parts of our results apply to compact almost-Hermitian homogeneous spaces, and to other connections besides the Levi-Civita connection. For these other connections we give a criterion that is both necessary and sufficient for their Dirac operator to be formally self-adjoint. We hope to use the detailed results given here to clarify statements in the literature of high-eneregy physics concerning "Dirac operators" for matrix algebras that converge to coadjoint orbits. To facilitate this we employ here only global methods -- we never use local coordinate charts, and we use the cross-section modules of vector bundles.


Field sensitivity to L^p variations of a scatterer

Abstract: For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational Frechet derivative of the scattered field measured in the H^1 (root-mean-square-gradient) norm as a function of the material coefficients measured in an L^p (p-power integral) norm, with 2<p<infinity, as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also establish the variational derivative of the transmitted energy with respect to the material coefficients in L^p.


Super-sequences in the arc component of a compact connected group

Abstract: Let G be an abelian topological group. The symbol \hat{G} denotes the group of all continuous characters \chi : G --> T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that \chi(E) \subseteq \phi([-1/4,1/4]) holds only for the trivial character \chi \in \hat{G}, where \phi : R --> T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Aussenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : \hat{G} --> \hat{G}_a defined by r(\chi) = \chi\restriction_{G_a} for \chi \in \hat{G}, is a topological isomorphism. We also show that an infinite compact group G is connected if and only if its arc component G_a contains a super-sequence S converging to the identity e that generates a dense subgroup of G (equivalently, S \setminus {e} is an infinite suitable set for G in the sense of Hofmann and Morris).


Parametric Bing and Krasinkiewicz maps: revisited

Abstract: Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitation topology contains a dense $G_\delta$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\in Y$, are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension.


On the non-quadraticity of values of the q-exponential function and related q-series

Abstract: We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;\lambda)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-\lambda)}, \qquad |q|>1, \quad \lambda\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($\lambda=0$) and the $q$-exponential function ($\lambda=1$). In particular, we prove the non-quadraticity of the numbers $F_q(\alpha;\lambda)$ for integral $q$, rational $\lambda$ and $\alpha\notin-\lambda q^{\mathbb Z_{>0}}$, $\alpha\ne0$.


Abstract: In this paper, we investigate Jordan *-homomorphisms on $C^*$-algebras associated with the following functional inequality $\|f(\frac{b-a}{3})+f(\frac{a-3c}{3})+f(\frac{3a+3c-b}{3})\| \leq \|f(a)\|.$ We moreover prove the superstability and the generalized Hyers-Ulam stability of Jordan *-homomorphisms on $C^*$-algebras associated with the following functional equation $$f(\frac{b-a}{3})+f(\frac{a-3c}{3})+f(\frac{3a+3c-b}{3})=f(a).$$


Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

Abstract: In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\eqno {2 cm}$$for fixed integers $k$ with $k\neq0,\pm1$ in the quasi-Banach spaces.


Stability of homomorphisms and derivations in $C^*$-ternary algebras

Abstract: In this paper, we investigate homomorphisms between $C^*$-ternary algebras and derivations on $C^*$-ternary algebras, associated with the following functional equation $$f(\frac{x_2-x_1}{3})+f(\frac{x_1-3x_3}{3})+f(\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$ Moreover, we prove the generalized Hyers-Ulam -Rassias stability of homomorphisms in $C^*$-ternary algebras and of derivations on $C^*$-ternary algebras.


n-Jordan homomorphisms

Abstract: Let $n\in \Bbb N,$ and let $A,B$ be two rings. An additive map $h: A\to B$ is called n-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a \in {A}$. Every Jordan homomorphism is an n-Jordan homomorphism, for all $n\geq 2,$ but the converse is false, in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Indeed some results related to continuity are given as well.


From Schoenberg to Pick-Nevanlinna: Toward a complete picture of the variogram class

Abstract: We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-L\'{e}vy type and demonstrate some important properties of rotationally invariant variograms.


On the chromatic number of random d-regular graphs

Abstract: In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal.


Stability of a mixed type quadratic, cubic and quartic functional equation

Abstract: In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation $$3(f(x+2y)+f(x-2y))=12(f(x+y)+f(x-y))+4f(3y)-18f(2y)+36f(y)-18f(x).$$


Entropy vs volume for pseudo-Anosov maps

Abstract: We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichm\"uller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments.


Zero entropy invariant measures for some skew product diffeomorphisms

Abstract: In this paper we study some skew product diffeomorphisms with nonuniformly hyperbolic structure along fibers. We show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers.


Generalizations of Chung-Feller Theorem

Abstract: The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of three classes of lattice paths and give the generalizations of the above two theorems. We prove the Chung-Feller theorems of Dyck type for these three classes of lattice paths and the Chung-Feller theorems of Motzkin type for two of these three classes. From the obtained results, we find an interesting fact that many lattice paths have the Chung-Feller properties of both Dyck type and Motzkin type.


The L_\infty-deformation complex of diagrams of algebras

Abstract: The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_\infty-algebra which induces a graded Lie bracket on cohomology. As an example, the L_\infty-algebra structure on the deformation complex of an associative algebra morphism g, with the underlying cochain complex isomorphic to the Gerstenhaber-Schack complex of g, is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse morphisms of vector spaces. Its L_\infty-deformation complex has a nontrivial constant term. Explicit formulas for the L_\infty-operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fully-fledged L_\infty-algebra with nontrivial higher operations.


Hazard Estimation under Generalized Censoring

Abstract: This paper focuses on the problem of the estimation of the cumulative hazard function of a distribution on a general complete separable metric space when the data points are subject to censoring by an arbitrary adapted random set. A problem involving observability of the estimator proposed in [8] and [9] is resolved and a functional central limit theorem is proven for the revised estimator. Several examples and applications are discussed, and the validity of bootstrap methods is established in each case.


Recent Developments in Nonregular Fractional Factorial Designs

Abstract: Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.


Transformations of locally conformally K\"ahler manifolds

Abstract: We consider several transformation groups of a locally conformally K\"ahler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperk\"ahler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally K\"ahler manifold which is neither Weyl-reducible nor locally conformally hyperk\"ahler are holomorphic and conformal


Asymptotics for the size of the largest component scaled to "log n" in inhomogeneous random graphs

Abstract: We study the inhomogeneous random graphs in the subcritical case. We derive an exact formula for the size of the largest connected component scaled to $\log n$ where $n$ is the size of the graph. This generalizes the recent result for the "rank 1 case". Here we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, plays the crucial role in the subcritical case as well. But now these are the negative solutions which come into play.


On the classification of degree 1 elliptic threefolds with constant $j$-invariant

Abstract: We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in $\Ps(2,3,1,1,1)$ that admit no variation of Hodge structure.


The number of Hecke eigenvalues of same signs

Abstract: We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals.


Mathematical model for resistance and optimal strategy

Abstract: We propose a mathematical model for one pattern of charts studied in technical analysis: in a phase of consolidation, the price of a risky asset goes down $\xi$ times after hitting a resistance level. We construct a mathematical strategy and we calculate the expectation of the wealth for the logaritmic utility function. Via simulations, we compare the strategy with the standard one.


Strong Gaussian approximations of product-limit and Quantile Processes for Strong mixing and censored data

Abstract: In this paper, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate $O((\log n)^{-\lambda})$ for some $\lambda>0,$. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.


Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions

Abstract: We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we further describe.


Towards finite generation of the canonical ring without the MMP

Abstract: This paper is the first of two steps in a project to prove finite generation of the log canonical ring without Mori theory.


Semiparametric regression estimation using noisy nonlinear non invertible functions of the observations

Abstract: We investigate a semiparametric regression model where one gets noisy non linear non invertible functions of the observations. We focus on the application to bearings-only tracking. We first investigate the least squares estimator and prove its consistency and asymptotic normality under mild assumptions. We study the semiparametric likelihood process and prove local asymptotic normality of the model. This allows to define the efficient Fisher information as a lower bound for the asymptotic variance of regular estimators, and to prove that the parametric likelihood estimator is regular and asymptotically efficient. Simulations are presented to illustrate our results.


Deformation of symmetric functions and the rational Steenrod algebra

Abstract: In 1999, Reg Wood conjectured that the quotient of Q[x_1,...,x_n] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group S_n. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the quotient in this case. We introduce a non-commutative q-deformation of the ring of symmetric functions, which specializes at q=1 to the rational Steenrod algebra. We use this formalism to obtain some partial results. Finally, we describe several conjectures based on an extensive computer exploration. In particular, we extend Reg Wood's conjecture to q formal and to any q complex not of the form -a/b, with a in {1,...,n} and b a positive natural number.


Existence and uniqueness of constant mean curvature spheres in Sol_3

Abstract: We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere S_H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real numbers H such that there exists a solution of the isoperimetric problem with mean curvature H.


A geometric approach to Conn's linearization theorem

Abstract: We give a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple.


Maximal inequalities for dual Sobolev spaces $W^{-1,p}$ and applications to interpolation

Abstract: We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev spaces.


Invariants alg\'ebriques de graphes et reconstruction

Abstract: We report on results about a study of algebraic graph invariants, based on computer exploration, and motivated by graph-isomorphism and reconstruction problems.


Algebraic invariants of graphs; a study based on computer exploration

Abstract: We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction conjecture. There is a huge body of literature on invariant theory which provides both general results and algorithms. However, there is a combinatorial explosion in the computations involved and, to our knowledge, the ring I_n has only been completely described for n<=4. This led us to study the ring I_n in its own right. We used intensive computer exploration for small n, and developed PerMuVAR, a library for MuPAD, for computing in invariant rings of permutation groups. We present general properties of the ring I_n, as well as results obtained by computer exploration for small n, including the construction of a medium sized generating set for I_5. We address several conjectures suggested by those results (low degree system of parameters, unimodality), for I_n as well as for more general invariant rings. We also show that some particular sets are not generating, disproving a conjecture of Pouzet related to reconstruction, as well as a lemma of Grigoriev on the invariant ring over digraphs. We finally provide a very simple minimal generating set of the field of invariants.


Normal approximation for coverage models over binomial point processes

Abstract: We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over a binomial point process of $n$ points in a toroidal spatial region of volume $n$. The proof is based on Stein's method via size-biased couplings.


Hyperbolic (1,2)-knots in S^3 with crosscap number two and tunnel number one

Abstract: A knot in S^3 is said to have crosscap number two if it bounds a once-punctured Klein bottle but not a Moebius band. In this paper we give a method of constructing crosscap number two hyperbolic (1,2)-knots with tunnel number one which are neither 2-bridge nor (1,1)-knots. An explicit infinite family of such knots is discussed in detail.


Binomial coefficients and the ring of p-adic integers

Abstract: Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.


Arithmetical rank of toric ideals associated to graphs

Abstract: Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$.


Patching and local-global principles for homogeneous spaces over function fields of p-adic curves

Abstract: This is the final version, to appear in Commentarii Mathematici Helvetici.


The structure of a local embedding and applications to Chern classes of weighted blow-ups

Abstract: For a proper local embedding between two Deligne--Mumford stacks Y and X, we find, under certain mild conditions, a new (possibly non-separated) Deligne--Mumford stack X', with an etale, surjective and universally closed map to the target X, and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to Y. Moreover, a natural set of weights on the substacks of X' allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of X' and X. We apply the construction above to the computation of the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We describe the stack X' in the case when X is the moduli space of stable maps with local embeddings at the boundary. We apply the methods above to find the Chern classes of the stable map spaces.


Parameter estimation for rough differential equations

Abstract: We construct the "expected signature matching" estimator for differential equations driven by rough paths and we prove its consistency and asymptotic normality. We use it to estimate parameters of a diffusion and a fractional diffusions, that is, a differential equation driven by fractional Brownian motion.


Orders of Finite Reductive Monoids

Abstract: We show four formulas for calculating the orders of finite reductive monoids with zero. As applications, these formulas are then used to calculate the orders of finite reductive monoids induced from the $F_q$-split $\J$-irreducible monoids $\overline {K^*\rho(G_0)}$ where $G_0$ is a simple algebraic group over the algebraic closure of $F_q$, and $\rho: G_0\to GL(V)$ is the irreducible representation associated with any dominant weight. Finally, we give an explicit formula for the orders of finite symplectic monoids associated with the last fundamental dominant weight of type $C_l$; the connections to $H$-polynomials and Betti numbers are shown.


Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{\frak {sl}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis

Abstract: We further define two-parameter quantum affine algebra $U_{r,s}(\widehat{\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of importance for the quantum {\it affine} cases is that we can work out the compatible two-parameter version of the Drinfel'd realization as a quantum affinization of $U_{r,s}(\frak{sl}_n)$ and establish the Drinfel'd isomorphism Theorem in the two-parameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum {\it affine} Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel'd generators).


Accurate numerical linear algebra with Bernstein-Vandermonde matrices

Abstract: The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein-Vandermonde matrix is considered. Bernstein-Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to $n$ the Bernstein basis, a widely used basis in Computer Aided Geometric Design, instead of the monomial basis. Our approach is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix (or its inverse) by means of Neville elimination. The explicit expressions obtained for the determinants involved in the process makes the algorithm both fast and accurate.


Accurate computations with Said-Ball-Vandermonde matrices

Abstract: A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.


Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond

Abstract: We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in $W^{1,2}$, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.


Constant Mean Curvature Hypersurfaces Condensing to Geodesic Segments and Rays in Riemannian Manifolds

Abstract: We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a `friction term' which permits the usual analytic gluing construction to be carried out.


Choosing a penalty for model selection in heteroscedastic regression

Abstract: We consider the problem of choosing between several models in least-squares regression with heteroscedastic data. We prove that any penalization procedure is suboptimal when the penalty is a function of the dimension of the model, at least for some typical heteroscedastic model selection problems. In particular, Mallows' Cp is suboptimal in this framework. On the contrary, optimal model selection is possible with data-driven penalties such as resampling or $V$-fold penalties. Therefore, it is worth estimating the shape of the penalty from data, even at the price of a higher computational cost. Simulation experiments illustrate the existence of a trade-off between statistical accuracy and computational complexity. As a conclusion, we sketch some rules for choosing a penalty in least-squares regression, depending on what is known about possible variations of the noise-level.


Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface and related surfaces

Abstract: We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.


Non-colliding Jacobi processes as limits of Markov chains on Gelfand-Tsetlin graph

Abstract: We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite-dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on Gelfand-Tsetlin graph. We compute finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with Doob h-transform of a family of independent diffusions. Space-time correlation functions of the limit process have a determinantal form.


Operator Algebras with Unique Preduals

Abstract: We show that every free semigroup algebras has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak*-closed unital operator operator algebra containing a weak* dense subalgebra of compact operators has a unique Banach space predual.


On the Geometry of the Second Fundamental Form of Translation Surfaces in E3

Abstract: In this paper we study the second fundamental form of translation surfaces in E3. We give a non-existence result for polynomial translation surfaces in E3 with vanishing second Gaussian curvature KII. We classify those translation surfaces for which KII and H are proportional. Finally we obtain that there are no II-minimal translation surfaces in the Euclidean 3-space.


An L^1 Ergodic Theorem for Sparse Random Subsequences

Abstract: We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that a certain deterministic condition implies a weak maximal inequality for a sequence of \ell^1 convolution operators.


Donkin-Koppinen filtration for general linear supergroup

Abstract: We consider a generalization of Donkin-Koppinen filtrations for coordinate superalgebras of general linear supergroups. More precisely, if $G=GL(m|n)$ is a general linear supergroup of (super)degree $(m|n)$, then its coordinate superalgebra $K[G]$ is a natural $G\times G$-supermodule. For every finitely generated ideal $\Gamma\subseteq \Lambda\times\Lambda$, the largest subsupermodule $O_{\Gamma}(K[G])$ of $K[G]$, which has all composition factors of the form $L(\lambda)\otimes L(\mu)$ where $(\lambda, \mu)\in\Gamma$, has a decreasing filtration $O_{\Gamma}(K[G])=V_0\supseteq V_1\supseteq...$ such that $\bigcap_{t\geq 0}V_t=0$ and $V_t/V_{t+1}\simeq V_-(\lambda_t)^*\otimes H_-^0(\lambda_t)$ for each $t\geq 0$. Here $H_-^0(\lambda)$ is a costandard $G$-supermodule, and $V_-(\lambda)$ is a standard $G$-supermodule, both of highest weight $\lambda\in\Lambda$ (see \cite{z}). We deduce the existence of such a filtration from more general facts about standard and costandard filtrations in certain highest weight categories which will be proved in Section 4. Until now, analogous results were known only for highest weight categories with finite sets of weights. We believe that the reader will find the results of Section 4 interesting on its own. Finally, we apply our main result to describe invariants of (co)adjoint action of $G$.


Microlocal analysis and evolution equations

Abstract: Lecture notes from 2008 CMI/ETH Summer School on Evolution Equations. These notes are an informal introduction to the applications of microlocal methods in the study of linear evolution equations and spectral theory. Calculi of pseudodifferential operators and Fourier integral operators are discussed and axiomatized, but not constructed: the focus is on how to apply these tools.


Estimating a monotone trend

Abstract: Motivated by global warming issues, we consider a time se- ries that consists of a nondecreasing trend observed with station- ary fluctuations, nonparametric estimation of the trend under monotonicity assumption is considered. The rescaled isotonic es- timators at an interior point are shown to converge to Chernoff's distribution under minimal conditions on the stationary errors. Since the isotonic estimators suffer from the spiking problem at the end point, two modifications are proposed. The estima- tion errors for both estimators of the boundary point are shown to have interesting limiting distributions. Approximation accu- racies are assessed through simulations. One highlight of our treatment is the proof of the weak convergence results which involve several recent techniques developed in the study of con- ditional central limit questions. These weak convergences can be shown to hold conditionally given the starting values.


Ultrahigh dimensional variable selection: beyond the linear model

Abstract: Variable selection in high-dimensional space characterizes many contemporary problems in scientific discovery and decision making. Many frequently-used techniques are based on independence screening; examples include correlation ranking (Fan and Lv, 2008) or feature selection using a two-sample t-test in high-dimensional classification (Tibshirani et al., 2003). Within the context of the linear model, Fan and Lv (2008)showed that this simple correlation ranking possesses a sure independence screening property under certain conditions and that its revision, called iteratively sure independent screening (ISIS), is needed when the features are marginally unrelated but jointly related to the response variable. In this paper, we extend ISIS, without explicit definition of residuals, to a general pseudo-likelihood framework, which includes generalized linear models as a special case. Even in the least-squares setting, the new method improves ISIS by allowing variable deletion in the iterative process. Our technique allows us to select important features in high-dimensional classification where the popularly used two-sample t-method fails. A new technique is introduced to reduce the false discovery rate in the feature screening stage. Several simulated and two real data examples are presented to illustrate the methodology.


On the intersections of solvable Hall subgroups in finite groups

Abstract: In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. The minimal counter example is shown to be an almost simple group of Lie type.


Halfway Up To the Mathematical Infinity: On the Ontological and Epistemic Sustainability of Georg Cantor's Transfinite Design

Abstract: Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on \emph{ad hoc} axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic \emph{sustainability} of this design has been irremediably compromised by the underlying it peremptory, Reductionist mindset of the XIXth century's ideology of science.


Time and Space Varying Copulas

Abstract: In this article we review existing literature on dynamic copulas and then propose an n-copula which varies in time and space. Our approach makes use of stochastic differential equations, and gives rise to a dynamic copula which is able to capture the dependence between multiple Markov diffusion processes. This model is suitable for pricing basket derivatives in finance and may also be applicable to other areas such as bioinformatics and environmental science.


On Hofmann's bilinear estimate

Abstract: Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace's equation on Lipschitz domains, The first one is a certain bilinear estimate for a class of weak solutions and the second is a criterion which allows to identify the domain of the generator of the semi-group yielding such solutions.


Upper bounds on Rubinstein distances on configuration spaces and applications

Abstract: In this paper, we provide upper bounds on several Rubinstein-type distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two well-known gradients, in the sense of Malliavin calculus, which can be defined on this space. Actually, we show that depending on the distance between configurations which is considered, it is one gradient or the other which is the most effective. Some applications to distance estimates between Poisson and other more sophisticated processes are also provided, and an application of our results to tail and isoperimetric estimates completes this work.


The elliptic threefold y^2=x^3+16s^6+16t^6-32(t^3s^3+t^3+s^3)+16

Abstract: We present a method to calculate the rank of $E(\oQ(s,t))$ for the elliptic curve mentioned in the title. This method uses a generalization of a method from Van Geemen and Werner to calculate $h^4(Y)$ for nodal hypersurfaces $Y$.


Relevement de formes modulaires de Siegel

Abstract: Dans cette note, nous montrons que certaines formes modulaires de Siegel de caract\'eristique p et de genre 2 ou 3 se rel\`event en caract\'eristique 0. Ce r\'esultat g\'en\'eralise un th\'eor\`eme classique obtenu par Katz pour les formes de genre 1. Nous utilisons des r\'esultats de Shepherd-Barron et de Hulek et Sankaran, ainsi que des th\'eor\`emes d'annulation de la cohomologie coh\'erente d\^us \`a Deligne, Illusie et Raynaud et \`a Esnault et Viehweg. ----- In this note, we show that cuspidal Siegel modular forms of characteristic p and genus 2 or 3 can be lifted to characteristic 0. This result extends a classical theorem proved by Katz for genus 1 modular forms. We use ampleness results due to Shepherd-Barron, Hulek and Sankaran, and vanishing theorems due to Deligne, Illusie, Raynaud, Esnault and Viehweg.


On the nonexistence of ternary extremal self-dual codes

Abstract: In this note, we give a new nonexistence result of ternary extremal self-dual codes.


On the Siegel-Weil Theorem for Loop Groups (I)

Abstract: We proved a new Siegel-Weil formula for orthogonal and symplectic groups, which will be used later to prove a generalization of Siegel-Weil formula for loop groups.


Non parametric estimation of the structural expectation of a stochastic increasing function

Abstract: This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the small variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So we define the structural expectation of the underlying stochastic function. Then we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved.


Estimation of the distribution of random shifts deformation

Abstract: Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution $\mu$, modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Building on results of Dalalyan et al. (2006), semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided.


Kernel Inverse Regression for spatial random fields

Abstract: In this paper, we propose a dimension reduction model for spatially dependent variables. Namely, we investigate an extension of the \emph{inverse regression} method under strong mixing condition. This method is based on estimation of the matrix of covariance of the expectation of the explanatory given the dependent variable, called the \emph{inverse regression}. Then, we study, under strong mixing condition, the weak and strong consistency of this estimate, using a kernel estimate of the \emph{inverse regression}. We provide the asymptotic behaviour of this estimate. A spatial predictor based on this dimension reduction approach is also proposed. This latter appears as an alternative to the spatial non-parametric predictor.


Plane sextics via dessins d'enfants

Abstract: We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type $\bold E_7$ singular point and compute their fundamental groups. All groups found are finite; one of them is nonabelian.


A Clebsch-Gordan formula for SL_3 and applications to rationality

Abstract: If R, S, T are irreducible SL_3-representations, we give an easy and explicit description of a basis of the space of equivariant maps from R tensor S to T. We apply this method to the rationality problem for invariant function fields. In particular, we prove the rationality of the moduli space of plane curves of degree 34. This uses a criterion which ensures the stable rationality of some quotients of Grassmannians by an SL-action.


A numerical algorithm for zero counting II: Randomization and Condition

Abstract: This paper was witdrawn by the authors.


Symmetric quasi-hereditary envelopes

Abstract: We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary structure with respect to two opposite orders, that they have strong exact Borel and $\Delta$-subalgebras and the corresponding triangular decompositions.


Evolution by mean curvature in sub-Riemannian geometries: A stochastic approach

Abstract: We study the phenomenon of evolution by horizontal mean curvature flow in sub-Riemannian geometries. We use a stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow.


Unityped algebras

Abstract: The paper is essentially a continuation of B.Plotkin, G.Zhitomirski, "Some logical invariants of algebras and logical relations between algebras", St.Peterburg Math. J., {19:5}, (2008) 859 -- 879, whose main notion is that of logic-geometrical equivalence of algebras (LG-equivalence of algebras). This equivalence of algebras is more strict than elementary equivalence. In the paper we introduce the notion of unityped algebras and relate it to LG-equivalence. We show that these notions coincide. The idea of the type is one of the central ideas in M odel Theory. The correspondence introduced in the paper stimulates a bunch of problems which connect universal algebraic geometry and Model Theory. The paper consists of five sections: 1. General view 2. Logical noetherianity 3. Unitypeness and isomorphism 4. Logically perfect algebras 5. Some facts from algebraic logic. We provide a new general view on the subject, arising "on the territory" of universal algebraic geometry, which yield applications of algebraic logic and universal geometry in Model Theory.


K\"ahlerian Twistor Spinors

Abstract: On a K\"ahler spin manifold K\"ahlerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the K\"ahler structure, called K\"ahlerian twistor (Penrose) operator. We study K\"ahlerian twistor spinors and give a complete description of compact K\"ahler manifolds of constant scalar curvature admitting such spinors. As in the Riemannian case, the existence of K\"ahlerian twistor spinors is related to the lower bound of the spectrum of the Dirac operator.


Singular Riemannian Foliations: Exceptional Leaves; Tautness

Abstract: For a singular Riemannian foliation whose leaves are properly embedded, we show in the first part of this article the existence of global tubular neighbourhoods, and we develop a global description of the foliation as stratification by types of leaves. The second part deals with the further restriction to a foliation without horizontal conjugate points, introduced by Lytchak and Thorbergsson, which in the special case of an isometric group action equals the concept of variationally completeness. Therefrom, we deduce a global geometric description -- the focal points of the leaves are exactly the singular points -- as well as a topological one: tautness of the leaves.


On the Global Behavior of Solutions to a Planar System of Difference Equations

Abstract: We establish the relation between local stability of equilibria and slopes of critical curves for a specific class of difference equations. We then use this result to give global behavior results for nonnegative solutions of the system of difference equations \begin{equation*} %\tag{LGIN} \begin{array}{rcl} x_{n+1} & = & \displaystyle \frac{b_1 x_n}{1+x_n+c_1 y_{n}} +h_1 y_{n+1} & = & \displaystyle \frac{b_2 y_n}{1+y_n+c_2 x_{n}} +h_2 \end{array} \quad n=0,1,..., \quad (x_0,y_0) \in [0,\infty)\times [0,\infty) \end{equation*} with positive parameters. In particular, we show that the system has between one and three equilibria, and that the number of equilibria determines global behavior as follows: if there is only one equilibrium, then it is globally asymptotically stable. If there are two equilibria, then one is a local attractor and the other one is nonhyperbolic. If there are three equilibria, then they are linearly ordered in the south-east ordering of the plane, and consist of a local attractor, a saddle point, and another local attractor. Finally, we give sufficient conditions for having a unique equilibrium.


Non-ergodicity of Nose-Hoover dynamics

Abstract: The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a "thermostat" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nose-Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. It is also shown for some multidimensional systems that the averaged dynamics for the limit of infinite thermostat "mass" have many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.


Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees

Abstract: We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion).


On the restricted Verma modules at the critical level

Abstract: We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-H"older multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients.


A syzygetic approach to the smoothability of zero-dimensional schemes

Abstract: We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this question for zero-dimensional schemes of regularity two. This invariant imposes obstructions for smoothability in general, and it completely answers the question of smoothability for certain zero-dimensional schemes of low degree. The tools of this paper also lead to other results about Hilbert schemes of points, including a characterization of nonsmoothable zero-dimensional schemes of minimal degree in every embedding dimension d\geq 4.


Convex PBW-Type Lyndon Bases and Restricted Two-parameter Quantum Groups of Type B

Abstract: We construct convex PBW-type Lyndon bases for two-parameter quantum groups U_{r,s}({so}_{2n+1}) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u_{r,s}({so}_{2n+1}) (r, s are roots of unity) is of Drinfeld double. All of Hopf isomorphisms of u_{r,s}({so}_{2n+1}), as well as u_{r,s}({sl}_n) are determined. Finally, necessary and sufficient conditions for u_{r,s}({so}_{2n+1}) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.


Arakelov (in)equalities

Abstract: We discuss several numerical conditions for families of projective varieties or variations of Hodge structures.


Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds

Abstract: Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally geodesic embedded Euclidean plane $\mathbb{R}^2$ (i.e., $M^n$ is a visibility manifold). Then Gromov's simplicial volume $\| M^n \|$ is non-zero. Consequently, $M^n$ is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if $Ric_g \ge -(n-1)$, then $vol(M^n, g) \ge \frac{1}{(n-1)^n n!} \| M^n \| > 0. Theorem B. (Perelman) Let $M^3$ be a closed a-spherical 3-manifold ($K(\pi, 1)$-space) with the fundamental group $\Gamma$. Suppose that $\Gamma$ contains no subgroups isomorphic to $\mathbb{Z}\oplus \mathbb{Z}$. Then $M^3$ is diffeomorphic to a compact quotient of real hyperbolic space $\mathbb{H}^3$, i.e., $M^3 \equiv \mathbb{H}^3/\Gamma$. Consequently, $MinVol(M^3) \ge {1/24}\| M^3 \| > 0$. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and {\it singular} spaces will also be discussed.


Floor decompositions of tropical curves : the planar case

Abstract: In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also \cite{FM} for combinatorial treatment of floor diagrams in the projective case.


The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Abstract: Given a projective scheme $X$ over a field $k$, an automorphism $\sigma$ of $X$, and a $\sigma$-ample invertible sheaf $L$, one may form the twisted homogeneous coordinate ring $B = B(X, L, \sigma)$, one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of $B$, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field $k$ of characteristic zero, we prove that that the primitive ideals of $B$ are characterized by the usual Dixmier-Moeglin conditions whenever the dimension of $X$ is no more than 2.


Twisted modules for quantum vertex algebras

Abstract: We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras.


On the singular scheme of codimension one holomorphic foliations in P^3

Abstract: In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in P^3 is locally free if and only the singular scheme is a curve, and that it splits if and only if that curve is arithmetically Cohen-Macaulay. On the other hand, we discuss when a split foliation in P^3 is determined by its singular scheme.


A remarkable sequence of integers

Abstract: A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.


Hartogs' extension theorems on Stein spaces

Abstract: We discuss various known generalizations of the classical Hartogs' extension theorem on Stein spaces with arbitrary singularities and present an analytic proof based on d-bar methods.


p-Density, exponential sums and Artin-Schreier curves

Abstract: In this paper we define the $p$-density of a finite subset $D\subset\ma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.


Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations

Abstract: For nonnegative real numbers $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\alpha+\beta+\gamma >0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \medskip \noindent Theorem 1. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \medskip \noindent Theorem 2. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}= \displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.}


Contact deformations of closed 1-forms on Torus bundles over the circle

Abstract: If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, every closed, nonsingular 1-form deforms linearly into contact forms.


Global Attractivity of the Equilibrium of a Difference Equation: An Elementary Proof Assisted by Computer Algebra System

Abstract: Let $p$ and $q$ be arbitrary positive numbers. It is shown that if $q < p$, then all solutions to the difference equation \tag{E} x_{n+1} = \frac{p+q x_n}{1+x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1}>0, x_0>0 converge to the positive equilibrium $\overline{x} = {1/2}(q-1 + \sqrt{(q-1)^2 + 4 p})$. \medskip The above result, taken together with the 1993 result of Koci\'c and Ladas for equation (E) with $q \geq p$, gives global attractivity of the positive equilibrium of (E) for all positive values of the parameters, thus completing the proof of a conjecture of Ladas.


Hopf Algebras of Graphs

Abstract: We define graded Hopf algebras with bases labeled by various types of graphs and hypergraphs, provided with natural embeddings into an algebra of polynomials in infinitely many variables. These algebras are graded by the number of edges and can be considered as generalizations of symmetric or quasi-symmetric functions.


$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras

Abstract: The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of degree $d$ is $d$-Koszul if and only if its associated monomial algebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class 2-$d$-Koszul algebras are introduced. In particular, it shown that 2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.