Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization

Abstract: Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing {\sf QR}, which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri. The method involves random sampling of columns, followed by a matrix factorization that exposes the well-conditioned subset of columns. This factorization, which is due to Grothendieck, is regarded as a central tool in modern functional analysis. The primary novelty in this work is an algorithm, based on eigenvalue minimization, for constructing the Grothendieck factorization. These ideas also result in a novel approximation algorithm for the $(\infty, 1)$ norm of a matrix, which is generally {\sf NP}-hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous {\sc maxcut} semidefinite program.


Andr\'e-Quillen cohomology of algebras over an operad

Abstract: We study the Andr\'e-Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For which operads is the associated Andr\'e-Quillen cohomology equal to an Ext-functor ? We give several criterion, based on the cotangent complex, to characterize this property. We apply it to homotopy algebras, which gives a new homotopy stable property for algebras over cofibrant operads.


Some New Examples of Non-K\"ahler Ricci Solitons

Abstract: We produce non-K\"ahler complete steady gradient Ricci solitons generalising those constructed by Bryant and Ivey.


Summing the curious series of Kempner and Irwin

Abstract: In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved, among other things, that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428 70807 47848 31968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.00745 x 10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol. Finally, we discuss a class of related series whose summation algorithm has not yet been developed.


Iitaka conjecture $C_{n,m}$ in dimension six

Abstract: We prove that the Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces holds up to dimension 6, that is, when $n\le 6$.


Nonabelian free group actions: Markov processes, the Abramov-Rohlin formula and Yuzvinskii's formula

Abstract: This paper introduces Markov chains and processes over nonabelian free groups and semigroups. We prove a formula for the $f$-invariant of a Markov chain over a free group in terms of transition matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the Abramov-Rohlin formula for skew-product actions and Yuzvinskii's addition formula for algebraic actions.


Split reductions of simple abelian varieties

Abstract: Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely, if the absolute endomorphism ring of X is noncommutative, then X_p is reducible for p in a set of positive density.


On the generalization of Gurland distribution

Abstract: In the present paper a generalization of Gurland distribution [3] is obtained as a beta mixture of the generalized Poisson distribution (GPD) of Consul and Jain [2]. The first two moments of the distribution and a recurrence relation among probabilities are obtained. The present distribution is supposed to be more general in nature and wider in scope.


On endomorphism rings and dimensions of local cohomology modules

Abstract: Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c = \height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$ (cf. \cite{HSt} and \cite{HS}). Here we prove that this is true if and only if $H^i_I(R) = 0, i = n, n -1$ provided $c \geq 2$ and $R/I$ has an isolated singularity resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$ for all $i > m, m$ a given integer, resp. an estimate of the dimension of $H^i_I(R).$


Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

Abstract: In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T], U(0) & = u_0. {aligned}. {equation} Here $(A(t))_{t\in [0,T]}$ are unbounded operators with domains $(D(A(t)))_{t\in [0,T]}$ which may be time dependent. We assume that $(A(t))_{t\in [0,T]}$ satisfies the conditions of Acquistapace and Terreni. The functions $F$ and $B$ are nonlinear functions defined on certain interpolation spaces and $u_0\in E$ is the initial value. $W_H$ is a cylindrical Brownian motion on a separable Hilbert space $H$. Under Lipschitz and linear growth conditions we show that there exists a unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk, to obtain space-time regularity results for the solution $U$ of \eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second order stochastic partial differential equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to several improvements of their result.


Probability and Statistics: Essays in Honor of David A. Freedman

Abstract: This volume is our tribute to David A. Freedman, whom we regard as one of the great statisticians of our time. He received his B.Sc. degree from McGill University and his Ph.D. from Princeton, and joined the Department of Statistics of the University of California, Berkeley, in 1962, where, apart from sabbaticals, he has been ever since. In a career of over 45 years, David has made many fine contributions to probability and statistical theory, and to the application of statistics. His early research was on Markov chains and martingales, and two topics with which he has had a lifelong fascination: exchangeability and De Finetti's theorem, and the consistency of Bayes estimates. His asymptotic theory for the bootstrap was also highly influential. David was elected to the American Academy of Arts and Sciences in 1991, and in 2003 he received the John J. Carty Award for the Advancement of Science from the U.S. National Academy of Sciences. In addition to his purely academic research, David has extensive experience as a consultant, including working for the Carnegie Commission, the City of San Francisco, and the Federal Reserve, as well as several Departments of the U.S. Government--Energy, Treasury, Justice, and Commerce. He has testified as an expert witness on statistics in a number of law cases, including Piva v. Xerox (employment discrimination), Garza v. County of Los Angeles (voting rights), and New York v. Department of Commerce (census adjustment). Lastly, he is an exceptionally good writer and teacher, and his many books and review articles are arguably his most important contribution to our subject.


Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh

Abstract: Jayanta Kumar Ghosh is one of the most extraordinary professors in the field of Statistics. His research in numerous areas, especially asymptotics, has been groundbreaking, influential throughout the world, and widely recognized through awards and other honors. His leadership in Statistics as Director of the Indian Statistical Institute and President of the International Statistical Institute, among other eminent positions, has been likewise outstanding. In recognition of Jayanta's enormous impact, this volume is an effort to honor him by drawing together contributions to the main areas in which he has worked and continues to work. The papers naturally fall into five categories. First, sequential estimation was Jayanta's starting point. Thus, beginning with that topic, there are two papers, one classical by Hall and Ding leading to a variant on p-values, and one Bayesian by Berger and Sun extending reference priors to stopping time problems. Second, there are five papers in the general area of prior specification. Much of Jayanta's earlier work involved group families as does Sweeting's paper here for instance. There are also two papers dwelling on the link between fuzzy sets and priors, by Meeden and by Delampady and Angers. Equally daring is the work by Mukerjee with data dependent priors and the pleasing confluence of several prior selection criteria found by Ghosh, Santra and Kim. Jayanta himself studied a variety of prior selection criteria including probability matching priors and reference priors.


M-curves of degree 9 with three nests

Abstract: The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of a given degree $m$. For $m = 9$, the classification of the $M$-curves is still wide open. Let $C_9$ be an $M$-curve of degree 9 and $O$ be a non-empty oval of $C_9$. If $O$ contains in its interior $\alpha$ ovals that are all empty, we say that $O$ together with these $\alpha$ ovals forms a nest. The present paper deals with the $M$-curves with three nests. Let $\alpha_i, i = 1, 2, 3$ be the numbers of empty ovals in each nest. We prove that at least one of the $\alpha_i$ is odd. This is a step towards a conjecture of A. Korchagin, claiming that at least two of the $\alpha_i$ should be odd.


Numerical simulation of BSDEs using empirical regression methods: theory and practice

Abstract: This article deals with the numerical resolution of backward stochastic differential equations. Firstly, we consider a rather general case where the filtration is generated by a Brownian motion and a Poisson random measure. We provide a simulation algorithm based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. We state fully explicit error bounds. Secondly, restricting to the case of a Brownian filtration, we consider reflected BSDEs and adapt the previous algorithm to that situation. The complexity of the algorithm is very competitive and allows us to treat numerical results in dimension 10.


Trees of definable sets over the p-adics

Abstract: To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and for one-dimensional sets.


On generating relative and absolute invariants of linear differential equations

Abstract: A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative invariants, and it is explicitly shown how to generate fundamental sets of relative and absolute invariants of all orders for the general linear equation. Explicit constructions are made for the linear ODE of order five. The approach used for the explicit determination of invariants is based on an infinitesimal method.


On Positive Integers Represented as Arithmetic Series

Abstract: The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In particular, we shall be concerned with the representation of positive integers as arithmetic series of the simplest kind, i.e., either as sums of successive odd positive numbers, or as sums of successive even positive numbers (both treated as Problem 1), or as sums of consecutive positive integers (treated as Problem 2).


Simultaneous packing and covering in sequence spaces

Abstract: We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal.


Bootstrap percolation in three dimensions

Abstract: By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they have at least $r\in\mathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the $d$-dimensional grid $[n]^d$: with $2\leq r\leq d$ fixed, it was proved by Cerf and Cirillo (for $d=r=3$), and by Cerf and Manzo (in general), that \[p_c([n]^d,r)=\Theta\biggl(\frac{1}{\log_{(r-1)}n}\biggr)^{d-r+1},\] where $\log_{(r)}$ is an $r$-times iterated logarithm. However, the exact threshold function is only known in the case $d=r=2$, where it was shown by Holroyd to be $(1+o(1))\frac{\pi^2}{18\log n}$. In this paper we shall determine the exact threshold in the crucial case $d=r=3$, and lay the groundwork for solving the problem for all fixed $d$ and $r$.


Confinement of matroid representations to subsets of partial fields

Abstract: Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem. A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields. We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6. Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.


Seshadri constants on the self-product of an elliptic curve

Abstract: The purpose of this paper is to study Seshadri constants on the self-product $E\times E$ of an elliptic curve $E$. We provide explicit formulas for computing the Seshadri constants of all ample line bundles on the surfaces considered. As an application, we obtain a good picture of the behaviour of the Seshadri function on the nef cone.


Gap in Nonlinear Equivalence for Numerical Methods for PDEs

Abstract: For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and convergence.


The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors

Abstract: To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated to plane germs. Here we show that the Poincare series of a set of divisorial valuations on the ring of germs of functions of two variables defines "the topology of the set of the divisors" in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which proves that the Alexander polynomial is equivalent to the embedded topology.


Generators of simple Lie algebras II

Abstract: This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic (see arXiv:0708.1711). We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S,H or K never enjoy the 'one-and-a-half generation' property. The methods rely on a study of centralisers in Cartan type Lie algebras.


Minimax State Estimation for a Dynamic System Described by a Differential-Algebraic Equation

Abstract: In this report we address the linear state estimation problem: to estimate a linear transformation $\ell(\varphi)$ of the state $\varphi$ through an algorithm $\widehat{\ell(\varphi)}$ operating on measurements $y$, where $L\varphi=f,y=H\varphi+\eta$. We study the estimation problem in terms of the minimax estimation framework: to find a linear algorithm $\widehat{\widehat{\ell(\varphi)}}$ that minimizes the worst case error $\sup_{\varphi,\eta}d(\ell(\varphi),\widehat{\ell(\varphi)}) $. A key feature of the presented estimation approach is to fix a class of linear operators $L$, $H$; given any pair $L,H$ from that class we describe a class $\mathcal L$ of all solution operators $\ell$ such that the worst case error is finite. We formulate a duality theorem (like Kalman duality principle) that is the estimation problem is equal to the optimal control problem if $G$ is convex bounded subset of the corresponding Hilbert space, $L$ is a closed linear mapping. We obtain optimal estimations as solutions of the linear operator equations if $G$ is an ellipsoid. Then we apply this to the state estimation for the linear differential-algebraic equations (DAE). The minimax observer for DAE is represented in the form of the minimax filter. For discrete time DAEs we present the online minimax estimator.


Aronszajn Compacta

Abstract: We consider a class of compacta X such that the maps from X onto metric compacta define an Aronszajn tree of closed subsets of X.


The Brauer algebra and the symplectic Schur algebra

Abstract: Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\ge1, r\ge0 and m\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. Finally, we deduce from the previous results a new proof of the geometric description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.


Cauchy problem for viscous rotating shallow water equations

Abstract: We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.


Families of equivariant differential operators and anti de Sitter spaces

Abstract: We prove existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.


Ergodic actions of S_\mu U(2) on C*-algebras from II_1 subfactors

Abstract: To a proper inclusion N\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_\mu U(2). The deformation parameter is determined by -1<\mu<0 and [M:N]=|\mu+\mu^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_\mu U(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. \mu is negative as M is a real bimodule.


The second iterate for the Navier-Stokes equation

Abstract: We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak strong uniqueness for Koch-Tataru solutions; and finally an instability result in $B^{-1}_{\infty,q}$, for q>2.


Poset Resolutions of Monomial Ideals

Abstract: We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $\eta:P\lra\mbb{N}^n$ a sequence of multigraded modules and maps.


A Note on Algebraic Multigrid Methods for the Discrete Weighted Laplacian

Abstract: In recent contributions, algebraic multigrid methods have been designed and studied from the viewpoint of the spectral complementarity. In this note we focus our efforts on specific applications and, more precisely, on large linear systems arising from the approximation of weighted Laplacian with various boundary conditions. We adapt the multigrid idea to this specific setting and we present and critically discuss a wide numerical experimentation showing the potentiality of the considered approach.


Worst Case to Average Case Reductions for Polynomials

Abstract: A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [gt07] have shown that bias imply low rank over large fields (i.e. for the case $d < |\F|$). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be {\em approximated} by few polynomials of bounded degree, can be {\em computed} by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.


On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

Abstract: We prove the ill-posedness in $ H^s(\T) $, $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from $H^s(\T) $ into itself for any fixed $ t\neq 0 $. This result is slightly stronger than the one obtained by Christ-Colliander-Tao where the discontinuity of the solution map is established. Moreover our proof is different and clarifies the ill-posedness phenomena. Our approach relies on a new result on the behavior of the associated flow-map with respect to the weak topology of $ L^2(\T) $.


Fundamental classes not representable by products

Abstract: We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.


Note on a theorem of Bousfield and Friedlander

Abstract: We examine the proof of a classical localization theorem of Bousfield and Friedlander and we remove the assumption that the underlying model category be right proper. The key to the argument is a lemma about factoring in morphisms in the arrow category of a model category.


Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations

Abstract: We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $\gamma$ and $\kappa$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters.


Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

Abstract: We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $\tau_\alpha$ from a wedge with apex at the origin and interior half-angle $\alpha$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \to \infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $\tau_\alpha < \infty$ a.s. for any $\alpha$ (while for a stronger drift field $\tau_\alpha$ is infinite with positive probability). Here we study the more difficult problem of the existence and non-existence of moments $E[\tau_\alpha^s]$, $s>0$. Assuming (in common with much of the literature) a uniform bound on the walk's increments, we show that for $\alpha < \pi/2$ there exists $s_0 \in (0,\infty)$ such that $E[\tau_\alpha^s]$ is finite for $s < s_0$ but infinite for $s > s_0$; under specific assumptions on the drift field we show that we can attain $E[\tau_\alpha^s] = \infty$ for any $s > 1/2$. We show that for $\alpha \leq \pi$ there is a phase transition between drifts of magnitude $O(1/|x|)$ (the critical regime) and $o(1/|x|)$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.


Springer theory via the Hitchin fibration

Abstract: In this paper, we translate the Springer theory of Weyl group representations into the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient g/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the Geometric Langlands program.


Global attractor and asymptotic smoothing effects for the weakly damped cubic Schr\"odinger equation in $L^2(\T)$

Abstract: We prove that the weakly damped cubic Schr\"odinger flow in $L^2(\T)$ provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak $ L^2(\T) $-convergence inspired by a previous work of the author. Combining the compactness in $ L^2(\T) $ of the attractor with the approach developed by Goubet, we show that the attractor is actually a compact set of $ H^2(\T) $. This asymptotic smoothing effect is optimal in view of the regularity of the steady states.


Consecutive integers in high-multiplicity sumsets

Abstract: Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is as follows. Let $k,l\ge 1$ and $n\ge 3$ be integers, and suppose that $A_1,...,A_k\subset[0,l]$ are integer sets of size at least $n$, none of which is contained in an arithmetic progression with difference greater than 1. If $k\ge 2\lceil(l-1)/(n-2)\rceil$, then the sumset $A_1+...+A_k$ contains a block of consecutive integers of length $k(n-1)$.


Generating Simplicial Complexes

Abstract: In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented.


Rational Symplectic Field Theory for Legendrian knots

Abstract: We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.


An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial

Abstract: Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing "deficiency" with multiplicity of $\theta$ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai-Edmonds Structure Theorem. Consequently, the matching polynomial of a vertex transitive graph has simple roots.


Algebraic cycles and motivic iterated integrals II

Abstract: This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret the multiple logarithm as an element of this Hopf algebra as long as the products of consecutive arguments are not 1.


Predicting Regional Classification of Levantine Ivory Sculptures: A Machine Learning Approach

Abstract: Art historians and archaeologists have long grappled with the regional classification of ancient Near Eastern ivory carvings. Based on the visual similarity of sculptures, individuals within these fields have proposed object assemblages linked to hypothesized regional production centers. Using quantitative rather than visual methods, we here approach this classification task by exploiting computational methods from machine learning currently used with success in a variety of statistical problems in science and engineering. We first construct a prediction function using 66 categorical features as inputs and regional style as output. The model assigns regional style group (RSG), with 98 percent prediction accuracy. We then rank these features by their mutual information with RSG, quantifying single-feature predictive power. Using the highest- ranking features in combination with nomographic visualization, we have found previously unknown relationships that may aid in the regional classification of these ivories and their interpretation in art historical context.


Norm one idempotent cb-multipliers with applications to the Fourier algebra in the cb-multiplier norm

Abstract: For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{Mcb}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{Mcb}(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which $A_{Mcb}(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)


Discrete spectrum and Weyl's asymptotic formula for incomplete manifolds

Abstract: Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl's asymptotic formula.


Uniform Local Existence for Inhomogeneous Rotating Fluid Equations

Abstract: We investigate the equations of anisotropic incompressible viscous fluids in $\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniformlocal existence result with respect to the Rossby number in the same functional spaces under the additional assumption that $B=B(t,x_1)$ or $B=B(t,x_2)$. We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.


Lower bounds for index of Wente tori

Abstract: We show numerically that any of the constant mean curvature tori first found by Wente must have index at least eight.


On the index of constant mean curvature 1 surfaces in hyperbolic space

Abstract: We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the index of the catenoid cousins and some other examples. Secondly, it allows us to be able to apply a method similar to that of Choe (using Killing vector fields on minimal surfaces in Euclidean 3-space) to our case as well, resulting in lower bounds of index for other examples. And thirdly, it allows us to give a more direct proof of the result by do Carmo and Silveira that if a constant mean curvature 1 surface in hyperbolic 3-space has finite total curvature, then it has finite index. Finally, we show that for any constant mean curvature 1 surface in hyperbolic 3-space that has been constructed via a correspondence to a minimal surface in Euclidean 3-space, we can take advantage of this correspondence to find a lower bound for its index.


Power sums of Hecke's eigenvalues and application

Abstract: We sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi's recent results on higher order symmetric powers. As an application, we improve Kohnen, Lau & Shparlinski's lower bound for the number of Hecke eigenvalues of same signs.


Ehrhart Theory for Lawrence Polytopes and Orbifold Cohomology of Hypertoric Varieties

Abstract: We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart $\delta$-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart $\delta$-polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the $\delta$-polynomial.


Limit Surfaces of Riemann Examples

Abstract: The family of embedded, singly periodic minimal surfaces of Riemann have as limit-surfaces the helicoid, the catenoid, a single plane, or an infinite set of equally-spaced parallel planes.


Nonsingular Ricci flow on a noncompact manifold in dimension three

Abstract: We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.


Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

Abstract: We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in $L^p$-norm of the unit circle, $p\geq2$. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\lambda$ relative to the unit disk, that the approximants themselves converge in capacity to $F$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more.


On the Degree Sequence and its Critical Phenomenon of an Evolving Random Graph Process

Abstract: In this paper we focus on the problem of the degree sequence for the following random graph process. At any time-step $t$, one of the following three substeps is executed: with probability $\alpha_1$, a new vertex $x_t$ and $m$ edges incident with $x_t$ are added; or, with probability $\alpha-\alpha_1$, $m$ edges are added; or finally, with probability $1-\a$, $m$ random edges are deleted. Note that in any case edges are added in the manner of preferential attachment. we prove that there exists a critical point $\alpha_c$ satisfying: 1) if $\alpha_1<\alpha_c$, then the model has power law degree sequence; 2) if $\alpha_1>\alpha_c$, then the model has exponential degree sequence; and 3) if $\alpha_1=\alpha_c$, then the model has a degree sequence lying between the above two cases.


Remnant inequalities and doubly-twisted conjugacy in free groups

Abstract: We give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms $\phi$ and $\psi$ such that certain remnant words from $\phi$ are longer than the images of generators under $\psi$. Our first result is a remnant inequality condition which implies that two words $u$ and $v$ are not doubly-twisted conjugate. Further we show that if $\psi$ is given and $\phi$, $u$, and $v$ are chosen at random, then the probability that $u$ and $v$ are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if $\phi$, $u$, and $v$ are chosen at random, then $u$ and $v$ are not in the same singly-twisted conjugacy class with probability 1. Our second result generalizes Kim's "bounded solution length". We give an algorithm for deciding doubly-twisted conjugacy relations in the case where $\phi$ and $\psi$ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if $\phi$ has remnant words of length at least 2. As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism.


Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)

Abstract: We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabilized. As a corollary, we produce a manifold that has distance one Heegaard splittings of arbitrarily high genus. Finally, we show that in a 3-manifold formed by a sufficiently complicated gluing, a low genus, unstabilized Heegaard splitting can be expressed in a unique way as an amalgamation over the gluing surface.


Band-Dominated Fredholm Operators and a Question of Rabinovich, Roch and Silbermann

Abstract: Withdrawn due to a likely error with the homeomorphism at line (4). Old abstract: In the monograph 'Limit Operators and their Applications in Operator Theory', the authors define the operator spectrum of a band-dominated operator T and prove that T is Fredholm if and only if all of the operators in its operator spectrum are invertible with uniformly bounded inverses. They also ask whether the uniform boundedness condition can in fact be dispensed with. In this note we answer this question affirmatively.


Graph polynomials and their applications II: Interrelations and interpretations

Abstract: This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore how an understanding of one polynomial can guide research into others. We also discuss multivariable generalizations of some of these polynomials and the theory facilitated by this. We conclude with two examples, one from biology and one from physics, that illustrate the applicability of graph polynomials in other fields. This is the second chapter of a two chapter series, and concludes Graph Polynomials and Their Applications I: The Tutte Polynomial, arXiv:0803.3079


Functions with isolated singularities on surfaces

Abstract: Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of stabilizers and orbits of f with respect to the right action of the diffeomorphisms group of M. The present paper extends those calculations to a large class of maps M-->P with degenerate singularities satisfying certain set of axioms.


Kahler manifolds with quasi-constant holomorphic curvature

Abstract: The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.


Retarded integral inequalities of Gronwall-Bihari type

Abstract: We establish two nonlinear retarded integral inequalities. Bounds on the solution of some retarded equations are then obtained.


Some Examples of Blackadar and Kirchberg's MF Algebras

Abstract: In the paper, we provide some examples of MF algebras by considering minimal or maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C$^*$-algebras, whose BDF extension semigroup is not group. These examples include, for example, $ C_r^*(F_n)\otimes_{max} C^*(F_n) $, $ C_r^*(F_n)\otimes_{min} C^*(F_n) $, $C_r^*(H_1\ast H_2)$ with $2\le |H_1|<\infty$ and $3\le |H_2|<\infty$ where $|H_1|$, $|H_2|$ are the orders of thr groups $H1$, or $H_2$ respectively, and many others.


Promotion and Evacuation

Abstract: Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schutzenberger. This paper surveys the basic properties of these two operations and discusses some generalizations.


First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials

Abstract: A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form S_D(f) = {x \in D: f(x) >= 0}. Here D={x\in R^n: g_1(x) >= 0, ..., g_m(x) >= 0} is a convex domain defined by some "nice" concave polynomials g_i(x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over \mc{D}, we prove that S_D(f) has some explicit semidefinite representations under certain conditions called preordering concavity or q-module concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria. When f(x) is a polynomial or rational function having singularities on the boundary of S_D(f), a perspective transformation is introduced to find some explicit semidefinite representations for S_D(f) under certain conditions. In the particular case n=2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that S_D(f) always admits an explicitly constructible semidefinite representation.


A version of smooth K-theory adapted to the total Chern class

Abstract: A version of smooth K-theory is constructed, which is adapted to the total Chern class instead of the Chern character (contrarily to previous theories). Some total Chern class morphism from this K-theory to Cheeger-Simons differential characters is constructed. This answers a question raised by U. Bunke.


On the strength of Hausdorff's gap condition

Abstract: Hausdorff's gap condition was satisfied by his original 1936 construction of an (omega-1,omega-1) gap in P(N)/Fin. We solve an open problem in determining whether Hausdorff's condition is actually stronger than the more modern indestructibility condition, by constructing an indestructible (omega-1,omega-1) gap not equivalent to any gap satisfying Hausdorff's condition, from uncountably many random reals.


Reasonable non--Radon--Nikodym ideals

Abstract: For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N.


About the almost everywhere convergence of the spectral expansions of functions from $L_1^\a(S^N)$

Abstract: In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space $L_1^\a$ is used. For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we proved the positive results on summability of Fourier-Laplace series. Note that when order $\alpha$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series, which based on spectral expansions property of self-adjoint Laplace-Beltrami operator on the unit sphere.


Maximum principle for viscosity solutions on Riemannian manifolds

Abstract: In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity version.


Analytic Disks and the Projective Hull

Abstract: Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves the polynomial hull of gamma. When X = P^n, complex projective space, the projective hull hat{c} of c comes into play. One always has V contained in hat{c}, and for analytic curves they conjecturally coincide. In this paper we establish an approximate analogue of this idea which holds without the analyticity of c. We characterize points in hat{c} as those which lie on a sequence of analytic disks whose boundaries converge down to c. This is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose work is essential here. The results are applied to construct a remarkable example of a closed curve c in P^2, which is real analytic at all but one point, and for which the closure of hat{c} is W \cup L where L is a projective line and W is an analytic (non-algebraic) subvariety of P^2 - L. Furthermore, hat{c} itself is the union of W with only two points on L.


"Iff" is not expressible in independence-friendly logic

Abstract: Ordinary first-order logic has the property that two formulas \phi and \psi have the same meaning in a structure if and only if the formula ``\phi iff \psi'' is true in the structure. We prove that independence-friendly logic does not have this property.


Automata generating free products of groups of order 2

Abstract: We construct a family of automata with n states, n>3, acting on a rooted binary tree that generate the free products of cyclic groups of order 2.


Computing j-multiplicity

Abstract: The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the j-multiplicity, giving another proof for the length formula and the additive formula.


Selberg's zeta functions for congruence subgroups of modular groups in SL(2,R) and SL(2,C)

Abstract: It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).


$n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions

Abstract: The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of $L$-functions. Under the assumption of the Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case that the Fourier transform $\hat{f}(u)$ of any test function $f$ is supported in the region $\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein's result on computing the $n$-level density for the Fourier transform $\hat{f}(u)$ being supported in the region $\sum^n_{j=1}u_j < 2$.


Graph polynomials and Tutte-Grothendieck invariants: an application of elementary finite Fourier analysis

Abstract: This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), and a description of a family of polynomials containing the graph polynomial which yield Tutte-Grothendieck invariants in a similar way.


Weighted pluricomplex energy

Abstract: We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy $\mathcal{E}_\chi (\Omega)$ in the general case ($\chi(0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator $(dd^c \cdot)^n$ on the classes $\mathcal{E}\chi(\Omega).$


The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Abstract: Let $L$ be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra $\hat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1... x_lv_0$, where $l\le m$, $x_i\in \hat{\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\rm gr}$ with respect to the PBW filtration. The "top-down" description deals with a structure of $L^{\rm gr}$ as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field $e_\theta(z)^2$, which corresponds to the longest root $\theta$. The "bottom-up" description deals with the structure of $L^{\rm gr}$ as a representation of the current algebra ${\mathfrak g}\otimes {\mathbb C}[t]$. We prove that each quotient $F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of ${\mathfrak g}$.


Products of Functions in Hardy and Lipschitz or Bmo Spaces

Abstract: We define as a distribution the product of a function (or distribution) h in some Hardy space Hp with a function b in the dual space of Hp. Moreover, we prove that the product bxh may be written as the sum of an integrable function with a distribution that belongs to some Hardy-Orlicz space, or to the same Hardy space Hp, depending on the values of p.


New estimates and tests of independence in some copula models

Abstract: We introduce new estimates and tests of independence in copula models with unknown margins using $\phi$-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an interior or a boundary value of the parameter space. Simulation results show that the choice of $\chi^2$-divergence has good properties in terms of efficiency-robustness.


Algorithm for computing local Bernstein-Sato ideals

Abstract: Given $p$ polynomials of $n$ variables over a field $k$ of characteristic 0 and a point $a \in k^n$, we propose an algorithm computing the local Bernstein-Sato ideal at $a$. Moreover with the same algorithm we compute a constructible stratification of $k^n$ such that the local Bernstein-Sato ideal is constant along each stratum. Finally, we present non-trivial examples computed with our algorithm.


A Satake type theorem for Super Automorphic forms

Abstract: Aim of this article is a Satake type theorem for super automorphic forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. This theorem - roughly speaking - says that for large weight k and all p from 1 to infinity (both including) a super automorphic form on B is a super cusp form if and only if its p-norm with respect to a certain measure on the quotient of B is finite. And so in particular all these Lp-spaces coincide! We will give a proof of this theorem using an unbounded realization of B and Fourier decomposition at the cusps of the quotient mapped to infinity via a partial Cayley transform.


Self-similar Focusing in Porous Media: An Explicit Calculation

Abstract: We consider a porous mediaum flow in which the gas is initially distributed in the exterior of an empty region (a hole) and study the final stage of the hole-filling process. Hole-filling is asymptotically described by a self-similar solution which depends on a constant determined by the initial configuration. In general, this constant must be found numerically.. Here we give an example of a one-dimensional symmetric flow where the constant is obtained explicitly.


Geometry of syzygies via Poncelet varieties

Abstract: We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\P^1},\O_{\P^1}(n))$. We define $\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\P^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of $\frak{X}_{k,r,d}$. In the second part we make a link between $\frak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties.


Instability results for an elliptic equation on compact Riemannian manifolds with non-negative Ricci curvature

Abstract: We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals.


Smoothing Effects for Navier-Stokes Equations

Abstract: We prove some smoothing effects for the 3-D Navier-Stokes equations for initial data belonging to the critical Sobolev space $H^{1/2}(\R^3)$. Asymptotic behavior of the global solution when the time goes to infinity is studied. We also obtain a new energy estimate. Other results in this direction and with different methods can be found in \cite{C4}.


Asymptotic Analysis of MHD Systems

Abstract: In this paper, we study the convergence of strong solutions of a Magneto-Hydro-Dynamic system. On the torus ${\mathbb{T}}^3$, the proof is based on Schochet's methods, whereas in the case of the whole space ${\bf \mathbb{R}^3}$, we use Strichartz's type estimates and a product law's $2D\times3D$.


The Problem with the Linpack Benchmark Matrix Generator

Abstract: We characterize the matrix sizes for which the Linpack Benchmark matrix generator constructs a matrix with identical columns.


Diffusive stability of oscillations in reaction-diffusion systems

Abstract: We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations decay algebraically with the diffusive rate t^{-n/2} in space dimension n. We also compute the leading order term in the asymptotic expansion of the solution, and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation, at the expense of making the whole system quasilinear. Stability is then obtained by a global fixed point argument in temporally weighted Sobolev spaces.


Constructing arithmetic subgroups of unipotent groups

Abstract: Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.


Discrete Tracy-Widom Operators

Abstract: Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper considers discrete Tracy-Widom operators, and gives sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.


Automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras

Abstract: In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by B. Kostant and which is discussed by A. Fialowski using Lie algebra cohomology in case of affine type.


Matrix factorizations and intertwiners of the fundamental representations of quantum group U_q (sl_n)

Abstract: We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional data which is a sequence naturally induced by coloring, we define matrix factorizations, and then we define a matrix factorization for planar diagram obtained by gluing the essential colored planar diagrams as tensor product of the matrix factorizations for the essential planar diagrams. Moreover, we show that some matrix factorizations deribed from tensor product of the essential matrix factorizations have homotopy equivalences corresponding to MOY relations.


Injectives in residuated algebras

Abstract: Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops.


Examples of finite $p$-divisible sets of MHS

Abstract: We provide some examples which give evidence to the conjectures contained in my paper "Finiteness of $p$-Divisible Sets of Multiple Harmonic Sums" (math.NT/0303043). All the main theoretical results can be found in that paper.


Pavelka-style completeness in expansions of \L ukasiewicz logic

Abstract: An algebraic setting for the validity of Pavelka style completeness for some natural expansions of \L ukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard MV-algebra on the real segment $[0, 1]$ is an injective MV-algebra. In particular the logics associated with MV-algebras with product and with divisible MV-algebras are considered.


On the boundary of the moduli spaces of log Hodge structures: triviality of the torsor

Abstract: In this paper we will study the moduli spaces of log Hodge structures introduced by Kato-Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. We treat the following 2 cases: (A) the case where the period domain is Hermitian symmetric, (B) the case where the Hodge structures are of the mirror quintic type. Especially we study a property of the torsor.


Bounds for solution of linear diophantine equations

Abstract: Given linear diophantine equation Ax=b, rank A=m. Let d be the maximum of absolute values of the mxm minors of the matrix (A | b). It is shown that if M={x : Ax=b, x nonnegative and integer} is nonempty, then there exists x=(x1,...,xn) in M, such that xi does not exceed d (i=1,2,..,n).


On Galois representations and Hilbert-Siegel modular forms

Abstract: In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local Langlands correspondence for GSp(4) studied recently in a preprint of Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at p to the dimensions of the parahoric fixed spaces at p. The Galois representations are constructed by first passing to GL(4) over a CM extension, then applying the book of Harris-Taylor plus a refinement due to Taylor-Yoshida, and finally descending to F by a delicate patching argument. This is a variation of the techniques used by Blasius and Rogawski in order to attach motives to Hilbert modular forms.


Weakly almost periodic functionals on the measure algebra

Abstract: It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C$^*$-algebra. This implies that the weakly almost periodic functionals on $M(G)$, the measure algebra of a locally compact group $G$, is a C$^*$-subalgebra of $M(G)^* = C_0(G)^{**}$. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.


Growth of intersection numbers for free group automorphisms

Abstract: For a fully irreducible automorphism \phi of the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'\phi^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'\phi^n for n large.


Geometry of configuration spaces of tensegrities

Abstract: Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in R^d to admit a tensegrity with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in R^d. In particular we find surgeries on graphs that give relations between different strata. Based on numerous examples we give a description of geometric conditions defining the strata for plane tensegrities, we conjecture that the list of such conditions is sufficient to describe any stratum. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices.


Singular Points of Real Quintic Curves Via Computer Algebra

Abstract: There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for reducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, a major component of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.


Zeros of partial sums of the Riemann zeta-function

Abstract: We investigate the distribution of the zeros of partial sums of the Riemann zeta-function, sum_{n\leq X}n^{-s}, estimating the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of their horizontal distribution.


Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Abstract: Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.


Deformations of linear Poisson orbifolds

Abstract: Let $\Gamma$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $\Gamma$ action. We study generalized quadratic relations on the tensor algebra $T(V)\rtimes \Gamma$. We prove that the quotient algebras of $T(V)\rtimes \Gamma$ by such relations satisfy PBW property. Such quotient algebras can be viewed as quantizations of linear or constant Poisson structures on $S(V)\rtimes \Gamma$, and are natural generalizations of symplectic reflection algebras.


Equidistribution of Fekete points on complex manifolds

Abstract: We prove the several variable version of the classical equidistribution theorem for Fekete points of a compact subset of the complex plane, which settles a well-known conjecture in pluri-potential theory. The result is obtained as a special case of a general equidistribution theorem for Fekete points in the setting of a given holomorphic line bundle over a compact complex manifold. The proof builds on our recent work "Capacities and weighted volumes for line bundles".


Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem

Abstract: The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltotic symplectic invariant. A generalisation of the Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed.


A construction of symmetric linear functions of the restricted quantum group \overline{U}_q (sl_2)

Abstract: In this paper we construct all the primitive idempotents of the restricted quantum group $\overline{U}_q (sl_2)$ and also determine the multiplication rules among a basis given by the action of generators of $\bar{U}_q (sl_2)$ to the idempotents. By using this result we construct a basis of the space of symmetric linear functions of $\overline{U}_q (sl_2)$ and determine the decomposition of the integral of the dual of $\overline{U}_q (sl_2)$ twisted by the balancing element to the basis of the space of symmetric linear functions.


Frequentist and Bayesian measures of confidence via multiscale bootstrap for testing three regions

Abstract: A new computation method of frequentist $p$-values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region. We introduce new parametric models for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for one-sided test, can also computes confidence measures of two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation is improved by the two-step multiscale bootstrap and also by including higher-order terms. Model selection is important not only as a motivating application of our method, but also as an essential ingredient in the method. A compromise between frequentist and Bayesian is attempted by showing that the Bayesian posterior probability with an noninformative prior is interpreted as a frequentist $p$-value of ``zero-sided'' test.


Discontinuous Superprocesses with Dependent Spatial Motion

Abstract: We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting-branching particle systems where the spatial motions of the particles are not independent. The main work is to solve the martingale problem. When we turn to the uniqueness of the process, we generalize the localization method introduced by [D.W. Stroock, Diffusion processes associated with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32(1975) 209--244] to the measure-valued context. As for existence, we use particle system approximation and a perturbation method. This work generalizes the model introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab. 6(2001), no.25, 33 pp. (electronic)] where quadratic branching mechanism was considered. We also investigate some properties of the process.


Asymptotic formulas for partial sums of class numbers of indefinite binary quadratic forms

Abstract: Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group.


Equivariant differential characters and symplectic reduction

Abstract: We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction.


Some Remarks on the Braided Thompson Group BV

Abstract: Matthew Brin and Patrick Dehornoy independently discovered a braided version BV of Thompson's group V. In this paper, we discuss some properties of BV that might make the group interesting for group based cryptography. In particular, we show that BV does not admit a non-trivial linear representation.


Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications

Abstract: We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Gu\'edon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.


A homogeneous Gibbons-Hawking ansatz and Blaschke products

Abstract: A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, complete. Our construction also gives rise to incomplete metrics on 3-dimensional contact manifolds that induce complete Carnot-Caratheodory distances.


A proof of the Riemann hypothesis

Abstract: This paper has been withdrawn by the author, due to a mistake on page 29.


Sets of non-differentiability for conjugacies between expanding interval maps

Abstract: We study differentiability of topological conjugacies between expanding piecewise $C^{1+\epsilon}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the type of conjugacies under consideration.


Weyl groupoids of rank two and continued fractions

Abstract: A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices. Key words: Cartan matrix, continued fraction, Nichols algebra, Weyl groupoid


A note on the Ricci flow on noncompact manifolds

Abstract: Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with initial data $(M^3,g_0)$ has a long time solution. This extends a recent result of Ma and Zhu. We also have a higher dimensional version, and we reprove a K$\ddot{a}$hler analogy due to Chau, Tam and Yu.


Isochronicity conditions for some real polynomial systems

Abstract: This paper focuses on isochronicity of linear center perturbed by a polynomial. Isochronicity of a linear center perturbed by a degree four and degree five polynomials is studied, several new isochronous centers are found. For homogeneous isochronous perturbations, a first integral and a linearizing change of coordinates are presented. Moreover, a family of Abel polynomial systems is also considered. By investigations until degree 10 we prove the existence of a unique isochronous center. These results are established using a computer implementation based on Urabe theorem.


An elementary proof of the Briancon-Skoda theorem

Abstract: We give a new elementary proof of the Brian\c{c}on-Skoda theorem, which states that for an $m$-generated ideal $\mathfrak{a}$ in the ring of germs of analytic functions at $0\in \C^n$, the $\nu$:th power of its integral closure is contained in $\mathfrak{a}$, where $\nu = \min(m,n)$.


Nonhomogeneous analytic families of trees

Abstract: We consider a dichotomy for analytic families of trees stating that either there is a colouring of the nodes for which all but finitely many levels of every tree are nonhomogeneous, or else the family contains an uncountable antichain. This dichotomy implies that every nontrivial Souslin poset satisfying the countable chain condition adds a splitting real. We then reduce the dichotomy to a conjecture of Sperner Theory. This conjecture is concerning the asymptotic behaviour of the product of the sizes of the m-shades of pairs of cross-t-intersecting families.


Sur Les Suites D'Interpolation Pour Les Espaces De Bergman a Poids Dans la Boule De $\mathbb{C}^n$}

Abstract: Let $A$ be a sequence of points of $\mathbb{B}^n$ the unit ball in $\mathbb{C}^n.$ In terms of interpolating vectorial function (or Amar's function)[1], we give a necessary condition on $A$ to be interpolating for the weighted Bergman space $B^p_\alpha (\mathbb{B}^n). In the particular case of Hardy space $H^p (\mathbb{B}^2)$, this condition is sufficient no optimal. In the main theorem proof, we resolve Gleason's problem (vectorial form) in $B^p_\alpha (\mathbb{B}^n)$


On the unitarization of linear representations of primitive partially ordered sets

Abstract: We describe all weights which are appropriated for the unitarization of linear representations of primitive partially ordered sets of finite type.


Classifying Camina groups: a theorem of Dark and Scoppola

Abstract: Recall that a group $G$ is a Camina group if every nonlinear irreducible character of $G$ vanishes on $G \setminus G'$. Dark and Scoppola classified the Camina groups that can occur. We present a different proof of this classification using Theorem 2, which strengthens a result of Isaacs on Camina pairs. Theorem 2 is of independent interest.


A conjecture on homotopy groups of spheres, details on the algebra of higher cohomology operations

Abstract: The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the Adams spectral sequence is described.


Properties of Design-Based Functional Principal Components Analysis

Abstract: This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA' estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.


Une minoration du minimum essentiel sur les varietes abeliennes

Abstract: We extend to the general codimension a lower bound for the essential minimum on abelian varieties found in a previous work, under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, ``up to an epsilon'', in the degree of the subvariety. Following a strategy of Amoroso, we change the transcendance phasis in order to simplify the zero lemma and its combinatorics. The last argument is a descent procedure on varieties, which is far more intricate in the abelian setting since there is not, in general, a lifting of the Frobenius in characteristic zero.


The mapping class group cannot be realized by homeomorphisms

Abstract: Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface $M$ of genus $\g \ge 2$, there is no homomorphic section $\E:\MC(M) \to \Homeo(M)$ of the standard projection map $\Proj:\Homeo(M) \to \MC(M)$.


Covers of surfaces with fixed branch locus

Abstract: Given a connected smooth projective surface X over the complex numbers, together with a simple normal crossings divisor D on it, we study finite normal covers Y of X that are unramified outside D. Given moreover a fibration of X onto a curve C, we prove that the `height' of Y over C is bounded linearly in terms of the degree of Y over X. We indicate how an arithmetic analogue of this result, if true, can be auxiliary in proving the existence of a polynomial time algorithm that computes the mod-l Galois representations associated to a given smooth projective geometrically connected surface over the rational numbers. A precise conjecture is formulated.


Singular Points of Reducible Sextic Curves

Abstract: There are 106 individual types of singular points for reducible complex sextic curves.


Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and vectors

Abstract: Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior in the tangent linear dynamics and to determine the associated manifold structure in the flow of a finite-dimensional nonlinear autonomous dynamical system. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting is defined using finite-time Lyapunov exponents and vectors, guided by the asymptotic theory of partially hyperbolic sets, with important modifications for the finite-time case; for example, finite-time Lyapunov analysis relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. The splitting is used to locate points on normally hyperbolic center manifolds. Determining manifolds from tangent bundle structure is more generally applicable than approaches, such as the singular perturbation method, that require special normal forms or other a priori knowledge. The use, features, and accuracy of the approach are illustrated via several detailed examples.


Selfdual representations of division algebras and Weil groups: A contrast

Abstract: Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations \sigma of the Weil group of a local field k of dimension n with the irreducible representations \pi of the invertible elements of a division algebra D over k of index n, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for n even, \sigma is symplectic if and only if \pi is orthogonal. Our results treat more generally the case of GL_m(B), for B a division algebra over k of index r, and n=mr.


A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure

Abstract: We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we consider maximal truncations T_# of the Hilbert transform. The norm estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms of an A_p condition on the weights and two testing conditions. The first is the norm condition above, but the function f varies over bounded functions supported on a cube. The second is a dual weak-type condition, for arbitrary functions. This result should be compared to the result of Nazarov, Treil and Volberg, arXiv:math/0702758. Additional results are obtained for 2<p<\infty, and for the weak type inequality.


Discrete Symbol Calculus

Abstract: This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space $x$ and frequency $\xi$. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fast-converging, non-asymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution $N$ very weakly, typically only through $\log N$ factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into one-way components and 3) depth-stepping in reflection seismology.


Random systems of polynomial equations. The expected number of roots under smooth analysis

Abstract: We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0$, $t\in\mathbb{R}^m$, $i=1,...,m$, where the $P_i$'s are non-random polynomials having degrees $d_i$'s (the "signal") and the $X_i$'s (the "noise") are independent real-valued Gaussian centered random polynomial fields defined on $\mathbb{R}^m$, with a probability law satisfying some invariance properties. For each $i$, $P_i$ and $X_i$ have degree $d_i$. The problem is the behavior of the number of roots for large $m$. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large nor too small, it follows that the quotient between the expected value of the number of roots of the perturbed system and the expected value corresponding to the centered system (i.e., $P_i$ identically zero for all $i=1,...,m$), tends to zero geometrically fast as $m$ tends to infinity. In particular, this means that the behavior of this expected value is governed by the noise part.


Tridiagonal pairs of $q$-Racah type

Abstract: Let $K$ denote an algebraically closed field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i\rbrace_{i=0}^\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq \delta$, where $V^*_{-1}=0$ and $V^*_{\delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=\delta$. For $0 \leq i \leq d$ let $\theta_i$ (resp. $\theta^*_i$) denote the eigenvalue of $A$ (resp. $A^*$) associated with $V_i$ (resp. $V^*_i$). The pair $A,A^*$ is said to have {\it $q$-Racah type} whenever $\theta_i = a + b q^{2i-d}+ c q^{d-2i}$ and $\theta^*_i = a^* + b^*q^{2i-d}+c^*q^{d-2i}$ for $0 \leq i \leq d$, where $q, a,b,c,a^*,b^*,c^*$ are scalars in $K$ with $q,b,c,b^*,c^*$ nonzero and $q^2 \not\in \lbrace 1,-1\rbrace$. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over $K$ that have $q$-Racah type. Our proof involves the representation theory of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$.


Stratification of Unfoldings of Corank 1 Singularities

Abstract: In the study of equisingularity of families of mappings Gaffney introduced the crucial notion of excellent unfoldings. This definition essentially says that the family can be stratified so that there are no strata of dimension 1 other than the parameter axis for the family. Consider a family of corank 1 multi-germs with source dimension less than target. In this paper it is shown how image Milnor numbers can ensure some of the conditions involved in being excellent. The methods used can also be successfully applied to cases where the double point set is a curve. In order to prove the results the rational cohomology description of the disentanglement of a corank 1 multi-germ is given for the first time. Then, using a simple generalization of the Marar-Mond Theorem on the multiple point space of such maps, this description is applied to give conditions which imply the upper semi-continuity of the image Milnor number. From this the main results follow.


The symmetries of the 2phi1

Abstract: We show that the only symmetries of the 2phi1 within a large class of possible transformations are Heine's transformations. The class of transformations considered consists of equation of the form 2phi1(a,b;c;q,z)= f(a,b,c,z) 2phi1(L(a,b,c,q,z)), where f is a q-hypergeometric term and L a linear operator on the logarithms of the parameters. We moreover prove some results on q-difference equations satisfied by 2phi1, which are used to prove the main result.


Melnikov theory to all orders and Puiseux series for subharmonic solutions

Abstract: We study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even in general other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of the perturbation parameter. To obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy, we need further assumptions. The method we use to construct the solution -- when this is possible -- is based on a combination of the Newton-Puiseux algorithm and the tree formalism. This leads to a graphical representation of the solution in terms of diagrams. Finally, if the subharmonic Melnikov function is identically zero, we show that it is possible to introduce higher order generalisations, for which the same kind of analysis can be carried out.


Effective Construction of a Positive Operator which does not admit Triangular Factorization

Abstract: We have constructed a concrete example of a non-factorable positive operator. As a result, for the well-known problems (Ringrose, Kadison and Singer problems) we replace existence theorems by concrete examples.


Hypercontact structures and Floer homology

Abstract: We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact 3-manifold M and a hyperkaehler manifold X.


Optimal consumption policies in illiquid markets

Abstract: We investigate optimal consumption policies in the liquidity risk model introduced in Pham and Tankov (2007). Our main result is to derive smoothness results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a second-order ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given.


Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry

Abstract: David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain $\Omega$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of $\Omega$ and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological weak Finsler metric, in which the unit ball at each tangent space is naturally identified with the domain $\Omega$ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure on $\Omega$, and the unit ball at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.