Harmonic functions via restricted mean-value theorems

Abstract: Let $f$ be a function on a bounded domain $\Omega \subseteq \mathbb{R}^n$ and $\delta$ be a positive function on $\Omega$ such that $B(x,\delta(x))\subseteq \Omega$. Let $\sigma(f)(x)$ be the average of $f$ over the ball $B(x,\delta(x))$. The restricted mean-value theorems discuss the conditions on $f,\delta,$ and $\Omega$ under which $\sigma(f)=f$ implies that $f$ is harmonic. In this paper, we study the stability of harmonic functions with respect to the map $\sigma$. One expects that, in general, the sequence $\sigma^n(f)$ converges to a harmonic function. Among our results, we show that if $\Omega$ is strongly convex (respectively $C^{2,\alpha}$-smooth for some $\alpha\in [0,1]$), the function $\delta(x)$ is continuous, and $f\in C^0(\bar \Omega)$ (respectively, $f\in C^{2,\alpha}(\bar \Omega)$), then $\sigma^n(f)$ converges to a harmonic function uniformly on $\bar \Omega$.


Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory

Abstract: Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration.


Line crossing problem for biased monotonic random walks in the plane

Abstract: In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$, where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and $\beta/(\beta + 1)$, respectively. Among our results, we show that if $\beta \geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random walk, starting from the origin, crosses the line $y=\alpha x+d$ for all $d\geq 0$ with probability 1.


The Abelian Monodromy Extension Property for Families of Curves

Abstract: Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U of S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily-Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.


Multiplier ideal sheaves, Nevanlinna theory, and diophantine approximation

Abstract: This note states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does not provide anything new, it may be a more convenient formulation for some applications.


Hochschild homology and cohomology of {\ell}^1({\mathbb Z}_+^k)

Abstract: Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\ell}^1({\mathbb Z}_+^k)$ [Gourdeau, Lykova, White, 2005] we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology.


Rearrangements and radial graphs of constant mean curvature in hyperbolic space

Abstract: We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques.


The Classification of Higher Order Modular Forms and their Cohomology

Abstract: We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.


On posterior distribution of Bayesian wavelet thresholding

Abstract: We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \citet{abramovich04} where the authors studied the convergence of several Bayesian estimators.


On the Schwartz space isomorphism theorem for rank one symmetric space

Abstract: In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0< p\leq 2)$ under the Fourier transform for the class of functions of left $\delta$-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker's \cite{A} proof of the corresponding result in the case of left $K$-invariant functions on $X$. Thus we give a proof which relies only on the Paley--Wiener theorem.


On the cohomology of orbit space of free \pmb{${\ZZ}_{p}$}-actions on lens spaces

Abstract: Let $G = \ZZ_p$, $p$ an odd prime, act freely on a finite-dimensional CW-complex $X$ with mod $p$ cohomology isomorphic to that of a lens space $L^{2m-1} (p;q_1,...,q_m)$. In this paper, we determine the mod $p$ cohomology ring of the orbit space $X/G$, when $p^2\nmid m$.


Approximation of functions of two variables by certain linear positive operators

Abstract: We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an $r$th order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.


On an inequality concerning the polar derivative of a polynomial

Abstract: In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.


Abstract: Let $\phi$ be an analytic self-map and $u$ be a fixed analytic function on the open unit disk $D$ in the complex plane $\CC.$ The weighted composition operator is defined\break by \begin{equation*} uC_\phi f =u \cdot (f\circ \phi), f \in H(D). \end{equation*} Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps.


A generalization of d'Alembert formula

Abstract: In this paper we find a closed form of the solution for the factored inhomogeneous linear equation \begin{equation*} \prod_{j=1}^{n}(\frac{\hbox{d}}{\hbox{d}t}-A_{j}) u(t) =f(t). \end{equation*} Under the hypothesis $A_{1},A_{2}, ..., A_{n}$ are infinitesimal generators of mutually commuting strongly continuous semigroups of bounded linear operators on a Banach space $X$. Here we do not assume that $A_{j}$s are distinct and we offer the computational method to get explicit solutions of certain partial differential equations.


A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

Abstract: Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of $M$.


Equivariant embeddings of Hermitian symmetric spaces

Abstract: We prove that equivariant, holomorphic embeddings of Hermitian symmetric spaces are totally geodesic (when the image is not of exceptional type).


Some notes on the equivalence of first-order rigidity in various geometries

Abstract: These pages serve two purposes. First, they are notes to accompany the talk "Hyperbolic and projective geometry in constraint programming for CAD" by Walter Whiteley at the "Janos Bolyai Conference on Hyperbolic Geometry", 8--12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will present the equivalence of the first-order rigidity theories of bar-and-joint frameworks in various geometries, including Euclidean, hyperbolic and spherical geometry. The bulk of the theory is outlined here, with remarks and comments alluding to other results that will make the final version of the paper.


On the Gauss Map with Vanishing Biharmonic stress-energy tensor

Abstract: We study the biharmonic stress-energy tensor $S_2$ of Gauss map. Adding few assumptions, the Gauss map with vanishing $S_2$ would be harmonic.


A non-monotone conservation law for dune morphodynamics

Abstract: We investigate a non-local non linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see \cite{Fow01, Fow02}. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well-posedness for initial data in $L^2$ and give explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results.


On a property of the number 977731833235239280

Abstract: We solve a theoretical arithmetics problem stated by Wac{\l}aw Sierpi\'nski. The problem has remained open for a couple of decades.


Hyperbolicity of general deformations: proofs

Abstract: We modify the deformation method explored previously in a joint work of B. Shiffman and the author, in order to construct further examples of Kobayashi hyperbolic surfaces in the projective 3-space of any even degree starting with degree 8.


Local equivalence of symmetric hypersurfaces in $\mathbb C^2$

Abstract: The Chern-Moser normal form and its analog on finite type hypersurfaces in general do not respect symmetries. Extending the work of N. K. Stanton, we consider the local equivalence problem for symmetric Levi degenerate hypersurfaces of finite type in $\mathbb C^2$. The results give for all such hypersurfaces a complete normalization which respects the symmetries. In particular, they apply to tubes and rigid hypersurfaces, providing an effective classification. The main tool is a complete normal form constructed for a general hypersurface with a tube model. As an application, we describe all biholomorphic maps between tubes, answering a question posed by N. Hanges.


Compatible Geometric Matchings

Abstract: This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings $M$ and $M'$ of the same set of $n$ points, for some $k\in\Oh{\log n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound of $k\leq n-2$. We then study the conjecture: \emph{every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching}. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with $n$ edges has an edge-disjoint compatible matching with approximately $4n/5$ edges.


Algebraic causality: Bayes nets and beyond

Abstract: The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of Causal Bayesian Networks has also been investigated in this context. After reviewing these newer relationships, we proceed to demonstrate that many of the ideas embodied in the concept of a ``causal model'' can be more generally expressed directly in terms of a partial order and a family of polynomial maps. The more conventional graphical constructions, when available, remain a powerful tool.


The causal manipulation of chain event graphs

Abstract: Discrete Bayesian Networks have been very successful as a framework both for inference and for expressing certain causal hypotheses. In this paper we present a class of graphical models called the chain event graph (CEG) models, that generalises the class of discrete BN models. It provides a flexible and expressive framework for representing and analysing the implications of causal hypotheses, expressed in terms of the effects of a manipulation of the generating underlying system. We prove that, as for a BN, identifiability analyses of causal effects can be performed through examining the topology of the CEG graph, leading to theorems analogous to the back-door theorem for the BN.


n-Dimensional geometric-shifted global bilinear correspondences of Langlands on mixed motives III

Abstract: This third paper,devoted to global correspondences of Langlands,bears more particularly on geometric-shifted bilinear correspondences on mixed (bi)motives generated under the action of the products,right by left,of differential elliptic operators.The mathematical frame,underlying these correspondences,deals with the categories of the Suslin-Voevodsky mixed (bi)motives and of the Chow mixed (bi)motives which are both in one-to-one correspondence with the functional representation spaces of the shifted algebraic bilinear semigroups.A bilinear holomorphic and supercuspidal spectral representation of an elliptic bioperator is then developed.


Non-commutative residue of projections in Boutet de Monvel's calculus

Abstract: Using results by Melo, Nest, Schick, and Schrohe on the K-theory of Boutet de Monvel's calculus of boundary value problems, we show that the non-commutative residue introduced by Fedosov, Golse, Leichtnam, and Schrohe vanishes on projections in the calculus. This partially answers a question raised in a recent collaboration with Grubb, namely whether the residue is zero on sectorial projections for boundary value problems: This is confirmed to be true when the sectorial projections is in the calculus.


Discrete Koenigs nets and discrete isothermic surfaces

Abstract: We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.


Sure Wins, Separating Probabilities and the Representation of Linear Functionals

Abstract: We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$ admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions


Characterization of quasi-coherent modules that are module schemes

Abstract: The R-module functors that are essential for the development of the theory of the linear representations of an affine R-group are the quasi-coherent R-modules and the R-module schemes. The aim of this paper is to study when a quasi-coherent R-module is an R-module scheme. We will prove that it is equivalent to giving a characterization of projective R-modules of finite type.


Deconvolution for an atomic distribution

Abstract: Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\sigma Z_i$ and $Y_i$ and $Z_i$ are independent. Assume that unobservable $Y$'s are distributed as a random variable $UV,$ where $U$ and $V$ are independent, $U$ has a Bernoulli distribution with probability of zero equal to $p$ and $V$ has a distribution function $F$ with density $f.$ Furthermore, let the random variables $Z_i$ have the standard normal distribution and let $\sigma>0.$ Based on a sample $X_1,..., X_n,$ we consider the problem of estimation of the density $f$ and the probability $p.$ We propose a kernel type deconvolution estimator for $f$ and derive its asymptotic normality at a fixed point. A consistent estimator for $p$ is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.


A normal distribution for the disturbance term in regression theory

Abstract: In regression theory, it is stated that the disturbance term follows the normal distribution when the sample size is large. In Professor J.Johnston's words: "In view of the many factors involved, an appeal to the Central Limit Theorem would further suggest a normal distribution for u." This paper includes an elementary proof that the disturbance term follows the normal distribution when n is large.


Towards a good definition of algebraically overtwisted

Abstract: Symplectic field theory (SFT) is a collection of homology theories that provide invariants for contact manifolds. We give a proof that vanishing of any one of either contact homology, rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish "algebraically overtwisted".


G\'eom\'etrie, points entiers et courbes enti\`eres

Abstract: Let $X$ be a projective variety over a number field $K$ (resp. over $\mathbb{C}$). Let $H$ be the sum of ``sufficiently many positive divisors'' on $X$. We show that any set of quasi-integral points (resp. any integral curve) in $X-H$ is not Zariski dense.


Density modulo 1 of sublacunary sequences: application of Peres-Schlag's arguments

Abstract: Let the sequence $\{t_n\}_{n=1}^{\infty}$ of reals satisfy the condition $ \frac{t_{n+1}}{t_n} \ge 1+ \frac{\gamma}{n^\beta},0\le \beta <1, \gamma >0. $ Then the set $ \{\alpha \in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||t_n \alpha || > \frac{\varkappa}{n^\beta \log (n+1)} \} $ is uncountable. Moreover its Hausdorff dimension is equal to 1. Consider the set of naturals of the form $2^n3^m$ and let the sequence $ s_1=1, s_2=2, s_3=3, s_4=4, s_5=6, s_6 = 8,... $ performs this set as an increasing sequence. Then the set $ \{\alpha \in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||s_n \alpha || > \frac{\varkappa}{\sqrt{n}\log (n+1)} \} $ also has Hausdorff dimension equal to 1. The results obtained use an original approach due to Y. Peres and W. Schlag.


The complexity of the envelope of line and plane arrangements

Abstract: A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is hypothesized to be at least $d{n-2 \choose d-1}$. In this note we show that, for simple arrangements of 4 lines or more, the minimum number of external facets is equal to $2(n-1)$, and for simple arrangements of 5 planes or more, the minimum number of external facets is between $\frac{n(n-2)+6}{3}$ and $(n-4)(2n-3)+5$.


Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective

Abstract: The search for higher homotopy Hopf algebras (known today as A_\infty-bialgebras) began in 1996 during a conference at Vassar College honoring Jim Stasheff in the year of his 60th birthday. In a talk entitled "In Search of Higher Homotopy Hopf Algebras", I indicated that a DG Hopf algebra could be thought of as some (unknown) higher homotopy structure with trivial higher order structure and deformed using a graded version of Gerstenhaber and Schack's bialgebra deformation theory. In retrospect, the bi(co)module structure encoded in Gerstenhaber and Schack's differential defining deformation cohomology detects some (but not all) of the A_infty-bialgebra structure relations. Nevertheless, this motivated the discovery of A_infty-bialgebras by S. Saneblidze and myself in 2005.


On period spaces for p-divisible groups

Abstract: In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.


The integrals in Gradshteyn and Ryzhik. Part 10: the digamma function

Abstract: Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here.


Hodge cohomology of invertible sheaves

Abstract: v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $dim X\le p$, if $k$ has characteristic $p>0$. They show this if $k$ has characteristic 0 and if $n$ is prime to $p$ in characteristic $p>0$. We show the conjecture in characteristic $p>0$ if $n=p$ assuming in addition that $X$ is ordinary (in the sense of Bloch-Kato).


On the auxiliary particle filter

Abstract: In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by pitt and shephard (1999). Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.


An Auslander-type result for Gorenstein-projective modules

Abstract: An artin algebra $A$ is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective $A$-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type (\cite{A,A1}).


Iwasawa decompositions of split Kac-Moody groups

Abstract: We characterize all fields F for which a group with an F-locally split root group datum admits an Iwasawa decomposition. This class of groups in particular includes the split semisimple algebraic groups and the split Kac-Moody groups.


Ergodic behaviour of "signed voter models"

Abstract: We consider some questions raised by the recent paper of Gantert, L\"owe and Steif (2005) concerning ``signed'' voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed).


On geodesic homotopies of controlled width and conjugacies in isometry groups

Abstract: We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.


On Brill-Noether loci over Quot schemes and a Torelli theorem

Abstract: We prove a non abelian Torelli type result for smooth projective curves by working in the derived category of some associated polarized Quot schemes and defining Brill-Noether loci and Abel-Jacobi maps on them.


Quasi-fractal sets in space

Abstract: The focus here is on connected fractal sets with topological dimension 1 and a lot of topological activity, and their connections with analysis.


Homotopy type and v1-periodic homotopy groups of p-compact groups

Abstract: We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.


On the spectrum of infinite dimensional random products of compact operators

Abstract: We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic measure which is positive on non-empty open sets, we conclude that there is a residual subset of cocycles within which, for almost every x, either the Oseledets-Ruelle's decomposition along the orbit of x is dominated or has a trivial spectrum.


Distinguished tame supercuspidal representations

Abstract: This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data.


Spherical characters: the supercuspidal case

Abstract: We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $\pi$ of a connected reductive $p$-adic group, subject to the assumption that $\pi$ is obtained via induction from a representation of an open compact mod centre subgroup. We derive an integral formula for each spherical character belonging to the basis. This formula involves integration of a particular kind of matrix coefficient of $\pi$. We also obtain a similar formula for the function realizing the spherical character. In addition, we determine, subject to some conditions, which of these spherical characters vanish identically on an open neighbourhood of the identity. We verify that the requisite conditions are always satisfied for distinguished tame supercuspidal representations of groups that split over tamely ramified extensions.


Estimates on Monge-Amp\`ere operators derived from a local algebra inequality

Abstract: The goal of this short note is to relate the integrability property of the exponential $e^{-2\phi}$ of a plurisubharmonic function $\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\`ere mass of $(dd^c\phi)^n$. The inequality is valid locally or globally on an arbitrary open subset $\Omega$ in $\bC^n$. We show that $\int_\Omega(dd\phi)^n<n^n$ implies $\int_Ke^{-2\phi}<+\infty$ for every compact subset $K$ in $\Omega$, while functions of the form $\phi(z)=n\log|z-z_0|$, $z_0\in\Omega$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a posteriori to be equivalent to it, proved by A.Corti in dimension $n=2$, and later extended by L.Ein, T.De Fernex and M.Musta\c{t}\v{a} to arbitrary dimensions.


The log-linear group-lasso estimator and its asymptotic properties

Abstract: We define the group-lasso estimator for the natural parameters of the exponential families of distributions representing hierarchical log-linear models under multinomial sampling scheme. Such estimator arises as the solution of a convex penalized likelihood optimization problem based on the group-lasso penalty. We illustrate how it is possible to construct an estimator of the underlying log-linear model using the blocks of nonzero coefficients recovered by the group-lasso procedure. We investigate the asymptotic properties of the group-lasso estimator as a model selection method in a double-asymptotic framework, in which both the sample size and the model complexity grow simultaneously. We provide conditions guaranteeing that the group-lasso estimator is model selection consistent, in the sense that, with overwhelming probability as the sample size increases, it correctly identifies all the sets of nonzero interactions among the variables. Provided the sequences of true underlying models is sparse enough, recovery is possible even if the number of cells grows larger than the sample size. Finally, we derive some central limit type of results for the log-linear group-lasso estimator.


Autour de la cohomologie de Bott-Chern

Abstract: The goal of the memoir is to develop a new cohomology theory which encompasses De Rham and Dolbeault cohomology as well as Deligne Beilinson cohomology, in the context of general complex analytic manifolds. The special case of the Iwasawa manifold is investigated as a typical example of what occurs in the non K\"ahler case. Elementary applications to the Kodaira-Spencer deformation theory and to the calculation of Chern classes are given.


On the Hopf Lemma

Abstract: The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,\alpha}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.


Meridional Almost Normal Surfaces in Knot Complements

Abstract: Suppose $K$ is a knot in a closed 3-manifold $M$ such that $\bar{M-N(K)}$ is irreducible. We show that for any positive integer $b$ there exists a triangulation of $\bar{M-N(K)}$ such that any weakly incompressible bridge surface for $K$ of $b$ bridges or fewer is isotopic to an almost normal bridge surface.


Maximum Likelihood Estimation in Latent Class Models For Contingency Table Data

Abstract: Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximization of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the "100 Swiss Francs" problem, which we introduce and describe in detail.


Galois actions on torsion points of universal one-dimensional formal modules

Abstract: Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \in \{0,...,n-1\}$ let $U_h$ be the subscheme of $\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show that the representation of the fundamental group of $U_h$ on the Tate module of the etale quotient is surjective.


Locally Adaptive Nonparametric Binary Regression

Abstract: A nonparametric and locally adaptive Bayesian estimator is proposed for estimating a binary regression. Flexibility is obtained by modeling the binary regression as a mixture of probit regressions with the argument of each probit regression having a thin plate spline prior with its own smoothing parameter and with the mixture weights depending on the covariates. The estimator is compared to a single spline estimator and to a recently proposed locally adaptive estimator. The methodology is illustrated by applying it to both simulated and real examples.


Euclidean Jordan Algebras and Generalized Krein parameters of a strongly regular graph

Abstract: Let $\tau$ be a strongly $(n,p;a,c)$ regular graph,such that $0<c<p<n-1,$ $A$ his matrix of adjacency and let ${\cal V}_{n}$ be the Euclidean space spanned by the powers of $A$ over the reals where the scallar product $\bullet|\bullet$ is defined by $x|y={trace}(x \cdot y).$ In this work ones proves that ${\cal V}_{n}$ is an Euclidean Jordan algebra of rank 3 when one introduces in ${\cal V}_{n}$ the usual product of matrices. In this Euclidean Jordan algebra one defines the modulus of a matrix, and afterwards one defines $|A|^x \forall x\in \mathbb{R}.$ Working inside the Euclidean Jordan algebra ${\cal V}_{n}$ and making use of the properties of $|A|^x$ one defines the generalized krein parameters of the strongly $(n,p;a,c)$ regular graph $\tau$ and finally one presents necessary conditions over the parameters and the spectra of the $\tau$ strongly $(n,p;a,c)$ regular graph.


On The Density Estimation by Super-Parametric Method

Abstract: The super-parametric density estimators and its related algorism were suggested by Y. -S. Tsai et al [7]. The number of parameters is unlimited in the super- parametric estimators and it is a general theory in sense of unifying or connecting nonparametric and parametric estimators. Before applying to numerical examples, we can not give any comment of the estimators. In this paper, we will focus on the implementation, the computer programming, of the algorism and strategies of choosing window functions. B-splines, Bezier splines and covering windows are studied as well. According to the criterion of the convergence conditions for Parzen window, the number of the window functions shall be, roughly, proportional to the number of samples and so is the number of the variables. Since the algorism is designed for solving the optimization of likelihood function, there will be a set of nonlinear equations with a large number of variables. The results show that algorism suggested by Y. -S. Tsai is very powerful and effective in the sense of mathematics, that is, the iteration procedures converge and the rates of convergence are very fast. Also, the numerical results of different window functions show that the approach of super-parametric density estimators has ushered a new era of statistics.


The quantitative behaviour of polynomial orbits on nilmanifolds

Abstract: A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".


The shape of hyperbolic Dehn surgery space

Abstract: In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a surgery coefficient is measured using the Euclidean metric on a horospherical cross section to a cusp in the complete hyperbolic metric, rescaled to have area 1. We also obtain good estimates on the change in geometry (e.g. volumes and core geodesic lengths) during hyperbolic Dehn filling. This new harmonic deformation theory has also been used by Bromberg and his coworkers in their proofs of the Bers Density Conjecture for Kleinian groups.


Torische Ideale von Flusspolytopen

Abstract: In dieser Diplomarbeit werden einige Gradschranken f\"ur Erzeugendensysteme und Gr\"obnerbasen von torischen Idealen von Flusspolytopen bewiesen. Alle torischen Ideale von Flusspolytopen sind im Grad 3 erzeugt. Glatte (3x4)-Transportpolytope sind sogar im Grad 2 erzeugt. Die reduzierte Gr\"obnerbasis eines beliebigen (m \times n)-Transportpolytops bez\"uglich einer beliebigen umgekehrt lexikographischen Termordnung hat h\"ochstens Grad mn/2. Wir konstruieren auch ein Beispiel, f\"ur das diese Schranke ann\"ahernd scharf ist. ----- In this Diplomarbeit (Master's thesis), we prove some degree bounds for generating sets and Gr\"obner bases of toric ideals of flow polytopes. All toric ideals of flow polytopes are generated in degree three. Smooth (3x4)-transportation polytopes are even generated in degree 2. The reduced Gr\"obner basis of an arbitrary (m \times n)-transportation polytope with respect to an arbitrary reverse lexikographic term order has at most degree mn/2. We also construct an example, for which this bound is almost sharp.


On Popoviciu type tormulas for generalized restricted partition function

Abstract: Suppose that $a_1(n),a_2(n),...,a_s(n),m(n)$ are integer-valued polynomials in $n$ with positive leading coefficients. This paper presents Popoviciu type formulas for the generalized restricted partition function $$p_{A(n)}(m(n)):=#\{(x_1,...,x_s)\in \mathbb{Z}^{s}: all x_j\geqslant 0, x_1a_1(n)+...+x_sa_s(n)=m(n) \}$$ when $s=2$ or 3. In either case, the formula implies that the function is an integer-valued quasi-polynomial. The main result is proved by a reciprocity law for a class of fractional part sums and the theory of generalized Euclidean division.


A reciprocity map and the two variable p-adic L-function

Abstract: For primes p greater than 3, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of p-units to what is essentially the two-variable p-adic L-function of Mazur and Kitagawa.


Classical elliptic current algebras

Abstract: In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras on a complex torus leading to two different elliptic current algebras. Quantization of these classical current algebras give rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez-Felder-Rubtsov and Arnaudon-Buffenoir-Ragoucy-Roche-Jimbo-Konno-Odake-Shiraishi. Different degenerations of the classical elliptic algebras are considered. They yield different versions of rational and trigonometric current algebras. We also review the averaging method of Faddeev-Reshetikhin, which allows to restore elliptic algebras from the trigonometric ones.


Ubiquitous systems and metric number theory

Abstract: We investigate the size and large intersection properties of $$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|<{r_{i}}^t\text{for infinitely many}(i,k)\in I^{\mu,\alpha}\times\Z^d\},$$ where $d\in\N$, $t\geq 1$, $I$ is a denumerable set, $(x_{i},r_{i})_{i\in I}$ is a family in $[0,1]^d\times (0,\infty)$ and $I^{\mu,\alpha}$ denotes the set of all $i\in I$ such that the $\mu$-mass of the ball with center $x_{i}$ and radius $r_{i}$ behaves as ${r_{i}}^\alpha$ for a given Borel measure $\mu$ and a given $\alpha>0$. We establish that the set $E_{t}$ belongs to the class $\grint^h(\R^d)$ of sets with large intersection with respect to a certain gauge function $h$, provided that $(x_{i},r_{i})_{i\in I}$ is a heterogeneous ubiquitous system with respect to $\mu$. In particular, $E_{t}$ has infinite Hausdorff $g$-measure for every gauge function $g$ that increases faster than $h$ in a neighborhood of zero. We also give several applications to metric number theory.


Singularity sets of Levy processes

Abstract: We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well.


Random wavelet series based on a tree-indexed Markov chain

Abstract: We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.


Random fractals and tree-indexed Markov chains

Abstract: We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.


Liouville theorems for the Navier-Stokes equations and applications

Abstract: We study bounded ancient solutions of the Navier-Stokes equations. These are the solutions which are defined for all past time. In two space dimensions we prove that such solutions are either constant or functions of time only, depending on the exact definition of admissible solutions. The general three dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axi-symmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions.


Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups

Abstract: Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfaces introduced earlier are not actually surfaces. A simple example of seamed surface is book-like seamed surface: several rectangles glued by edges like sheets in a book. We prove that Hurwitz numbers for a class of regular coverings with action of fixed finite group $G$ on cover space such that stabilizers of generic points are conjugated to a fixed subgroup $K\subset G$ defines a new example of Klein Topological Field Theory (KTFT). It is known that KTFTs are in one-to-one correspondence with certain class of algebras, called in the work Cardy-Frobenius algebras. We constructed a wide class of Cardy-Frobenius algebras, including particularly all Hecke algebras for finite groups. Cardy-Frobenius algebras corresponding to regular coverings of surfaces by seamed surfaces are described in terms of group $G$ and its subgroups. As a result, we give an algebraic formula for introduced Hurwitz numbers.


On orbit closures for infinite type quivers

Abstract: For the Kronecker quiver, Zwara has found an example of a representation whose orbit closure is neither unibranch nor Cohen-Macaulay. In this note, we explain how to extend this example to all infinite type quivers without oriented cycles.


Lecture Notes on Equivariant Cohomology

Abstract: These are the lecture notes for the introductory graduate course I taught at Yale during Spring 2007. I mostly followed [GS], [BGV], [AB], [Par2], and there are no original results in these notes.


On maps which preserve equality of distance in F-spaces

Abstract: In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries are characterized through a simple property of their range.


\ell-adic class field theory for regular local rings

Abstract: In this paper, we prove the $\ell$-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is a $\ell$-adic variant of a result of Matsumi [13].


A Functorial Approach to the Infinitesimal Theory of Groupoids

Abstract: Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion of algebraic theory and Ehresmann's functorial notion of theory called sketches. We study totally intransitive Nishimura algebroids in detail. Finally we show that Nishimura algebroids naturally give rise to Lie algebroids.


A convenient category of locally preordered spaces

Abstract: As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits.


On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals

Abstract: We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry integral" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the "classical" level, up to level of distribution, say, lambda=log Q/log N < 2/3 (for enough large h). This time we don't apply Large Sieve inequality, as in our paper [C-S]. Precisely, our method is completely elementary.


A Support Theorem For The Radiation Fields On Asymptotically Euclidean Manifolds

Abstract: We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform on n-dimensional Euclidean space.


Einstein's equations and the embedding of 3-dimensional CR manifolds

Abstract: We prove several theorems concerning the connection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dimensional CR manifold defined by the fields. Using the reduced Einstein equations we construct two independent CR functions for the corresponding CR manifold. We also point out that the Einstein equations, imposed on spacetimes associated with a 3-dimensional CR manifold, imply that the spacetime metric, after an appropriate rescaling, becomes well defined on a circle bundle over the CR manifold. The circle bundle itself emerges as a consequence of Einstein's equations.


Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

Abstract: We analyze a class of weakly differentiable vector fields (\FF \colon \rn \to \rn) with the property that (\FF\in L^{\infty}) and (\div \FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field $\FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we establish a fundamental approximation theorem which states that, given a Radon measure $\mu$ that is absolutely continuous with respect to $\mathcal{H}^{N-1}$ on $\rn$, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure $\|\mu\|$. With this approximation theorem, we derive the normal trace of $\FF$ on the boundary of any set of finite perimeter, (E), as the limit of the normal traces of $\FF$ on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for $\FF$ holds on (E). With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an $(N-1)$-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.


Haagerup's Approximation Property and Relative Amenability

Abstract: A finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $% \tau $ has Haagerup's approximation property (relative to a von Neumann subalgebra $\mathcal{N}$) if there exists a net $(\phi_{\alpha})_{\alpha\in \Lambda}$ of normal completely positive ($\mathcal{N}$-bimodular) maps from $\mathcal{M}$ to $\mathcal{M}$ that satisfy the subtracial condition $% \tau \circ \phi_{\alpha}\leq \tau $, the extension operators $% T_{\phi_{\alpha}}$ are bounded compact operators (in $<\mathcal{M%},e_{\mathcal{N}}>$), and pointwise approximate the identity in the trace-norm, i.e., $\lim_{\alpha}||\phi_{\alpha}(x)-x||_{2}=0$ for all $% x\in \mathcal{M}$. We prove that the subtraciality condition can be removed, and provide a description of Haagerup's approximation property in terms of Connes's theory of correspondences. We show that if $\mathcal{N}\subseteq \mathcal{M}$ is an amenable inclusion of finite von Neumann algebras and $% \mathcal{N}$ has Haagerup's approximation property, then $\mathcal{M}$ also has Haagerup's approximation property. This work answers two questions of Sorin Popa.


Curvature estimates for minimal submanifolds of higher codimension

Abstract: We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman's result for the Bernstein type theorem.


Flatness-based pre-compensation of laser diodes

Abstract: A physical nonlinear dynamical model of a laser diode is considered. We propose a feed-forward control scheme based on differential flatness for the design of input-current modulations to compensate diode distortions. The goal is to transform without distortion a radio-frequency current modulation into a light modulation leaving the laser-diode and entering an optic fiber. We prove that standard physical dynamical models based on dynamical electron and photons balance are flat systems when the current is considered as control input, the flat output being the photon number (proportional to the light power). We prove that input-current is an affine map of the flat output, its logarithm and their time-derivatives up to order two. When the flat output is an almost harmonic signal with slowly varying amplitude and phase, these derivatives admit precise analytic approximations. It is then possible to design simple analogue electronic circuits to code approximations of the nonlinear computations required by our flatness-based approach. Simulations with the parameters of a commercial diode illustrate the practical interest of this pre-compensation scheme and its robustness versus modelling and analogue implementation errors.


The Chow rings of the algebraic groups E_6, E_7, and E_8

Abstract: We determine the Chow rings of the complex algebraic groups of the exceptional type E_6, E_7, and E_8, giving the explicit generators represented by the pull-back images of Schubert varieties of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SO_n, Spin_n, G_2, and F_4. Our method is based on Schubert calculus of the corresponding flag varieties, which has its own interest.


First Steps in Tropical Intersection Theory

Abstract: We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n.


Criteria for homotopic maps to be so along monotone homotopies

Abstract: The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.


Pulsating travelling fronts: Asymptotics and homogenization regimes

Abstract: This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations with Kolmogrov-Petrovsky-Piskunov (KPP) type nonlinearities in general periodic domains or in infinite cylinders with oscillating boundaries. Having a variational formula for the minimal speed of propagation involving eigenvalue problems ( proved in Berestycki, Hamel and Nadirashvili \cite{BHN1}), we consider the minimal speed of propagation as a function of diffusion factors, reaction factors and periodicity parameters. There we study the limits, the asymptotic behaviors and the variations of the considered functions with respect to these parameters. The last section treats a homogenization problem as an application of the results in the previous sections in order to find the limit of the minimal speed when the periodicity cell is very small.


Functorial Cartier duality

Abstract: In this paper we obtain the Cartier duality for k-schemes of commutative monoids functorially without providing the vector spaces of functions with a topology, generalizing a result for finite commutative algebraic groups by M. Demazure and P. Gabriel.


Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers

Abstract: This report is dedicated to the construction and analysis of so-called Generalized Impedance Boundary Conditions (GIBCs) used in electromagnetic scattering problems from imperfect conductors as higher order approximations of a perfect conductor condition. We consider here the 3-D case with Maxwell equations in a harmonic regime. The construction of GIBCs is based on a scaled asymptotic expansion with respect to the skin depth. The asymptotic expansion is theoretically justified at any order and we give explicit expressions till the third order. These expressions are used to derive the GIBCs. The associated boundary value problem is analyzed and error estimates are obtained in terms of the skin depth.


On O^*-representability and C^*-representability of *-algebras

Abstract: Characterization of the *-subalgebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. The applications to *-doubles, monomial *-algebras and several other classes of *-algebras are presented.


The Homogeneous Approximation Property and the Comparison Theorem for Coherent Frames

Abstract: We show that the homogeneous approximation property and the comparison theorem hold for arbitrary coherent frames. This observation answers some questions about the density of frames that are not covered by the theory of Balan, Casazza, Heil, and Landau. The proofs are a variation of the method developed by Ramanathan and Steger.


An elementary trigonometric equation

Abstract: A systematic study of the trigonometric equation A tan a + B sin b = C, where A, B and C^2 are rational numbers. The special case tan Pi/11 + 4 sin 3 Pi/11 = sqrt[11] appears in the classical literature.


Chen's double sieve, Goldbach's conjecture and the twin prime problem, 2

Abstract: For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).


1953: An unrecognized summit in human genetic linkage analysis

Abstract: This paper summarizes and discusses the methodological research in human genetic linkage analysis, leading up to and following from the paper of C. A. B. Smith presented as a Royal Statistical Society discussion paper in 1953. This paper was given as the Fisher XXVII Memorial Lecture, in Cambridge, December 4, 2006.


On Deformations of Pasting Diagrams

Abstract: We adapt the work of Power to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in $k$-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack, proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov.


A Field of Generalised Puiseux Series for Tropical Geometry

Abstract: In this paper we define a field K of characteristic zero with valuation whose value group is the real numbers, and we show that this field of generalised Puiseux series is algebraically closed and complete with respect to the norm induced by its valuation. We consider this field to be a good candidate for the base field for tropical geometry.


The j-invariant of a plane tropical cubic

Abstract: Several results in tropical geometry have related the j-invariant of an algebraic plane curve of genus one to the cycle length of a tropical curve of genus one. In this paper, we prove that for a plane cubic over the field of Puiseux series the negative of the generic valuation of the $j$-invariant is equal to the cycle length of the tropicalization of the curve, if there is a cycle at all.


Geometric incidence theorems via Fourier analysis

Abstract: We prove that non-trivial bounds for generalized Radon transforms imply correspondingly non-trivial discrete incidence theorems for manifolds and suitably regular point sets.


On the path structure of a semimartingale arising from monotone probability theory

Abstract: Let $X$ be the unique normal martingale such that $X_0=0$ and \[\mathrm{d}[X]_t=(1-t-X_{t-}) \mathrm{d}X_t+\mathrm{d}t\] and let $Y_t:=X_t+t$ for all $t\geq 0$; the semimartingale $Y$ arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of $Y$ are examined and various probabilistic properties are derived; in particular, the level set $\{t\geq 0\dvt Y_t=1\}$ is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of $Y$ are found to be trivial except for that at level 1; consequently, the jumps of $Y$ are not locally summable.


On Anosov diffeomorphisms with asymptotically conformal periodic data

Abstract: We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+\beta} smoothness of the Anosov splitting, and C^1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of a torus. As a corollary we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.


Global Structure of Locally Convex Hypersurfaces in Finsler-Hadamard Manifolds

Abstract: Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures


Morse theory and conjugacy classes of finite subgroups

Abstract: We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements.


Complete Reducibility and Separability

Abstract: Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module. Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and H is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.


Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes

Abstract: We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the mxn non-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.


New Digit Results and Several Problems

Abstract: We give some new relations for Newman digit sums respectively different modulos and put some problems. In particular, for the odd prime modulos we put an important conjecture.


An embedding theorem for automorphism groups of Cartan geometries

Abstract: We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an automorphism group; a result asserting local homogeneity and completeness of parabolic geometries admitting a maximal-rank group of automorphisms; and a local freeness theorem for actions additionally preserving a continuous volume form.


Estimating copula measure using ranks and subsampling: a simulation study

Abstract: We describe here a new method to estimate copula measure. From N observations of two variables X and Y, we draw a huge number m of subsamples (size n<N), and we compute the joint ranks in these subsamples. Then, for each bivariate rank (p,q) (0<p,q<n+1), we count the number of subsamples such that there exist an observation of the subsample with bivariate rank (p,q). This counting gives an estimate of the density of the copula. The simulation study shows that this method seems to gives a better than the usual kernel method. The main advantage of this new method is then we do not need to choose and justify the kernel. In exchange, we have to choose a subsample size: this is in fact a problem very similar to the bandwidth choice. We have then reduced the overall difficulty.


A lower bound on the subriemannian distance for H\"older distributions

Abstract: Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, H\"older continuous) distributions. These distributions are of great significance for the behavior of the parent dynamical system. The study of H\"older subriemannian geometries could therefore offer new insights into both dynamics and subriemannian geometry. In this paper we make a small step in that direction: we prove a H\"older-type lower bound on the subriemannian distance for H\"older continuous nowhere integrable codimension one distributions. This bound generalizes the well-known square root bound valid in the smooth case.


Open-closed moduli spaces and related algebraic structures

Abstract: We set up a Batalin-Vilkovisky Quantum Master Equation for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory and study the L_\infty and A_\infty algebraic structures associated to it.


S-integral preperiodic points for dynamical systems over number fields

Abstract: Given a rational map $\phi: {\mathbb P}^1\to {\mathbb P}^1$ defined over a number field $K$, we prove a finiteness result for $\phi$-preperiodic points which are $S$-integral with respect to a non-preperiodic point $P$, provided $P$ satisfies a certain local condition at each place. This verifies a special case of a conjecture of S. Ih.


Logarithmic vector fields and hyperbolicity

Abstract: In this article we prove that the complement of a very generic curve of degree at least equal to 14 in the complex projective plane is hyperbolic in the sense of Kobayashi. Thus, using a new method, we improve the former known bound obtained by El Goul. We also improve the case of a very generic curve with two components.


Hyperbolicit\'e des vari\'et\'es complexes

Abstract: These notes were written for the Cours Peccot given at the Coll\`ege de France in 2007.


Reduce Problems From Braid Groups To Braid Monoids

Abstract: This paper proposes for every $n$, linear time reductions of the word and conjugacy problems on the braid groups $B_n$ to the corresponding problems on the braid monoids $B_n^+$ and moreover only using positive words representations.


Mass transport and variants of the logarithmic Sobolev inequality

Abstract: We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.


Linearization of skew-periodic loops and $\mathbb S^1$-cocycles

Abstract: We discuss linearization of skew-periodic loops. We generalize the situation to linearization of non-commutative loops and $\mathbb S^1$-cocycles.


Variations and estimators for the selfsimilarity order through Malliavin calculus

Abstract: Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\in(1/2,2/3)$.


Noncentral convergence of multiple integrals

Abstract: Fix $\nu>0$, denote by $G(\nu/2)$ a Gamma random variable with parameter $\nu/2$ and let $n\geq2$ be a fixed even integer. Consider a sequence $\{F_k\}_{k\geq1}$ of square integrable random variables belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\nu$. As $k\to\infty$, we prove that $F_k$ converges in distribution to $2G(\nu/2)-\nu$ if and only if $E(F_k^4)-12E(F_k^3)\to12\nu^2-48\nu$.


On Voevodsky's algebraic K-theory spectrum BGL

Abstract: Under a certain normalization assumption we prove that the $\Pro^1$-spectrum $\mathrm{BGL}$ of Voevodsky which represents algebraic $K$-theory is unique over $\Spec(\mathbb{Z})$. Following an idea of Voevodsky, we equip the $\Pro^1$-spectrum $\mathrm{BGL}$ with the structure of a commutative $\Pro^1$-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over $\Spec(\mathbb{Z})$. For an arbitrary Noetherian scheme $S$ of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on $\mathrm{BGL}$. This monoidal structure is relevant for our proof of the motivic Conner-Floyd theorem. It has also been used by Gepner and Snaith to obtain a motivic version of Snaith's theorem.


Fast stable direct fitting and smoothness selection for Generalized Additive Models

Abstract: Existing computationally efficient methods for penalized likelihood GAM fitting employ iterative smoothness selection on working linear models (or working mixed models). Such schemes fail to converge for a non-negligible proportion of models, with failure being particularly frequent in the presence of concurvity. If smoothness selection is performed by optimizing `whole model' criteria these problems disappear, but until now attempts to do this have employed finite difference based optimization schemes which are computationally inefficient, and can suffer from false convergence. This paper develops the first computationally efficient method for direct GAM smoothness selection. It is highly stable, but by careful structuring achieves a computational efficiency that leads, in simulations, to lower mean computation times than the schemes based on working-model smoothness selection. The method also offers a reliable way of fitting generalized additive mixed models.


A Quiver Presentation for Solomon's Descent Algebra

Abstract: The descent algebra $\Sigma(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $\Sigma(W)$ is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct $\Sigma(W)$ as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of $S$, the set of simple reflections in $W$. From this construction we obtain some general information about the quiver of $\Sigma(W)$ and an algorithm for the construction of a quiver presentation for the descent algebra $\Sigma(W)$ of any given finite Coxeter group $W$.


Groebner bases for spaces of quadrics of codimension 3

Abstract: Let $R=\oplus_{i\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\dim R_2\leq 3$ and that $K$ is algebraically closed of characteristic $\neq 2$. We show that $R$ is defined by a Gr\"obner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.


Blind Minimax Estimation

Abstract: We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, one does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares estimator, i.e., they achieve lower mean-squared error for any value of the parameter vector. Both Stein's estimator and its positive-part correction can be derived within the blind minimax framework. Furthermore, our approach can be readily extended to a wider class of estimation problems than Stein's estimator, which is defined only for white noise and non-transformed measurements. We show through simulations that the BMEs generally outperform previous extensions of Stein's technique.


Homotopy nilpotent groups

Abstract: We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.


On tameness and growth conditions

Abstract: We study discrete subsets of C^d, relating "tameness" with growth conditions.


On Meromorphic Functions which are Brody Curves

Abstract: We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P_1, i.e., which have bounded spherical derivative.


Discrete behavior of Seshadri constants on surfaces

Abstract: Working over C, we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constant for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on P^2 and new results about ample divisors on blow ups of P^2 at general points.


Lyapunov Coefficients for Degenerate Hopf Bifurcations

Abstract: In this paper are studied the codimensions one, two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients. Algebraic expressions obtained with computer assisted calculations are displayed.


Intersecting Psi-classes on tropical M_{0,n}

Abstract: We apply the tropical intersection theory developed by L. Allermann and J. Rau to compute intersection products of tropical Psi-classes on the moduli space of rational tropical curves. We show that in the case of zero-dimensional (stable) intersections, the resulting numbers agree with the intersection numbers of Psi-classes on the moduli space of n-marked rational curves computed in algebraic geometry.


Combinatorial Gelfand Models

Abstract: A combinatorial construction of a Gelafand model for the symmetric group and its Iwahori-Hecke algebra is presented.


On Yamamuro's inverse and implicit function theorems in terms of calibrations

Abstract: For the Frechet space E=C^{\infty}(S^1) and for a smooth \phi: R to R, we prove that the associated map E to E given by x mapsto\phi\circ x satisfies the continuous B\Gamma--differentiability condition in Yamamuro's inverse function theorem only if \phi is affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps.


Plurisubharmonic polynomials and bumping

Abstract: We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain \Omega\subset C^n in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with bdy(\Omega), at the site of the bumping, are explicitly realised. Generally, when \Omega\subset C^n, n\geq 3, the known methods lead to bumpings with high orders of contact -- which are not explicitly known either -- at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C^3. These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity.


The circular law for random matrices

Abstract: We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.


Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces

Abstract: In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group.


Face enumeration - from spheres to manifolds

Abstract: We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes and the toric $h$-vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup's 3-dimensional constructions \cite{Wal}, allow us to give a complete characterization of the $f$-vectors of arbitrary simplicial triangulations of $S^1 \times S^3, \C P^2,$ $ K3$ surfaces, and $(S^2 \times S^2) # (S^2 \times S^2).$ We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the $g$-conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.


Automatically reduced degenerations of automatically normal varieties

Abstract: Let F be a flat family of projective schemes, whose geometric generic fiber is reduced and irreducible. We give conditions on a special fiber (a "limit" of the family) to guarantee that it too is reduced. These conditions often imply also that the generic fiber is normal. The conditions are particularly easy to check in the setup of a "geometric vertex decomposition" [Knutson-Miller-Yong '07]. The primary tool used is the corresponding limit _branchvariety_ [Alexeev-Knutson '06], which is reduced by construction, and maps to the limit subscheme; our technique is to use normality to show that the branchvariety map must be an isomorphism. As a demonstration, we give an essentially naive proof that Schubert varieties in finite type are normal and Cohen-Macaulay. The proof does not involve any resolution of singularities or cohomology-vanishing techniques (e.g. appeal to characteristic p).


Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

Abstract: A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\bdS$ is positively curved, and necessary condition: $\bdS$ has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on $\bdS$ where they vanish. The gaps between them are $\bdS$ having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ are sos-concave. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\pt_cT$, the maximum subset of $\pt T$ contained in $\pt \cv{T}$. We prove sufficient conditions for SDP representability: $\pt_cT$ is positively curved, and necessary conditions: $\pt_cT$ has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between them are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.


On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

Abstract: The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim X-1}{2}\rceil+cd(\pi_1(X)) for every complex $X$ with $cd(\pi_1(X))\le 2$.


Infinite products with strongly $B$-multiplicative exponents

Abstract: Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P :=\prod_{n \geq 0} (\frac{2n+1}{2n+2})^{(-1)^{N_{1,2}(n)}}$ and Robbins proved that $P = 1/\sqrt{2}$. Related products were studied by several authors. We show that a trick for proving that $P^2 = 1/2$ (knowing that $P$ converges) can be extended to evaluating new products with (generalized) strongly $B$-multiplicative exponents. A simple example is $$ \prod_{n \geq 0} (\frac{Bn+1}{Bn+2})^{(-1)^{N_{1,B}(n)}} = \frac{1}{\sqrt B}. $$


On Log Canonical Models of the Moduli Space of Stable Pointed Curves

Abstract: We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces.


A Generalization of De Vries Duality Theorem

Abstract: Generalizing Duality Theorem of H. de Vries, we define a category which is dually equivalent to the category of all locally compact Hausdorff spaces and all perfect maps between them.


Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation

Abstract: We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only if it is stable. We define consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical differentiation, numerical integration using quadrature rules and Monte Carlo integration.


Boring split links

Abstract: Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces in the exteriors of knots and links obtained by boring a split link. It is shown, for example, that if the boring operation is complicated enough, a split link or unknot cannot be obtained by boring a split link. Particular attention is paid to rational tangle replacement. If a knot is obtained by rational tangle replacement on a split link, and a few minor conditions are satisfied, the number of boundary components of a meridional planar surface is bounded below by a number depending on the distance of the rational tangle replacement. This result is used to give new proofs of two results of Eudave-Mu\~noz and Scharlemann's band sum theorem.


Locally toric manifolds and singular Bohr-Sommerfeld leaves

Abstract: When geometric quantization is applied to a manifold using a real polarization which is "nice enough", a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld leaves. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld leaves when studying the quantization of manifolds which are less "nice". In this paper, we examine the quantization of compact symplectic manifolds that can locally be modelled by a toric manifold, using a real polarization modelled on fibres of the moment map. We compute the results directly, and obtain a theorem similar to Sniatycki's, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kaehler polarization.


Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Abstract: Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.


The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

Abstract: We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component.


Stability of foliations induced by rational maps

Abstract: We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space $\mathbb P^r$, when $1\le q \le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.


Local Floer Homology and the Action Gap

Abstract: In this paper, we study the behavior of the local Floer homology of an isolated fixed point and the growth of the action gap under iterations. To be more specific, we prove that an isolated fixed point of a Hamiltonian diffeomorphism remains isolated for a certain class of iterations (the so-called admissible iterations) and that the local Floer homology groups for all such iterations are isomorphic to each other up to a shift of degree. Furthermore, we study the pair-of-pants product in local Floer homology, and characterize a particular class of isolated fixed points (the symplectically degenerate maxima), which plays an important role in the proof of the Conley conjecture. The proofs of these facts rely on an observation that for a general diffeomorphism, not necessarily Hamiltonian, an isolated fixed point remains isolated under all admissible iterations. Finally, we apply these results to show that for a quasi-arithmetic sequence of admissible iterations of a Hamiltonian diffeomorphism with isolated fixed points the minimal action gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem is a generalization of the Conley conjecture.


Homological flat dimensions

Abstract: For finitely generated module $M$ over a local ring $R$, the conventional notions of complete intersection dimension $\cid_R M$ and Cohen-Macaulay dimension $\cmdim_R M$ do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which for finitely generated modules, coincide with the corresponding classical ones.


Upper bounds of Hilbert coefficients and Hilbert functions

Abstract: Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring. In this note we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a $m$-primary ideal $I\subset R$ that improves all known upper bounds unless for a finite number of cases. We also provide new upper bounds of the Hilbert functions of $I$ extending the known bounds for the maximal ideal.


An Extension of Mok's Theorem on the Generalized Frankel Conjecture

Abstract: In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.


The Infinite in Sciences and Arts

Abstract: Actual infinity in its various forms is discussed, searched and not found.


Capacitary estimates of solutions of semilinear parabolic equations

Abstract: We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via Bessel capacities.


Factorization of Laurent series over commutative rings

Abstract: We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.


Cyclotomy Primality Proofs and their Certificates

Abstract: The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007.


Dual Elliptic Primes and Applications to Cyclotomy Primality Proving

Abstract: Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testing, some interest remains for the practical aspect of state of the art implementable proving algorithms.