Showing papers by "Institut Élie Cartan de Lorraine published in 2005"

Sommes des chiffres de multiples d'entiers

Abstract: Soit q ∈ N, q ≥ 2. Pour n ∈ N, on note s q (n) la somme des chiffres de n en base q. Nous donnons des majorations de sommes d'exponentielles de la forme G(x,y,θ;α,h) = Σ x

A Spillover Phenomenon in the Optimal Location of Actuators

Abstract: In this paper, we are interested in finding the optimal location and shape of the actuators in a stabilization problem. Namely, we consider the one-dimensional wave equation damped by an internal feedback supported on a subdomain $\omega$ of given length. The criterion we want to optimize represents the rate of decay of the total energy of the system. It theoretically involves all the eigenmodes of the operator. From an engineering point of view, it seems more realistic to consider only a finite number of modes, say the N first ones. In that context, we are able to prove existence and uniqueness of an optimal domain $\omega_N^*$: it is the better possible location for the actuators. We characterize this optimal domain and we point out the following strange phenomenon (at least for small lengths): the optimal domain $\omega_N^*$ which is the better one for the N first modes is actually the worse one for the N + 1th mode. This looks like the well-known spillover phenomenon in control theory. At last, we will give some possible extension and open problems in higher dimension.

Coexistence in two-type first-passage percolation models

Abstract: We study the problem of coexistence in a two-type competition model governedby first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation.Actually, we prove for a large class of ergodic stationary passage times that for distinct points $x,y\in\Zd$, there is a strictly positive probability that$\{z\in\Zd;d(y,z)d(x,z)\}$ are both infinite sets.We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by H{a}ggstr{o}m and Pemantle forindependent exponential times on the square lattice.

Convergence of the Lagrange--Galerkin Method for the Equations Modelling the Motion of a Fluid-Rigid System

Abstract: In this paper, we consider a Lagrange--Galerkin scheme to approximate a two-dimensional fluid-rigid body problem. The equations of the system are the Navier--Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme.

Bessel functions for root systems via the trigonometric setting

Abstract: By taking an appropriate limit, we obtain the Bessel functions related to root systems as limit of Heckman-Opdam hypergeometric functions . A more general class of Bessel functions is also investigated, which we shall call the \Theta-Bessel functions. Explicit formuals for the \Theta-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of non-compact causal symmetric spaces, were an integral representation for these special functions is shown.

Liapunov functions and stability in control theory. Second edition

Abstract: This book presents a modern and self-contained treatment of the Liapunov method for stability analysis, in the framework of mathematical nonlinear control theory. A Particular focus is on the problem of the existence of Liapunov functions (converse Liapunov theorems) and their regularity, whose interest is especially motivated by applications to automatic control. Many recent results in this area have been collected and presented in a systematic way. Some of them are given in extended, unified versions and with new, simpler proofs. In the 2nd edition of this successful book several new sections were added and old sections have been improved, e.g., about the Zubovs method, Liapunov functions for discontinuous systems and cascaded systems. Many new examples, explanations and figures were added making this book accessible and well readable for engineers as well as mathematicians.

Taille des valeurs de fonctions L de carrés symétriques au bord de la bande critique

Abstract: On prouve l'existence, pour tout poids $k$ et tout niveau $N$ sans facteur carre et sans petit facteur premier, de formes primitives $f_+$ et $f_-$ de poids $k$ et de niveau $N$ telles que $$ L(1,{\rm sym}^2f_+)\gg_{k}[\log\log(3N)]^{3} \quad{\rm et}\quad L(1,{\rm sym}^2f_-)\ll_{k}[\log\log(3N)]^{-1}. $$ L'existence de ces formes est deduite d'une etude minutieuse des moments de $L(1,{\rm sym}^2f)$. Cette etude permet aussi de tra{\^\i}ter le cas des niveaux sans facteur carre mais avec petits facteurs premiers. On en deduit un contre--exemple a l'equivalence entre moyennes harmonique et naturelle.

Sur certaines singularites non isolees d'hypersurfaces I

Abstract: The aim of this fisrt part is to introduce, for a rather large class of hypersurface singularities with 1 dimensionnal locus, the analog of the Brieskorn lattice at the origin (the singular point of the singular locus). The main results are the finitness theorem for the corresponding (a,b)-module obtained via Kashiwara's constructibility theorem, and non torsion results for a plane curve singularity (not nessarily reduced) and for the suspension of such non torsion cases with an isolated singularity.

Percolating paths through random points

Abstract: We prove consistency of four different approaches to formalizing the idea of minimum average edge-length in a path linking some infinite subset of points of a Poisson process. The approaches are (i) shortest path from origin through some m distinct points; (ii) shortest average edge-length in paths across the diagonal of a large cube; (iii) shortest path through some specified proportion ? of points in a large cube; (iv) translation-invariant measures on paths in R^d which contain a proportion ? of the Poisson points. We develop basic properties of a normalized average length function c(?) and pose challenging open problem

Spectres euclidiens et inhomogènes des corps de nombres

Abstract: Cette these vise a repondre a certaines questions relatives aux notions de spectres euclidien et inhomogene (pour la norme) d'un corps de nombres, et notamment a celles qui concernent son minimum euclidien. Nous etablissons que pour tout corps de nombres K, son minimum euclidien M(K), est egal a son minimum inhomogene, et que si le rang r du groupe des unites de K verifie r>1, les spectres euclidiens et inhomogenes de K sont egaux et rationnels lorsque K n'est pas CM. Les resultats que nous etablissons sous l'hypothese r>1, ont pour consequence la decidabilite de l'euclidianite de K pour la norme. Nous montrons egalement comment calculer explicitement M(K). Nous decrivons un algorithme pour le cas totalement reel, qui a permis de construire des tables jusqu'au degre 8, et nous indiquons comment le transposer a des corps de nombres quelconques. Cet algorithme a permis de trouver de nombreux exemples de corps de nombres principaux, non euclidiens pour la norme et euclidiens en 2 etapes.

Chevalley cohomology for Kontsevich's graphs

Abstract: We introduce the Chevalley cohomology for the graded Lie algebra of polyvector ?elds on Rd. This cohomology occurs naturally in the problem of construction and classi?cation of formalities on the space Rd. Considering only graph formalities, that is, formalities de?ned with the help of graphs as in the original construction of Kontsevich, we de?ne (as the ?rst and third authors did earlier for the Hochschild cohomology) the Chevalley cohomology directly on spaces of graphs. More precisely, observing ?rst a noteworthy property for Kontsevich?s explicit formality on Rd, we restrict ourselves to graph formalities with that property. With this restriction, we obtain some simple expressions for the Chevalley coboundary operator; in particular, we can write this cohomology directly on the space of purely aerial, nonoriented graphs. We also give examples and applications.

Modules de Brieskorn et formes hermitiennes pour une singularité isolée d'hypersurface.

Abstract: This article intends to give a synthetic survey about the canonical hermitian form of a germ of holomorphic function $f$ with an isolated singularity at the origin in $\Bbb C ^{n+1}$. The link between this canonical hermitian form, the hermitian Poincare duality on the Milnor ' fiber of $f$ and the variation map, proved by F. Loeser in [Lo.86] is presented in the (a,b)-module setting with purely local arguments.

An adaptive particle -in-cell method using multi-resolution analysis.

Abstract: In this paper, we introduce a new PIC method based on an adaptive multi-resolution scheme for solving the one dimensional Vlasov–Poisson equation. Our approach is based on a description of the solution by particles of unit weight and on a reconstruction of the density at each time step of the numerical scheme by an adaptive wavelet technique: the density is firstly estimated in a proper wavelet basis as a distribution function from the current empirical data and then “de-noised” by a thresholding procedure. The so-called Landau damping problem is considered for validating our method. The numerical results agree with those obtained by the classical PIC scheme, suggesting that this multi-resolution procedure could be extended with success to plasma dynamics in higher dimensions.

Ambient Dirac eigenvalue estimates and the Willmore functional

Abstract: We prove a lower bound for the Willmore functional with the help spinors and the Dirac operator. As a corollary we verify the Willmore conjecture in a certain region of the spin-conformal moduli space for tori with sufficiently small $L^p$-norms of the Gauss curvature.

Applications du calcul stochastique à l'étude de certains processus

Abstract: Ce document contient la synthese des travaux de recherche effectues entre 1996 et 2005, apres la these de doctorat de l'auteur, et concerne l'etude fine de certains processus stochastiques : mouvement brownien lineaire ou plan, processus de diffusion, mouvement brownien fractionnaire, solutions d'equations differentielles stochastiques ou d'equations aux derivees partielles stochastiques. La these d'habilitation s'articule en six chapitres correspondant aux themes suivants : etude des integrales par rapport aux temps locaux de certaines diffusions, grandes deviations pour un processus obtenu par perturbation brownienne d'un systeme dynamique depourvu de la propriete d'unicite des solutions, calcul stochastique pour le processus gaussien non-markovien non-semimartingale mouvement brownien fractionnaire, etude des formules de type Ito et Tanaka pour l'equation de la chaleur stochastique, etude de la duree de vie du mouvement brownien plan reflechi dans un domaine a frontiere absorbante et enfin, estimation non-parametrique et construction d'un test d'adequation a partir d'observations discretes pour le coefficient de diffusion d'une equation differentielle stochastique. Les approches de tous ces themes sont probabilistes et basees sur l'analyse stochastique. On utilise aussi des outils d'equations differentielles, d'equations aux derivees partielles et de l'analyse.

A Berry–Esseen theorem for Feynman–Kac and interacting particle models

Abstract: In this paper we investigate the speed of convergence of the fluctuations of a general class of Feynman-Kac particle approximation models. We design an original approach based on new Berry-Esseen type estimates for abstract martingale sequences combined with original exponential concentration estimates of interacting processes. These results extend the corresponding statements in the classical theory and apply to a class of branching and genealogical path-particle models arising in non linear filtering literature as well as in statistical physics and biology.

Sur les variétés de Fano obtenues par éclatement d'une courbe lisse dans une variété projective

Abstract: Cette these a pour but de classifier les varietes de Fano X (c'est-a-dire les varietes algebriques dont le fibre anticanonique est ample) obtenues par eclatement le long d'une courbe lisse C dans une variete projective complexe et lisse Y. D'apres la theorie de Mori, les varietes de Fano de nombre de Picard superieur ou egal a 2 possedent au moins deux morphismes dits contractions extremales. Dans notre situation, on peut assurer l'existence d'une contraction extremale dont les fibres rencontrent le diviseur exceptionnel de l'eclatement le long de C. Lorsque cette autre contraction est fibrante, on a une classification complete des pairs (Y,C) (en admettant une hypothese supplementaire dans le cas ou les fibres generales de la contraction sont de dimension 1). Si l'autre contraction est birationnelle, on n'a que des resultats partiels qui montrent neanmoins que des situations tres variees peuvent se produire.

Mathematical challenges in shape optimization

Abstract: This paper covers some theoretical investigations performed in France, in the framework of the CNRS programme GDR {\it Applications Nouvelles de l'Optimisation de Forme}. The programme included also some activities in Poland. We do not restrict the presentation to the French community in the research field, the list of references includes all recent monographs on the shape optimization. The outline of the paper is the following. First we present some main fields of the activity in shape optimization. To present some precise results, from mathematical point of view, we include two sections. The first is devoted to the eigenvalues, the second to the drag minimization. Many theoretical questions related to these problems are still open.

Un Critere D'Isotrivialite Pour Les Familles De Varietes Hyperkähleriennes Sans Facteur Algebrique

Abstract: We show the isotriviality of families of simply-connected hyperkahler manifolds without algebraic factor, and general tori,provided the total space of the family is compact Kahler.

Transformation de Poisson sur un arbre localement fini

Abstract: Dans cet article on etudie en premier lieu la resolvante (le noyau de Green) d'un operateur agissant sur un arbre localement fini. Ce noyau est suppose invariant par un groupe G d'automorphismes de l'arbre. On donne l'expression generique de cette resolvante et on etablit des simplifications sous differentes hypotheses sur G. En second lieu on introduit la transformation de Poisson qui associe a une mesure additive finie sur l'espace Ω des bouts de l'arbre une fonction propre de 1' operateur. On montre que la bijectivite de cette transformation se deduit de la non nullite de certains determinants et on montre celle-ci pour des cas assez generaux.

Semi-Riemannian Symmetric spaces

Abstract: Jordan Holder decompositions of representations and applications to orthogonal Lie algebras semi-riemannian symmetric spaces and holonomy problems.

A Stabilization Phenomenon for a Class of Stochastic Partial Differential Equations

Abstract: In this paper we investigate the long-time behavior of solutions to a class of semilinear, stochastic partial differential equations defined on a bounded domain D ⊂ R with smooth boundary ∂D and driven by an infinite-dimensional noise derived from an L(D)-valued Wiener process. We consider the case of a noise with nuclear covariance and we derive the almost sure convergence of solutions to one of the two possible asymptotic states, under some suitable conditions. We also present partial stabilization results regarding the case of a one-dimensional heat equation driven by a space-time white noise. Mathematics Subject Classification: 60H15, 35R60

A Berry-Esseen theorem for Feynman-Kac and interacting particle models

Abstract: In this paper we investigate the speed of convergence of the fluctuations of a general class of Feynman-Kac particle approximation models. We design an original approach based on new Berry-Esseen type estimates for abstract martingale sequences combined with original exponential concentration estimates of interacting processes. These results extend the corresponding statements in the classical theory and apply to a class of branching and genealogical path-particle models arising in nonlinear filtering literature as well as in statistical physics and biology.

Tangential deformations on fibred Poisson manifolds

Abstract: In a recent article, Cattaneo, Felder and Tomassini explained how the notion of formality can be used to construct flat Fedosov connections on formal vector bundles on a Poisson manifold M and thus a star product on M through the original Fedosov method for symplectic manifolds. In this paper, we suppose that M is a fibre bundle manifold equipped with a Poisson tensor tangential to the fibers. We show that in this case the construction of Cattaneo-Felder-Tomassini gives tangential star products.

Approximate controllability of a semi-discrete 1-D wave equation

Abstract: This article deals with the approximate boundary controllability problem for a semi-discrete 1-D wave equation. By using a Fourier method, we show that there exists a sequence of approximate controls for the semi-discrete wave equation which remains uniformly bounded when the mesh size tends to zero. These approximate controls will be explicitly constructed and estimates for their norms will be given in function of the discretization step. 2000 Mathematics Subject Classification. Primary 65M06; Secondary 35L05.

Percolating paths through random points

Abstract: We prove consistency of four different approaches to formalizing the idea of minimum average edge-length in a path linking some infinite subset of points of a Poisson process The approaches are (i) shortest path from origin through some $m$ distinct points; (ii) shortest average edge-length in paths across the diagonal of a large cube; (iii) shortest path through some specified proportion $\delta$ of points in a large cube; (iv) translation-invariant measures on paths in $\Reals^d$ which contain a proportion $\delta$ of the Poisson points We develop basic properties of a normalized average length function $c(\delta)$ and pose challenging open problem

Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds

Abstract: Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator D in the metric $\tilde{g}$. A previous result stated that $\lambda(M,[g],\si) \leq \lambda(\mS^n) =\frac{n}{2} \om_n^{{1/n}}$ where \om_n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n \geq 2 such that D is invertible. E.g. we show that strict inequality holds in dimension $n\equiv 0,1,2\mod 4$ if a certain endomorphism does not vanish. Because of its tight relations to the ADM mass in General Relativity, the endomorphism will be called mass endomorphism. We apply the strict inequality to spin-conformal spectral theory and show that the smallest positive Dirac eigenvalue attains its infimum inside the enlarged volume-1-conformal class of g.

The first eigenvalue of the Dirac operator in a conformal class

Abstract: In this overview article, we study the first positive eigenvalue of the Dirac operator in a unit volume conformal class. In particular, we discuss the question whether the infimum is attained. In the first part, we explain the corresponding variational problem. In the following parts we discuss the relation to the spinorial mass endomorphism and an application to surfaces of constant mean curvature. The article also mentions some open problems and work in progress.