Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif

This work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group, defined over the field of algebraic numbers . Let G be such a group. Let W be a hyperplane of the tangent space at the origin of G, defined over , and u be a complex point of this tangent space, such that the image of u by the exponential map of the Lie group G(ℂ) is an algebraic point. Then we obtain a lower bound for the distance between u and W⊗ℂ, which improves the results known before and which is, in particular, the best possible for the height of the hyperplane W. The proof rests on Baker’s method and Hirata’s reduction as well as a new arithmetic argument (Chudnovsky’s process of variable change) which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof.

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Mathematics Subject Classification (2000) 11J86, 11J20, 14L10     

Parties semi-algébriques d'une variété algébrique <Emphasis Type="Italic">p</Emphasis>-adique

Let k be a field complete with respect to an ultrametric absolute value, let A be a finitely generated k-algebra and let X be its spectrum. We denote by X^an the analytification à la Berkovich of the algebraic variety X. We say that a subset of X^an is semi-algebraic if it can be defined by a boolean combination of inequalities |f||g| where f and g are in A, where is a symbol belonging to {<,>,,} and where is a positive real number. In this text we show that the image of any semi-algebraic subset under an algebraic map between affine varieties is semi-algebraic, and that a semi-algebraic subset has only finitely many connected components, each of which is semi-algebraic. In fact our results concern not only algebraic varieties, but also analytic families of algebraic varieties which are parametrized by an affinoid space.     

Un théorème de Harris et Segal pour un anneau d'entiers algébriques exceptionnel

Let K be a number field, O_K the ring of the integers of K, ℓ a prime integer and Z_(ℓ) the localisation of Z at ℓ. Harris and Segal [4] proved that there exists infinitely many primes p of O_K such that the natural morphism K_i(O_K) ⊗ Z_(ℓ) → K_i(O_K /p) ⊗ Z_(ℓ) in algebraic K-theory is split surjective for i > 0, except if ℓ = 1 and K is exceptional. In this Note, we prove that the Harris-Segal theorem is still true for ℓ = 2 in the exceptional case, if we replace algebraic K-theory by orthogonal K-theory defined by Karoubi [5]. Thanks to [3], we can then determine a direct summand of the 2-torsion of KO_n(O_K).     

Un théorème sur les séries de Dirichlet

We considerk Dirichlet series a _j (n)n ^–s (1jk),k2. We suppose that for eachj the series a _j (n)n ^–s converges fors=s _j =j+it _j , and that Max j<1/(k–1). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line es = (₁+...+ _k )/k+1–1/k and we estimate the rate of convergence. The number 1–1/k cannot be replaced by a smaller one.     

Une méthode de Γ-convergence pour un modèle de membrane non linéaire

We consider a cylindrical three-dimensional nonlinear hvperelastic body whose stored energy goes to infinity when the jacobian of the deformation gradient goes to zero. We show, under appropriate hypotheses on the applied loads, that the three-dimensional energies Γ-converge to the energy of a nonlinear membrane when the thickness of the cylinder goes to zero. The Γ-convergence takes place in the weak topology of a Sobolev space. As a consequence, we show that the sequence of minimizers also converges toward a minimizer of the limit energy.     

Sur l'unicité pour le système de Boussinesq avec diffusion non linéaire

In this article we prove global existence and uniqueness result for the 2D Boussinesq system with non linear thermal diffusion in some critical spaces.     

Un théorème de convergence presque-sŭre pour des familles subordonnées à un semi groupe discret

LetP be a positive contraction ofL ¹, we say that a sequence (Q _n ) _n of positive operators is subordinated toP if $$Q_0 = Id, \forall n \in \mathbb{N} Q_n P \geqslant Q_{n + 1} .$$ Let (Q _n,k ) be a double sequence of operators such that for each integerk the sequence (Q _n,k ) _n is subordinated toP. We prove a result of almost everywhere convergence for the quotientsW ^k f/W _k g, whereW ^k f=∑ _n=0 ^∞ Q _n,k f.     

Estimations a posteriori d’un schéma de volumes finis pour un problème non linéaire

Summary. RésuméDans cet article, nous développons des estimations a posteriori de lerreur commise en approchant un problème elliptique non linéaire, par une Méthode de Volumes Finis (MVF) de type Vertex-Centered. Les estimateurs obtenus mesurent, le résidu de léquation forte et lirrégularité de la solution discrète, qui se traduit par des sauts à travers les inter-élélments. Une condition sur la fonction flux numérique sera imposée pour traiter la non conformité du problème, au cours de lévaluation de lestimation. Summary.We derive a posteriori error estimates for vertex-centered finite volume discretizations of a class of non-linear elliptic pdes. The error estimates are of residual type and incorporate residuals of the pde in its strong form on the control volumes and jumps of the pdes prinicipal part across the element boundaries. The non-conformity of the discretization is taken into account by a condition on the numerical flux function.

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Mathematics Subject Classification (2000):35J60, 65N15, 65N30, 65N50, 74S10Recherche menée dans le cadre du projet AUPELF-FICU N° 2000/PAS/38 et du projet DAAD     

Existence et unicité de la solutions faible-renormalisée pour un système non linéaire de Boussinesq

We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq systems. We establish regularity results for the heat equation which we combine with the usual techniques for Navier–Stokes equations mixed with the tools involved for renormalized solutions. To cite this article: N. Bruyère, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Sur un théorème de Kronecker concernant les variétés algébriques

A classical result of Kronecker, stated at the end of the Section 10 of Kronecker (J. Reine Angew. Math. 92 (1882) 1–123), is that any radical of a finitely generated ideal in a polynomial ring of n variables is the radical of an ideal generated by n+1 elements. We give a constructive and elementary proof of a generalisation presented in (Michigan Math. J. 31 (1984) (2) 167–180): in a ring of Krull dimension $\text{?}\text{n}$ a radical of a finitely generated ideal is the radical of an ideal generated by n+1 elements. To cite this article: T. Coquand, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes

Résumé Nous étendons la méthode de démonstration du théorème de Berry-Esseen proposée par Bergström aux suites de variables aléatoires faiblement dépendantes. En particulier, nous montrons que, pour les suites stationnaires de variables aléatoires réelles bornées, la vitesse de convergence dans le théorème limite central en distance de Lévy est de l'ordre den ^–1/2 dès que la suite ( _p)p>0 des coefficients de mélange uniforme satisfait la condition _p>0 p _p <

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About the Berry-Esseen Theorem for weakly dependent sequences

We extend the method of Bergström for the rates of convergence in the central limit theorem to weakly dependent sequences. In particular, we prove that, for stationary and uniformly mixing sequences of real-valued and bounded random variables, the rate of convergence in the central limit theorem is of the order ofn ^–1/2 as soon as the sequence ( _p)p>0 of uniform mixing coefficients satisfies _p>0 p _p <.

     

Un problème mixte pour l’équation de Korteweg-de Vries sur un intervalle borné

The Korteweg-de Vries equation occurs as a model for unidirectional propagation of small amplitude long waves in numerous physical systems. The aim of this work is to propose a well posed mixed initial-boundary-value problem when the spacial domain is of finite extent. More precisely, we establish local existence of solutions for arbitrary initial data in the Sobolev space H¹, and global existence for small initial data in this space.     

Une approche intrinsèque d'un modèle non linéaire de la théorie des coques

We consider the model of a nonlinearly elastic “shallow” shell proposed by L.H. Donnell, V.Z. Vlasov, K.M. Mushtari & K.Z. Galimov, and W.T. Koiter. We show that the linearized change of curvature and nonlinear strain tensor fields appearing in the energy of this model can be taken as the sole unknowns of the problem, instead of the displacement field as is customary. In order to justify this “intrinsic approach” to this nonlinear model, we identify nonlinear compatibility conditions that these new unknowns must satisfy. These conditions are of Donati type, in the sense that they take the form of integral orthogonality relations against divergence-free tensor fields.