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     

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).     

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.     

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 Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Un théorème de finitude (A Theorem of Finitude)

We prove that, for an inner form of GL _n , the number of classes of cuspidal automorphic representations with fixed central character and fixed factor at almost every place is finite. We also prove, in the local situation, relations between the level and the -factor of an irreducible smooth representation.     

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     

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 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.     

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

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Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

Une version non linéaire du théorème de représentation de Riesz

We give a necessary and sufficient condition for a nondecreasing functional on the set of the continuous nonnegative functions on a compact metrizable space to have a unique extension as a functional Choquet's capacity on the set of Borel nonnegative functions.