Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

Sur une décomposition topologique des fonctions de Sobolev et sur la localisation de l'énergie dans les plasmas idéaux

Motivated by properties of “frozen topological invariants” in ideal plasmas theory, we consider the problem of an intrinsic characterization of the weak closure in H₀¹ (Ω) of all functions ψ₀ ∘ ϕ, with ϕ : Ω → Ω a diffeomorphism. We show that one may associate with ψ₀ a class W^weak (ψ₀) of functions which captures robust topological properties, of the level sets of {ψ₀} and, moreover, is closed under weak limits.     

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H=(a,b)_F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over (H,τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h_K over K. This gives a map of Witt groups ρ:W⁻¹(H,τ)→W(K) induced by ρ(h)=h_K. When K is a generic splitting field of H we prove in this note that the map ρ is injective.     

Une méthode de décomposition en temps avec des schémas dʼintégration réversible pour la résolution de système dʼéquations différentielles ordinaires

We propose a time domain decomposition method that breaks the sequentiality of the integration scheme for systems of ODE. Under the condition of differentiability of the flow, we transform the initial value problem into a well-posed boundary values problem using the symmetrization of the interval of time integration and time-reversible integration scheme. For systems of linear ODE, we explicitly construct the block tridiagonal system satisfied by the solutions at the time sub-intervals extremities. We then propose an iterative algorithm of Schwarz type for updating the interfaces conditions which can extend the method to systems of nonlinear ODE.     

Théorie de la réduction des formes quadratiques définies positives dans un corps algébriqueK fini

Sans résumé     

Construction dʼun champ continu de métriques

Adaptive computation using adaptive meshes is now recognized as essential for solving complex PDE problems. This computation requires, at each step, the definition of a continuous metric field to govern the generation of the adapted meshes. In practice, via an appropriate a posteriori error estimation, metrics are calculated at the vertices of the computational domain mesh. In order to obtain a continuous metric field, the discrete field is interpolated in the whole domain mesh. In this Note, a new method for interpolating discrete metric fields, based on a so-called “natural decomposition” of metrics, is introduced. The proposed method is based on known matrix decompositions and is computationally robust and efficient. Some qualitative comparisons with classical methods are made to show the relevance of this methodology.     

Paires élémentaires de corps pseudo-finis: dénombrement des complétions

Let II be a complete theory of pseudo-finite fields. In this Note we prove that, in the langage of fields to which we add a unary predicate for a substructure, the theory of non-trivial elementary pairs of models of II has 2^No completions, that is, the maximum that could exist. For a more detailed presentation, see [5].     

Sur l'' application de réciprocité pour une surface fibrée en coniques définie sur un corps local

For a proper smooth variety X defined over a local field k, unramified class field theory investigates the reciprocity map _X: SK₁(X) ^ab ₁(X) as introduced by S. Saito. We study this map in the case when X is a surface admitting a proper surjection onto a smooth geometrically connected curve C with a smooth conic as generic fibre. Without any assumption on the reduction of C, we prove that _X is injective modulo n for all n invertible in k and its cokernel is the same as that of _C.     

Perturbation des résolvantes et représentation des mesures excessives

Résumé Etant donnée une résolvante V=(V )_>0 sous-markovienne sur un espace mesurable (X, B) de noyau initial V propre; on étudie alors le balayage des mesures surmédianes au moyen de la résolvante perturbée V de V par une function mesurable positive bornée sur X.Dans le cas où (X, E _v) est un espace de balayage, on montre que toute mesure excessive vérifiant , s'écrit d'une manière unique sous la forme =V où est une mesure positive sur X.     

Estimation de la fonction de régression pour variable explicative et réponse fonctionnelles dépendantes

We consider a kernel estimate of the regression when the response variable is in a Banach space and the explanatory variable takes its values in a semi-metric space. Our main result states the almost complete convergence (with rate) of the constructed estimate when the sample considered is a β-mixing sequence.     

L'estimateur de ≪ mélange progressif ≫ appliqué aux arbres de régression

We present a version of Catoni's “progressive mixture estimator” [3] suited for a general regression framework. Following basically Catoni's steps, we derive strong non-asymptotic upper bounds for the Kullback-Leibler risk in this framework. We give a more explicit form for this bound when the models considered are regression trees, present a modified version of the estimator in an extended framework and propose an approximate computation using a Metropolis algorithm.     

Sur les démonstrations arithmétiques dans la théorie du corps de classes

Sans résumé     

Lemmes de décomposition appliqués à des inégalités de type Hardy–Sobolev

Decomposition (or concentration-compactness) lemmas have already shown their efficience in order to show existence of minimizers or ground state solutions. The aim of this paper is to apply new version of these lemmas to minimisation problems involving Hardy–Sobolev type inequalities on a specific class of unbounded domains. More precisely, we shall find ground state solution for the following quotient, where value of real numbers ,b,q and are given.