Approximants de Padé simultanés de logarithmes

Simultaneous Padé-approximants of logarithms give simultaneous diophantine approximations for logarithms of rational numbers close to 1. Lower bounds for linear forms of logarithms with integer coefficients are derived.     

The computation of non-perfect Padé-Hermite approximants

We describe a simple and efficient algorithm to generate a number of polynomial vectors which can be used to describe all possible solutions for a type I Padé-Hermite problem. If denotes the order of approximation, which is a measure for the size of the Padé-Hermite problem, it uses only order ² operations, even if the given system is not perfect. To this end, the problem is considered as a special case of a generalized Padé-Hermite problem which is also defined and analysed.     

Un théorème de dualité

Sans résumé Herrn H. Hopf zum sechzigsten Geburtstag gewidmet     

Approximants de Padé et séries hypergéométriques équilibrées

In this paper, we present and solve some very general new Padé approximant problems, whose solutions can be expressed with hypergeometric series. These series appear in the proofs of the irrationality of ζ(3), of infinitely many ζ(2n+1), and in essentially all results of this kind in the literature. We also prove two new Diophantine results with this method.     

Le problème de Brill–Noether et le théorème de Teixidor

Let C be a smooth projective algebraic curve of genus g. The Brill–Noether problem is to determine at which conditions on n, d and k, there exists a stable vector bundle of rank n, degree d and with at least k independent global sections. Teixidor gave a rather general answer about the existence in her paper [Tei]. The result was available only for a generic curve, in a very imprecise sense, and the proof was really difficult. We show here, that the result is in fact true for all curves and a little more. Received: 5 September 1997 / Revised version: 15 June 1998     

Sur un théorème de prolongement fonctionnel de Keldych concernant le problème de Dirichlet

Sans résumé Ces recherches ont fait l'objet d'exposés, d'abord au Séminaire de théorie du potential (Paris, javier 1960).     

Homogénéisation d'un problème de capillarité

We study the behavior of the height of a liquid contained in a vertical cylindrical tube where are immerged a periodic distribution (period ɛ) of cylindrical fibers with radius r < ɛ and whose material-liquid coefficient is λ. It is showed that the homogenized problem depends on the limit of λr/ɛ² as (ɛ, λ, r) → (0,0, 0). The second problem is when the border is made of two alternate periodic bands, the first of length r being incrusted in the second of length ɛ − r. Here, the homogenized problem depends on the limit of r/ɛ as (, r) → (0, 0). For this, we use epi-convergence techniques.     

Sur un problème de Rényi

Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev _q (x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv _q (x); this improves on previous results.

     

Extension d'un théorème de Jacobi

Sans résuméExtrait d'une lettre adressée à M. Lerch.S. auch Journal f. Math., Bd. 72.     

Sur un théorème de Choquet

Résumé Le but de ce travail est de caractériser les éspaces topologiques qui possèdent la propriété suivant: pour toute famille(f _t ) _{t I} de fonctions numériques définies dansX il existe une partie dénombrableI ₀ de l'ensemble d'indicesI telle que les fonctions inf _tI f _i et inf _tI0 f _i ont la meme régularisation inférieurement semi-continue.On démontre que l'espace topologiqueX possède cette propriété si et seulement si pour tout sous-ensembleA deX il existe une sous-ensemble dénombrableA ₀,A ₀ A tel queA ₀.On généralise un résultat topologique de G. Choquet.

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Sur un Théorème de Steiner

Sans résumé     

Caractère bien posé du problème de Cauchy pour le système de Bérenger

The Bérenger perfectly matched layer is used in computational electromagnetism as an absorbing layer in scattering problems. It raises delicate mathematical issues. In this Note we show, for regular data, the existence and uniqueness of strong solutions to the Cauchy problem derived from the PML method. The result is presented in the 2-D case. The key to the proof is an appropriate control of a mixed H¹- L² norm of the solution by the same norm of the initial data. Beside a paper is in preparation about extensions of this results (L² estimates, 3-D case) (see also [5]).     

Sur un théorème de la théorie de la sommabilité

Sans résumé     

Théorème de Gabrielov et fonctions log-exp-algébriques

We give a new proof of Wilkie 's theorem on log-exp algebraic functions. The tools used here are “explicit” Gabrielov's theorem and our geometric presentation of the theorem of van den Dries, Macintyre, and Marker on log-exp analytic functions.