Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

Actions infinitésimales dans la correspondance de Langlands locale p-adique

Let V be a two-dimensional absolutely irreducible ${\overline{\mathbb Qp}}$ -representation of ${{\rm Gal}(\overline{\mathbb Qp}/\mathbb Qp)}$ and let ${\prod(V)}$ be the ${{\rm GL}_2(\mathbb Qp)}$ Banach representation associated by Colmez??s p-adic Langlands correspondence. We establish a link between the action of the Lie algebra of ${{\rm GL}_2(\mathbb Qp)}$ on the locally analytic vectors ${\prod(V)^{\rm an}}$ of ${\prod(V)}$ , the connection ${\nabla}$ on the ${(\varphi, \Gamma)}$ -module associated to V and the Sen polynomial of V. This answers a question of Harris, concerning the infinitesimal character of ${\prod(V)^{\rm an}}$ . Using this result, we give a new proof of a theorem of Colmez, stating that ${\prod(V)}$ has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge?CTate weights.     

Une propriété de continuité associée aux classes de cohomologie de Hodge

The obstructions, for a closed smooth differential form of bidimension $(p,p)$ on a projective manifold, to be cohomologous to an algebraic cycle with complex coefficients, are calculated in terms of the Chow transformation. They can be expressed as an orthogonality condition, on the manifold itself, with families parametrized by the Grassmannian of currents that are completely determined. Each of these currents is $\mathrm{d}{\mathrm{d}}^{\mathrm{c}}$-closed and with support in the intersection of the manifold and of the projective subspace associated with the parameter. By the theory of harmonic forms, a period is thus associated with that differential form for each parameter. We study the set of periods, obtained when the parameter varies, and we arrive at a continuity on the Grassmannian, when the cohomology class is rational. The same property can be obtained by going to the space of divisors of the Grassmannian and by using a characterization of Chow forms. We proceed here directly, by calculating the periods by means of the Atiyah–Hirzebruch theorem. This global continuity implies the orthogonality for all parameter.     

Estimation de la densité de probabilité améliorée par pre-testing

We consider the nonparametric problem of multidimensional probability density estimate. Using concept of minimax risk with random normalizing factor introduced by Lepski [Math. Methods Statist. 8 (1999) 441–486], by considering an independence hypothesis, we build an estimator which can be adaptive and whose accuracy, depending on the observation, is better than the minimax estimate, $n^{?\frac{\beta }{2\beta +d}}$, with prescribed confidence level. To cite this article: A.F. Yode, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur le problème d'équivalence de certaines structures infinitésimales

Résumé Après avoir rappelé la définition et quelques propriétés des pseudogroupes de Lie, on définit les structures infinitésimales régulières. On peut toujours associer des connexions affines à de telles structures; courbure et torsion de ces connexions. Equivalence locale de deux structures infinitésimales. Application à l'étude des structures presque complexes, presque paracomplexes, presque symplectiques, presque hermitiennes, presque parahermitiennes, presque quaternioniennes, presque quaternioniennes de deuxième espèce.     

Irrationalité de certaines sommes de séries

Résumé Let with |q| > 1, and a be a rational number such that a ² is not equal to for . In this note, we prove that the sum is irrational.     

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.     

Continuité de la densité de l'intégrale d'aire dans les espaces de Hardy

Nous estimons la dépendance en la fonction harmonique des fonctionnelles densité de l'intégrale d'aire. Nous obtenons notamment un contrôle en norme de sup _aR D u(,a)–D v(,a) ainsi qu'une inégalité aux bons- de type exponentielle.     

A propos de la propriété de Bergman

In a recent preprint, George M. Bergman has investigated the following property:for any generating set X of the group G there exists an integer n such that any element of G is a product of n elements of $\text{X}\cup {\text{X}}^{\mathrm{?1}}$. We will say in this case that G has the Bergman property.We have solved some of the questions asked in the above mentioned preprint and have found it suitable to investigate this property in a more general context, in particular for rings (essentially Boolean rings). To cite this article: A. Khelif, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

L'anneau de cohomologie d'une variété de Seifert

All manifolds M considered in this Note are orientable Seifert 3-manifolds with base surface S² and infinite fundamental group π₁ (M). Our goal is to compute the cohomology ring H^* (M; Z/2Z). The ring structure will enable us to determine whether M admits a degree 1 map into RP³ or not. We describe the equivariant chain complex for the universal cover M of M, and give a diagonal approximation. The cohomology ring H^* (M; Z/2Z) is computed.     

Variété de Picard d'une variété complète

Résumé On construit la variété dePicard d'une variété complète arbitraire U, c'est-à-dire, qu'on donne une structure canonique de variété de groupe au groupe des classes des diviseurs (localment principaux) de U algébriquement équivalents à zéro. à M. Enrico Bompiani pour son Jubilé scientifique.     

Test de primauté dans une extension de ore itérée

In this paper we prove a generalization of the primality test in [3] for a wide class of iterated Ore extensions of a commutative polynomial ring.     

Remarque sur la régularité locale de l'équation de Kadomtsev-Petviashvili-II

We give a short proof of a part of the Bourgain's result from [3]. More precisely, we prove that the Cauchy problem for the Kadomtsev-Petviashvili equation is locally well posed in Sobolev spaces with positive index.     

Sur l'inégalité logarithmique de Sobolev

We prove a strenghtened form of Gross's logarithmic Sobolev inequality (see [1]) with a “remainder term” added to the left-hand side. We use this result to determine the cases of equality on Rⁿ equipped with Lebesgue measure.     

Irrationalité de Valeurs Associées a l'Exponentielle de Carlitz

Lete and be the Carlitz-module analogues of their usual counterparts. We have proved in [4]-that these elements of are algebraically independent over whenq3. We study here the remaining caseq=2 and prove among other things that 1,e, are linearly independent over .

     

Régularité de la loi du maximum de processus gaussiens réguliers

Let X be a real-valued Gaussian process with a one-dimensional parameter. If the sample paths of X are of class C^2k, k integer and if the marginal distributions of X and its derivatives have no degeneracy, then the distribution function of the maximum on a bounded interval is of class C^k. We give a recurrence method allowing to express the derivatives by means of certain implicit formulae.