Uniformisation de l’espace des feuilles de certains feuilletages de codimension un

This paper deals with codimension one (may be singular) foliations on compact Kälher manifoldswhose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a “rigidly” embedded invariant hypersurface.     

Nombre de branchement d’un pseudogroupe

Lyons has defined an average number of branches per vertex of an infinite locally finite rooted tree. This number has an important role in several probabilistic processes such as random walk and percolation. In this paper, we extend the notion of branching number to any measurable graphed pseudogroup of finite type acting on a probability space. We prove that such a pseudogroup is Liouvillian (i.e. almost every orbit does not admit non-constant bounded harmonic functions) if its branching number is equal to 1. In order to prove that this actually generalizes results of C. Series and V. Kaimanovich on equivalence relations with polynomial and subexponential growth, we describe an example of minimal lamination whose holonomy pseudogroup has exponential growth and branching number equal to 1.     

Nombre de branchement d’un pseudogroupe

Lyons has defined an average number of branches per vertex of an infinite locally finite rooted tree. This number has an important role in several probabilistic processes such as random walk and percolation. In this paper, we extend the notion of branching number to any measurable graphed pseudogroup of finite type acting on a probability space. We prove that such a pseudogroup is Liouvillian (i.e. almost every orbit does not admit non-constant bounded harmonic functions) if its branching number is equal to 1. In order to prove that this actually generalizes results of C. Series and V. Kaimanovich on equivalence relations with polynomial and subexponential growth, we describe an example of minimal lamination whose holonomy pseudogroup has exponential growth and branching number equal to 1.     

Sur un probléme de Newman

We show that any polynomial P of degree n with coefficients ±1 satisfies the inequality. Under the same hypothesis, we show that the upper bound n + 0.8250041 holds for n large enough.     

Simulation de la propagation de fracture dans un solide élastique

We consider a brittle elastic solid (prone to develop fractures) as the limit of a damage model and propose a numerical method to determine its quasi-static evolution in the spirit of Francfort and Marigo [3] and Allaire et al. [1], [2].     

Sur la constante de Kummer-Leopoldt d’un corps de nombres

Kummers lemma for the cyclotomic field gives sufficient local conditions for a unit to be a global p-th power, and can thus be viewed as an early version of Leopoldts conjecture. To study generalizations and converse statements, we introduce the Kummer-Leopoldt constant (F) of a number field F and compute it in Iwasawa theoretic terms. The vanishing of (F) gives the desired generalization and converse of Kummers lemma.Mathematics Subject Classification(2000): Primary 11R23     

Décomposition de domaine pour un calcul hybride de l'équation de helmholtz

We prove an estimate for the Dirichlet-Neumann operator, and for the H¹ local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward     

Sur une note de Nagata relative à un problème de Krull

Sans résumé     

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.

     

Un critère de coupure de l'espace euclidien par un sous-ensemble arbitraire

Sans résuméA la mémoire deLeon Lichtenstein     

Sur l'étude de petites oscillations de deux liquides remplissant un domaine fixe de forme quelconque

Résumé On étudie un problme de valeurs propres concernant la vibration d'une interface séparant deux fluides non miscibles de densités différentes. Nous obtenons ici un théorème abstrait d'existence généralisant un résultat de J. Boujot [4]. Enfin, nous proposons une méthode d'approximation par pénalisation et nous donnons une estimation de l'erreur.

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We study an eigenvalue problem concerning the vibration of an interface separating two non miscible fluids with different densities. We obtain an abstract theorem of existence generalizing a result due to J. Boujot [4]. We propose a method of approximation by penalization and we give an estimation of the error.

     

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 problème de S. Ramanujan

In 1915, Srinivasa Ramanujan gives a lower limit for the maximum order of the iterated of the divisors function, $d(d(n))$. In 1989, Paul Erd?s and Aleksandar Ivi? give a upper bound. In this Note, we find the maximal order of the function $\omega (d(n))$, the number of prime divisors of $d(n)$, and we improve the result of Erd?s and Ivi? on the maximal order of $d(d(n))$. To cite this article: A. Smati, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Représentation intégrale de fonctionnelles convexes sur un espace de mesures

Résumé Partant d’un résultat abstrait de représentation intègrale pour une fonctionnelle convexe faiblement s.c.i. sur [M _b (Ω)] ^d (obtenu grace à [1]); on établit un résultat de convergence pour une suite de fonctionnelles du type μ _n εM _b ⁺ (Ω),f _n : Ω×R ^d →−∞, +∞] inté Quelques exemples motivés par des applications à la mécanique sont ensuite traités.

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Riassunto Partendo da un risultato astratto di rappresentazione integrale per un funzionale convesso debolmente s.c.i. su [M _b (Ω)] ^d (ottenuto grazie a [1]), si dimostra un risultato di convergenza per una successione di funzionali del tipo con μ _n εM _b ⁺ (Ω),f _n : Ω×R ^d →−∞, +∞] integrando normale convesso. Vengono inoltre trattati alcuni esempi motivati da applicazioni alla Meccanica.

     

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.