Complexes pondérés sur les compactifications de Baily-Borel: Le cas des variétés de Siegel

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety.

Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image.

More generally, we are able to define analogs in positive characteristic of the weighted cohomology complexes introduced by Goresky, Harder and MacPherson.

     

Une inégalité du type Payne-Polya-Weinberger pour le laplacien brut

Let us consider a riemannian vector bundle with compact basis and the rough laplacian associated to a connection on . We prove that the eigenvalues of are bounded above by a function of the first eigenvalue and of the geometry of , but independently of the choice of the connection .

     

Integration par parties dans l'espace de Wiener et approximation du temps local

Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u —, 0tT andX (t) = ( *X)(t),t 0, the regularization ofX by means of the convolution with the approximation of unity . The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, ) by means of a convenient normalization of the numberN ^X (u;) of crossings of the processX with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

---

---

Support partill du CICYT, N^o PB86-0238     

Approximation du temps local des surfaces gaussiennes

Summary LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d2),F its spectral measure, (x denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formX (t)=( *X)(t) where stands for an approximation of unity (as tends to zero) satisfying certain regularity conditions.The aim of this paper is to recover the local time ofX at a given levelu, as a limit of appropriate normalizations of the geometric measure of theu-level set of the regular approximating processesX . A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian processX can be a complex one as approaches O.The results are onL ²-convergence and include bounds for the speed of convergence.L ^presults may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values ofu do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original processX is irregular.Some technical results of analytic nature are included as appendices in Sect. 4.     

Une méthode de détermination du temps de transfert dans un système de deux zones multiplicatrices de neutrons

With a simplified delay time distribution function in the kinetic equations of a binode system, the neutron evolution is found to follow a two mode exponential decay. In the non-excited core, the evolution curve presents a maximum which time of occurrence takes place in a relation giving the interzonal transfer time. Values obtained from pulsed neutron experiments are discussed.     

Une remarque sur la symétrie asymptotique de la fonction de Green

Dans cette note, nous montrons que la symétrie asymptotique de la fonction de Green dans le cas des variétés compactes de courbure négative implique que la variété est localement symétrique.

     

Généralisation du critère de Beurling-Nyman pour l'hypothèse de Riemann

Nous généralisons dans cet article le critère de Beurling-Nyman, qui concerne la fonction de Riemann, à une large classe de séries de Dirichlet. Nous établissons donc une correspondance entre la densité d'un certain sous-espace de fonctions dans et la localisation des zéros d'une série de Dirichlet. Nous utilisons pour obtenir ce résultat la structure de l'espace de Hardy du demi-plan.

ABSTRACT. We generalise Beurling-Nyman's criterion, already known for the Riemann function, to a larger class of Dirichlet series. We reveal a link between the density of some subspace of functions in and the localization of the zeros of a Dirichlet series. To do so, we use the structure of the Hardy space of the half-plan.

     

Grandes déviations du temps local du mouvement brownien fractionnaire

The purpose of this paper is to prove a large deviation principle for a local time of fractional Brownian motion B^H for all H∈(0,1). To cite this article: E.H. Lakhel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 797–801.     

Une generalisation du theoreme d'extension de Radó

Using subharmonic functions we give an easy proof of E.L. Stout's generalization of Radó's extension theorem.

---

---

Travail partiellement subventionné par le Conseil National de Recherches du Canada (A 7668).     

Une extension du théorème de Borel-Weil

Sans résumé     

Une nouvelle estimation extrinsèque du spectre de l'opérateur de Dirac

We prove a new upper bound for the smallest eigenvalues of the Dirac operator on a compact hypersurface of the hyperbolic space. To cite this article: N. Ginoux, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Une generalisation du theoreme de Richardson sur l'existence de noyaux dans les graphes orientes

Richardson's theorem [4] asserts that if a digraph has no odd circuit, it possesses a kernel. We improve this theorem in the following manner: if each odd circuit has two short crossing chords, i.e. two chords of the type (X_i,X_i+2), (X_i+1,X_i+3), it possesses a kernel.     

On the increments of the local time of a Wiener sheet

{W(x, y), x≥0, y≥0} be a Wiener process and let η(u, (x, y)) be its local time. The continuity of η in (x, y) is investigated, i.e., an upper estimate of the process η(μ, [x, x + α) × [y, y + β)) is given when αβ is small.     

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness (in the sense of Meyer- Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.     

Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension

RÉSUMÉ. On considère dans un ouvert borné de , à bord régulier, le problème de Dirichlet

où , est positive et s'annule sur un ensemble fini de points de . On démontre alors sous certaines hypothèses sur et si est assez petit, que le problème (1) possède une solution convexe unique .

ABSTRACT. We consider in a bounded open set of , with regular boundary, the Dirichlet problem

where , is positive and vanishes on , a finite set of points in . We prove, under some hypothesis on and if is sufficiently small, that the problem (1) has a unique convex solution .

     

Minimizing multi-homogeneous Bézout numbers by a local search method

Consider the multi-homogeneous homotopy continuation method for solving a system of polynomial equations. For any partition of variables, the multi-homogeneous Bézout number bounds the number of isolated solution curves one has to follow in the method. This paper presents a local search method for finding a partition of variables with minimal multi-homogeneous Bézout number. As with any other local search method, it may give a local minimum rather than the minimum over all possible homogenizations. Numerical examples show the efficiency of this local search method.

     

Sur les transformées de Riesz dans le cas du Laplacien avec drift

We prove estimates for Riesz transforms with drift.