Note sur la réalité des points doubles des courbes gauches

A real curve max have three different types of nodes: real nodes with real branches, real nodes with imaginary branches, and imaginary nodes. We give a simple construction of space curves with a prescribed number of nodes of each type.     

Contrôlabilité exacte des approximations de Galerkin des équations de Navier-Stokes

We consider Navier-Stokes equations in a smooth domain of ℝⁿ, n = 2, 3, with Dirichlet boundary conditions. We introduce the classical Galerkin approximation and study its exact controllability when the control acts on an open non-empty subset of the domain. Under suitable assumptions on the elements of the Galerkin basis, we prove that this finite-dimensional system is exactly controllable. The proof combines HUM together with a classical fixed point argument, and relies on the cancellation properties of the non-linearity appearing in the Navier-Stokes equations.     

Rationalité des séries de poincaré et des fonctions zêta motiviques

Soient k un corps de caractéristique 0 et X une k[[t]]-variété (éventuellement singulière) plate, purement de dimension relative d. Nous prouvons la rationalité des séries de Poincaré motiviques et de fonctions Zêta dIgusa motiviques, associées à X, à laide de lintégration motivique, du théorème de désingularisation plongée dHironaka, de la théorie des modèles de Néron faibles pour les schémas formels et dun théorème délimination des quantificateurs en théorie des modèles.Revised version: 17 March 2004     

Sur la structure des espaces de probabilité conditionnelle

Sans résumé Présenté par A. Rényi     

Petits discriminants des corps de nombres totalement imaginaires de degré 8

In this paper we show that there are, up to isomorphisms, exactly 15 totally imaginary eight-degree number fields having a discriminant smaller than 1656110. The proof of this result was obtained in the following manner: For each totally imaginary eight-degree number field with a discriminant smaller than 1656110, we constructed a polynomial, one root of which was a generator of the number field. In order to circumscribe the number of polynomials to be studied, we used, on the one hand, methods issuing from the geometry of numbers and on the other, the method developed by Odlyzko, Poitou, and Serre for the determination of lower bounds for discriminants.     

Géométrie des espaces de Banach ayant des approximations de l'identité contractantes

Let a, c ≥ 0 and let B be a compact set of scalars. We introduce property M^* (a, B, c) of Banach spaces X which is a geometric property of Banach spaces generalizing property (M^*) due to Kalton. Using M^*(a, B, c) with max ¦B¦ + c > 1, we characterize intrinsically a large class of shrinking approximations of the identity, including those related to M-, u-, and h-ideals of compact operators. We also show that the existence of these approximations of the identity is separably determined. As an application, we study ideals of compact and approximable operators. In particular, this provides an alternative unified and easier approach to the theories of M-, u-, and h-ideals of compact operators.     

Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire

We present a simple proof of regularity and compactness phenomena induced by singular Boltzmann collision kernels.     

Sur la compacité des ensembles de Julia des polynômes p-adiques

Let R be a rational function, of degree 2, with complex coefficients. Then the Julia set of R is a closed subset of ¹(), and therefore compact. If one replace by the field _p (completion of an algebraic closure of the field _p of the p-adic numbers), then one can define also a Julia set for a rational function with p-adic coefficients. But as _p is not locally compact, the Julia set may or may not be compact. In this paper, we study the compactness of the Julia set of p-adic polynomials. Mathematics Subject Classification (2000):11S99, 37B99.     

Un modèle de Rotenberg avec des vitesses de maturité nulles

The Note [1] is devoted to the study of the Rotenberg's [8] model describing the growth of cell population when the maturation velocity belongs to a, b, with 0 < a < b < ∞. In this Note we bring new techniques for treating the case 0 = a < b < ∞.     

Formule de Cauchy-Crofton pour la densité des ensembles sous-analytiques

For the density of subanalytic sets, we give a formula analogous to the classical Cauchy-Crofton formula for the volume. To do this we define the local polar profiles of a subanalytic set and the corresponding finite sequence of local multiplicities which is the real counterpart of the multiplicity of complex analytic sets.