Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

Non-trivialité des points de Heegner

We give here a second proof of Mazur's conjecture on higher Heegner points, using a proven case of the André–Oort conjecture as a substitute for the main ingredient of our first proof [1], a theorem of M. Ratner. To cite this article: C. Cornut, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1039–1042.     

Algébricité des partitions markoviennes des automorphismes hyperboliques du tore

We prove that, to every Markov partition for a hyperbolic automorphism of the torus, one can associate an algebraic sofic coding similar to the one constructed in [6].     

Sur la régularité des trajectoires des Martingales à deux indices

Sans résuméA Leopold Schmetterer, à l'occasion de son soixantième anniversaireMembre du Laboratoire Associé au CNRS., n 224 «Processus Stochastiques et Applications»     

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.     

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.     

Seminégativité des courbes compactes des revêtements non compacts des surfaces projectives complexes

We show that the intersection form of a compact complex curveof any non compact cover ƒ:→ S of a smooth projective complex surface S is seminegative. Ifis not exceptional in , then C ≔ ƒ(C) is a fibre of the Albanese map α: S → B, whose image is a positive genus curve B (after possibly replacing S by a suitable finite cover), provided C is smooth and its normal bundle in S is torsion.These results agree with a question raised by I. Shafarevich in [1].     

Sur la structure des espaces de probabilité conditionnelle

Sans résumé Présenté par A. Rényi     

Sur une propriété des nombres naturels

Résumé L'auteur donne des conditions nécessaires et suffisantes pour qu'un nombre naturel n soit un ? nombre pratique ?, c'est-à-dire pour que tout nombre naturel <-n soit une somme de diviseurs naturels distincts du nombre n. à Mauro Picone pour son 70^me anniversaire.     

Dualité tannakienne pour les quasi-groupoïdes quantiques

This paper is the sequel of a previous one [2] where we extended the Tannaka-Krein duality results to the non-commutative situation, i.e. to ‘quantum groupoids’. Here we extend those results to the quasi-monoidal situation, corresponding to ‘quasi-quantum groupoids’ as defined in [3] (‘quasi-’ stands for quasi-associativity a la Drinfeld). More precisely, let B be a commutative algebra over a field k. Given a tensor autonomous category τ,. we define the notion of a quasi-fibre functor ω:τ-proj B (here, ‘quasi-’ means without compatibility to associativity constraints). On the other hand, we define the notion of a transitive quasi-quantum groupoid over B. We then show that the category of tensor autonomous categories equipped with a quasi-fibre functor (with suitable morphisms), is equivalent to the category of transitive quasi-quantum groupoids (5.4.2)

Moreover, we classify quasi-fibre functors for a semisimple tensor autonomous category (6.1.2), and give a few examples : a family of quantum groups having the same tensor category of representations as Sl²(C), but with non-isornorphic underlying coalgebras, constructed by means of an R-matrix introduced by Gurevich ([9]) in a manner suggested to the author by Lyubashenko (6.2.1 and 6.2.2), and quasi-quantum groups which cannot be obtained from quantum groups by a Drinfeld twist (6.2.1)     

Sur l'intégrabilité uniforme des martingales exponentielles

Sommaire Nous donnons des conditions suffisantes pour que la martingale locale (M) définie par C. Doléans-Dade soit une martingale uniformément intégrable. Le lien est établi avec la martingale locale exponentielle (N, z, ) de Kunita-Watanabe, ce qui permet de généraliser un théorème de Novikov.     

La Τ-régularité des mesures Gaussiennes

Summary We give a simple proof of the fact that a Radon gaussian measure on a locally convex vector space is carried by a countable union of metrisable compact sets. We show that a separable centered gaussian process with continuous covariance which is defined on a Polish space X, and is a.e. unbounded on any open set, has a.e. dense trajectories in X × . These results allow us to show that for any set I, any gaussian measure on ^I is -smooth.     

Frottement interne et anélasticité des solides

Summary Having recalled a few essential definitions in the theory of internal friction in solids, the authors, in a first chapter, study the propagation of longitudinal waves in a rod; they start from a simple law relating locally stress, strain and their first derivatives with respect to time. This law which is more general thanHooke's law has been proposed first byZener to explain the anelastic behaviour of a standard linear solid. The authors calculate the internal friction resulting from energy dissipation when the vibration is forced then free. The second chapter deals with problems arising when one tries to measure the internal friction in flexure of metallic specimens having the shape of rods or circular plates. The authors develop the basic flexure equations applied to the two types of specimen; then they determine the fundamental characteristics, which appear in internal friction measurement purposes.

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Recherche bénéficant d'un subside du Fonds National Suisses de la Recherche Scientifique (Commission pour la Science Atomique, FN A 176).     

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.     

Temps d'instabilité des systèmes hamiltoniens initialement hyperboliques

We compute the time of drift along a transition chain for an initially hyperbolic Hamiltonian system. We prove that this time is polynomial with respect to the opposite of the perturbing parameter.     

Friabilité des valeurs d’un polynôme

Let F(X) be an absolutely irreducible polynomial in \mathbbZ [X₁,..., X_n]{\mathbb{Z} [X_{1},\dots, X_{n}]}, with degree d. We prove that, for any δ < 4/3, for any sufficiently large x, there exists a positive density of integral n-tuples m = (m ₁, . . . , m _n ) in the hypercube max |m _i | ≤ x such that every prime divisor of F(m) is smaller than x ^d–δ . This result is improved when F satisfies some geometrical hypotheses.     

Une identité remarquable en théorie des partitions

We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials. The proof given is using -ring techniques. It would be interesting to obtain a bijective proof.

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Received January 18, 2000 / Revised January 23, 2000 / Published online July 20, 2000