Familles d'immersions holomorphes et formes de torsion analytique équivariantesFamilies of equivariant immersions and analytic torsion forms

In this Note, we extend the known results on the behaviour by immersion of the holomorphic analytic torsion forms to the equivariant case. To cite this article: J.-M. Bismut, X. Ma, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 893–897.     

Effondrement de quotients aléatoires de groupes hyperboliques avec torsion

We show that random quotients of hyperbolic groups with ‘harmful’ torsion collapse at densities smaller than expected. To cite this article: Y. Ollivier, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Familles de graphes expanseurs et paires de HeckeFamilies of expanding graphs and Hecke pairs

Let H be a subgroup of a group G. Suppose that (G,H) is a Hecke pair and that H is finitely generated by a finite symmetric set of size k. Then G/H can be seen as a graph (possibly with loops and multiple edges) whose connected components form a family (X_i)_i∈I of finite k-regular graphs. In this Note, we analyse when the size of these graphs is bounded or tends to infinity and we present criteria for (X_i)_i∈I to be a family of expanding graphs as well as some examples. To cite this article: M.B. Bekka et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 463–468.     

Construction inconditionnelle de groupes de Galois motiviquesAn unconditional construction of motivic Galois groups

We attach to any “classical” Weil cohomology theory over a field a motivic Galois group, defined up to an inner automorphism. We also study the specialisation of numerical motives and the behaviour of motivic Galois group by specialisation. To cite this article: Y. André, B. Kahn, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 989–994.     

The 3-XORSAT thresholdLe seuil de 3-XORSAT

We prove the existence of a threshold phenomenon regarding the random 3-XORSAT problem (or more generally k-XORSAT). We provide the value of the threshold as the solution of two transcendental equations. These results confirm rigorously those obtained by physicists using the one-step replica symmetry breaking method and thus give for the first time the proof of the validity of this method for a problem of the class of satisfiability problems. To cite this article: O. Dubois, J. Mandler, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 963–966.     

Nombres de Liouville et nombres normauxLiouville numbers and normal numbers

We prove that there exist Liouville numbers which are normal, as well as Liouville numbers which are non-normal to any base. To cite this article: Y. Bugeaud, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 117–120.     

Intersecting random half spaces with a cubeVolume de l'intersection d'un cube et de sous-espaces aleatoires

We compute the typical number of points of the discrete cube {?1,1}^N that belong to the intersection of M random half-spaces, when M is a small proportion of N. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 807–809.     

Invertibility of functional Galois connectionsInversibilité des correspondances de Galois fonctionnelles

We consider equations of the form Bf=g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann's covering theorem for max-plus linear equations. To cite this article: M. Akian et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 883–888.     

Espaces productivement de FréchetProductively Fréchet spaces

The class of topological spaces whose product with every strongly Fréchet space is also Fréchet is characterized internally. This solves a long standing problem. To cite this article: F. Jordan, F. Mynard, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 259–262.     

Invariants de Vassiliev et conjecture de PoincaréVassiliev invariants and the Poincaré conjecture

We prove the following result: if Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true, in other words every homotopy sphere is homeomorphic to the standard sphere. On the other hand, in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. To cite this article: M. Eisermann, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1005–1010.     

Besov spaces and Bergman projections on the ballLes espaces de Besov et les projections de Bergman dans la boule

A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball of $\text{C}{}^{\mathrm{N}}$. Precise conditions are given for the boundedness of Bergman projections from certain L^p spaces onto Besov spaces. Right inverses for these projections are also provided. Applications to complex interpolation are presented. To cite this article: H.T. Kaptano?lu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732.     

Complexité dynamique des réseaux de HopfieldDynamical complexity of the Hopfield networks

One considers the Hopfield networks. It is shown that this system can generate any structurally stable inertial dynamics, with a bounded memory. To cite this article: S. Vakulenko, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 639–642.     

Arbres de Markov couplePairwise Markov trees

The Hidden Markov Chain (HMC) models are widely applied in various problems. This succes is mainly due to the fact that the hidden model distribution conditional on observations remains a Markov chain distribution, and thus different processings, like Bayesian restorations, are handleable. These models have been recetly generalized to “Pairwise” Markov chains, which admit the same processing power and a better modeling one. The aim of this Note is to show that the Hidden Markov trees, which can be seen as extensions of the HMC models, can also be generalized to “Pairwise” Markov trees, which present the same processing advantages and better modelling power. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 79–82.     

Transfert lisse d'intégrales de KloostermanSmooth transfer of Kloosterman integrals

We establish the existence of smooth transfer between absolute Kloosterman integrals and Kloosterman integrals relative to a quadratic extension. To cite this article: H. Jacquet, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 229–232.     

Sur la stabilité du fibré de PicardOn the stability of the Picard bundle

We study the stability of the Picard bundle on the moduli space of stable vector bundles of fixed rank and determinant, over an irreducible, smooth, algebraic curve of genus g?2. To cite this article: V. Savin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 885–888.     

Poisson geometry and the Kashiwara–Vergne conjectureGéométrie de Poisson et la conjecture de Kashiwara–Vergne

We give a Poisson-geometric proof of the Kashiwara–Vergne conjecture for quadratic Lie algebras, based on the equivariant Moser trick. To cite this article: A. Alekseev, E. Meinrenken, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 723–728.     

Support of Virasoro unitarizing measuresSupport des mesures unitarisantes de l'algèbre de Virasoro

A unitarizing measure is a probability measure such that the associated L² space contains a closed subspace of holomorphic functionals on which the Virasoro algebra acts unitarily. It has been shown that the unitarizing property is equivalent to an a priori given formula of integration by parts, which has been computed explicitly. We show in this Note that unitarizing measures must be supported by the quotient of the homeomorphism group of the circle by the subgroup of Möbius transformations. To cite this article: H. Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626.     

The Neumann problem in the half-spaceLe problème de Neumannn dans le demi-espace

We study here the nonhomogeneous Neumann problem in the half-space $\text{R}{}^{\mathrm{N}}{}_{\mathrm{+}}$ with N?2. We give in L^p theory, with 1<p<∞, a basic existence and regularity results in weighted Sobolev spaces. To cite this article: C. Amrouche, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 151–156.     

Singular sets of Sobolev functionsEnsembles singuliers des fonctions de Sobolev

We are interested in finding Sobolev functions with “large” singular sets. Given $\text{N,k∈}\text{N}$, 1<p<∞, kp<N, for any compact subset A of $\text{R}{}^{\mathrm{N}}$, such that its upper box dimension is less than N?kp, we construct a Sobolev function $\text{u∈}\text{W}{}^{\mathrm{k,p}}\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ which is singular precisely on A. We introduce the notions of lower and upper singular dimensions of Sobolev space, and show that both are equal to N?kp. To cite this article: D. ?ubrini?, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 539–544.     

Groupes de Neretin et propriété (T) de KazhdanNeretin's groups and Kazhdan's property (T)

We consider Neretin's group of boundary extensions of piecewise isometries of a homogeneous simplicial tree. We prove that if a subgroup Γ of this group has Kazhdan's property (T), then a finite index subgroup of Γ stabilizes a finite collection of balls of the boundary of the tree, acting isometrically on each of them. To cite this article: A. Navas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 789–792.