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.     

Classification et mélanges de processusClustering and mixtures of processes

We propose a clustering method based on the estimation of mixtures of probability distributions, the new point being that the statistical units are described by probability distributions. The components of the mixtures are Dirichlet processes, normalized weighted Gamma processes, and Kraft processes. Mixtures obtained by applying some algorithms to the finite dimensional distributions of the components converge to the desired mixture as the dimension increases, since the components are mutually singular due to a theorem of Kakutani. The desired clusters are then the support of these components. To cite this article: R. Emilion, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 189–193.     

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.     

Chaı̂nes de Markov TripletTriplet Markov Chains

The Hidden Markov Chains (HMC) are widely applied in various problems. This succes is mainly due to the fact that the hidden process can be recovered even in the case of very large set of data. These models have been recetly generalized to ‘Pairwise Markov Chains’ (PMC) model, which admit the same processing power and a better modeling one. The aim of this note is to propose further generalization called Triplet Markov Chains (TMC), in which the distribution of the couple (hidden process, observed process) is the marginal distribution of a Markov chain. Similarly to HMC, we show that posterior marginals are still calculable in Triplets Markov Chains. We provide a necessary and sufficient condition that a TMC is a PMC, which shows that the new model is strictly more general. Furthermore, a link with the Dempster–Shafer fusion is specified. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 275–278.     

Petits espaces de Lebesgue et quelques applicationsOn small Lebesgue spaces and their applications

We prove some new properties of the small Lebesgue spaces introduced by Fiorenza [7]. Combining these properties with the Poincaré–Sobolev inequalities for the relative rearrangement (see [11]), we derive some new and precises estimates either for small Lebesgue–Sobolev spaces or for quasilinear equations with data in the small Lebesgue spaces. To cite this article: A. Fiorenza, J.-M. Rakotoson, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 23–26     

Actions moyennables et fonctions harmoniquesAmeanable actions and harmonic functions

We prove that the action of a countable discrete group on a locally compact invariant space of minimal harmonic functions is ameanable. To cite this article: P. Biane, E. Germain, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 355–358.     

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.     

Approximation pour la distance de WassersteinApproximation for the Wasserstein distance

Let $\text{C}$ be a set of probability-measures and P a probability on $\text{R}{}^{\mathrm{d}}$. Under some conditions, we show that we have a solution to the approximation problem of P by $\text{C}\text{.}$ There exists a probability $\text{Q}{}_0\text{∈}\text{C}$, such that $\text{l(P,Q}{}_0\text{)?l(P,Q),}\text{?Q∈}\text{C}$ where l is the square of the Wasserstein distance. To cite this article: N. Belili, H. Heinich, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 537–540.     

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.     

Estimation localement suroptimale et adaptative de la densitéLocally superoptimal and adaptive projection density estimators

We study a data-driven version of the density projection estimator in a general framework. We show that this estimator reaches a superoptimal rake on a dense set in the density class, and a quasi-optimal rake elsewhere. This set can be chosen by the statistician, and the superoptimal speed is reached for integrated quadratic error and almost sure uniform convergence. An adaptive version of the estimator is also considered. To cite this article: D. Bosq, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 591–595.     

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.     

Structures de Hodge mixtes et faisceaux réflexifs semistablesMixed Hodge structures and semistable reflexive sheaves

Let X be a smooth projective variety over an algebraically closed field of characteristic 0. We prove that the category of μ-semistable reflexive sheaves of slope μ equivariant for the action of some group on X is Abelian. The same claim for $\text{X=}\text{P}{}^2{}_{\mathrm{<mtext>C</mtext>}}$ and a stronger semistability condition gives us a geometric proof of the fact that the category of mixed Hodge structures is Abelian. To cite this article: O. Penacchio, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 475–480.     

Actions hamiltoniennes de tores et jacobiennes généraliséesTorus actions and generalized Jacobians

I describe the fibration in Liouville tori of an integrable system on the coadjoint orbits of SU(n) in terms of the (relative) generalized Jacobian of a family of singular curves. The integrable system contains the moment mapping of the maximal torus—these periodic Hamiltonians are responsible for the noncompact part of the Jacobian. To cite this article: M. Audin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 37–42     

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.     

Contractive liftings and the commutatorDilatations contractive et le commutateur

This paper presents the solution to a problem proposed by B.Sz.-Nagy about extending the commutant lifting theorem to the case when the underlying operators do not intertwine. The main theorem establishes minimal norm liftings of certain commutators. The proof is constructive. To cite this article: C. Foias et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 431–436.     

Pinceau intégral enlacéLinked integral pencils

Let γ₀ be a integral curve of an analytic vector field on a manifold of dimension 3. We suppose that γ₀ has oriented, iterated tangents. The integral pencil PI(γ₀) is the set of integral curves γ which have the same oriented, iterated tangent as γ₀. The curves of PI(γ₀) are either subanalitically separated or asymptotically linked. In this case PI(γ₀) has a formal axis which is divergent if and only if the curves of PI(γ₀) are not oscillating. To cite this article: F. Cano et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 855–858.     

S-arrangements avec répétitionsS-arrangements with repetitions

We study natural correspondances between the set of arrangements of subsets with repetitions and families of objects met in various areas of mathematics (walks in lattices, pre-order) and computer science (formal languages, Petri nets, Artificial Intelligence), using the framework of formal languages. For n=2, they are enumerated by Delannoy numbers. To cite this article: S.R. Schwer, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 261–266.     

Les formes de torsion holomorphes du complexe de de RhamThe holomorphic torsion forms of the de Rham complex

In this Note, we announce the vanishing of the holomorphic torsion forms of the relative de Rham complex of an equivariant fibration. To cite this article: J.-M. Bismut, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 243–247.     

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.