La fonction maximale de Hardy–Littlewood sur une classe d'espaces métriques mesurables

In this Note, we study the behavior of the Hardy–Littlewood maximal function M on cusp manifolds in terms of the growth of the volume of the base space. In particular, we prove that for all 1<p₀<+∞ fixed, there exists such a manifold on which M is bounded on L^p for p>p₀ but not for 1?p<p₀. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Intégrabilité uniforme semi-globale d'une classe de fonctions plurisousharmoniques

We prove that for the family $\text{G}{}_{\mathrm{s}}\text{={u∈PSH(}\text{B}\text{);u?0;}\text{max}{}_{\mathrm{<mtext>B</mtext><mtext>(0,s)</mtext>}}\text{u??1}}$, where s<exp(?1/2), the functions exp(?u), for $\text{u∈}\text{G}{}_{\mathrm{s}}$, are uniformly integrable on the ball $\text{B}\text{(0,ρ)}$ for $\text{ρ<ρ(s):=(1?s}\text{exp}\text{(}\text{1}\text{2}\text{))/(}\text{exp}\text{(}\text{1}\text{2}\text{)?s)}$. Furthermore ρ(s) is optimal. To cite this article: S. Benelkourchi, B. Jennane, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

La variété des classes analytiques d'équations aux q-différences dans une classe formelle

We describe the structure of affine algebraic variety of the set of analytical classes of q-difference equations within a given formal class. To cite this article: J.-P. Ramis et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Comportement asymptotique d'une grue

We study the asymptotic behaviour of a structure depending of two small parameters. The ratio of these two parameters plays a part in the limit problems. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Étude d'une forme volume naturelle sur l'espace de représentations du groupe d'un nœud dans SU(2)

For a knot K in S³, we construct in the line of Casson – or more precisely taking into account Lin's (J. Differential Geom. 35 (1992) 337–357) and Heusener's (Topology Appl. 127 (2003) 175–197) further works – a 1-volume form on the SU(2)-representation space of the group of K and we show how to interpret this volume form as a Reidemeister torsion. In the last part of this Note, we give an explicit computation of this volume form for torus knots. To cite this article: J. Dubois, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Une classification des flots solutions d'une EDS

Under general hypotheses, we show that the flows of kernels can be associated to a stochastic differential equation (SDE). We also show a classification theorem of the solutions of the SDE: they can be obtained through filtering the coalescing solution with respect to a sub-noise containing the white noise driving the SDE. The example of the isotropic flows is studied. To cite this article: Y. Le Jan, O. Raimond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

La construction bar d'une algèbre comme algèbre de Hopf E-infini

We prove that the bar construction of an E_∞ algebra forms an E_∞ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt–Eccles operad. (The surjection operad and the Barratt–Eccles operad are classical E_∞ operads.) To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Arithmétique d'une famille de corps cubiques

In this Note, we study the family of polynomials: P(X)=X³?nX²?n, with n=3^sp₁…p_t, where s=0 or 1 and where the p_i, for 1?i?t, are distinct prime numbers and all different from 3, and (4n²+27)/9^s is squarefree. For this family, we determine the arithmetic invariants of the number field $\text{K=}\text{Q}\text{(α),}$ where α is the only real root of the polynomial P(X), and we find the following results: $\text{O}{}_{\mathrm{K}}\text{=}\text{Z}\text{[α]}$ is the ring of integers of K, d_K=?n²(4n²+27) is the discriminant of K; ε=α²+1 is the fundamental unit of O_K and R_K=Log(α²+1) is the regulator of K. To cite this article: O. Lahlou, M. El Hassani Charkani, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Réflexion retardée pour des paquets d'ondes dispersives sur un réseau en forme d'étoile

We study Klein–Gordon equations on two half-axes with different dispersion relations. The reflection coefficient at the connecting point depends on the frequency. We obtain lower and upper bounds of the reflected part of the energy flow when the frequency band involved in the initial signal is sufficiently narrow. The reflection is delayed for low frequency wave packets and our expression of the delay is in accordance with the recent experiments of Haibel and Nimtz (Ann. Physik (Leipzig) 10 (2001) 707–712). The results are generalized to the case of a star-shaped network with n branches (n>2). To cite this article: F. Ali Mehmeti, V. Régnier, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Résolution en temps court d'une équation de Hamilton–Jacobi non locale décrivant la dynamique d'une dislocation

This Note studies a nonlocal geometric Hamilton–Jacobi equation that models the motion of a planar dislocation in a crystal. Within the framework of viscosity solutions and of the level-set approach, we show that the equation has a unique solution on a small time interval when the initial curve is the graph of a Lipschitz bounded function. To cite this article: O. Alvarez et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Estimation de l'évolution d'un total en présence d'information auxiliaire : une approche par splines de régression

We build an estimator for the evolution of a finite population total between two periods of time when auxiliary information is available. A superpopulation model is introduced in order to explain the relationship between the study and the auxiliary variables. The regression functions are estimated by regression splines and Horvitz–Thompson technique. Finally, an estimator for the evolution is derived and proved to be asymptotically unbiased and consistent and we compute a design-based variance formula. To cite this article: C. Goga, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Sur les distances mutuelles d'une chorégraphie à masses distinctes

We state some simple properties of a configuration of N bodies whose masses are not all equal, and whose motion is a ‘choreography’. In such a solution of Newton's equations, the bodies chase each other around the same curve, with the same phase shift between consecutive bodies. It follows from those properties that for any dimension of space, the masses of a choreography are the same for a logarithmic potential. A similar argument shows that the vorticities of a choreography are the same for N vortices which satisfy Helmholtz's equations (Philos. Mag. 33 (1858) 485–512). We prove a more general result for any potential. In particular, for a choreography with distinct masses, the ratio between the smallest and the largest mutual distances is bounded by a constant which does not depend on the masses. To cite this article: M. Celli, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Sur une classe d'espaces d'Einstein

Résumé Soit un espace pseudoriemannien M muni d'une 1-forme ω. Dans [11], chaque fibré F en droites sur M est considéré, de manière naturelle, comme un espace pseudoriemannien F(M, ω, ɛ) avec ɛ=± 1 et on étudie le cas où celui-ci est un espace d'Einstein. Dans la Section 1 nous déduisons quelques propriétés générales de ces espaces d'Einstein et nous indiquons une méthode de construction pour les 1-formes ω correspondantes. Dans la Section 2 nous déterminons tous les espaces d'Einstein F(M, ω, ɛ) pour lesquels M est une forme spatiale sphérique. Nous étudions aussi le cas dual où M est une certaine forme spatiale hyperbolique. Dans la Section 3 nous acceptons, comme conséquence de cette dualité, l'existence en relativité générale de l'ainsi nommée ? comatière ?, dans laquelle la vitesse de la lumière admet une limite inférieure non nulle. Nous proposons un modèle de coexistence matière-comatière, avec une frontière de séparation correspondant aux singularités du champ gravitationnel. Nous étudions la possibilité de communication entre ces deux zones opposées à l'aide de la lumière. A la mémoire du ProfesseurBruno Finzi Entrata in Redazione il 20 gennaio 1975.     

Approximation par éléments finis de problèmes elliptiques d'optimisation de forme

We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H¹ with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H¹ whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Sur une question d'équirépartition de nombres algébriques

We give an elementary counter-example for a conjecture of Pineiro, Szpiro and Tucker about the uniform distribution of algebraic numbers with small height. To cite this article: P. Autissier, C. R. Acad. Sci. Paris, Ser. I 342 (2006).