Existence globale pour une classe d'équations d'ondes perturbées

In this paper we prove a global well-posedness result for the following Cauchy problem:

where the initial data $\text{(f,g)∈}\text{H}\text{?}{}^1\text{(}\text{R}{}^3\text{)×L}{}^2\text{(}\text{R}{}^3\text{)}$ are compactly supported, 1?α<5, $\text{a}{}_{\mathrm{i}}\text{(t,x)∈L}{}^{\mathrm{\infty }}\text{(}\text{R}{}_{\mathrm{t}}\text{×}\text{R}{}_{\mathrm{x}}{}^3\text{)}$, $\text{V(t,x)∈L}{}^{\mathrm{\infty }}\text{(}\text{R}{}_{\mathrm{t}}\text{;L}{}^3\text{(}\text{R}{}^3{}_{\mathrm{x}}\text{))}$. To cite this article: N. Visciglia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Inégalités d'oracle pour l'estimation d'une densité de probabilité

We study the problem of the nonparametric estimation of a probability density in $L_2(\mathbb{R})$. Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Étude d'un modèle thermo-chimique de formation d'un matériau composite

We consider a composite material constituted of carbon or glass fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period $?>0$. First we prove the existence and uniqueness of a solution by using Schauder's fixed point theorem. Then, by using an asymptotic expansion, we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero and we obtain an error estimate in a case of weak non-linearity. Finally we solve numerically the homogenized problem. To cite this article: S. Meliani et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Analyse a posteriori d'erreur par reconstruction pour un modèle d'écoulement dans un milieu poreux fracturé

We present in this note a posteriori error estimates based on the postprocessing technique for the reduced model of flow in fractured porous media, introduced and analysed by V. Martin, J. Jaffré, and J. Roberts. This model is approximated by the Raviart–Thomas finite elements of lowest order. In this type of approximation, the velocity is well approximated. A postprocessing of the pressure appears to be necessary since it does not belong to $H_0^1(\mathrm{\Omega })$. We give an upper bound for the error in the energy norm, with some indicators that are expressed in terms of the reconstruction of the pressure. Numerical results show that all indicators converge to zero when the mesh size goes to zero, with the same speed as the error. One of these indicators can be interpreted as both a discretization indicator and an indicator of the reduced model validity.     

Sur l'algèbre enveloppante d'une algèbre pré-Lie

We construct an associative product on the symmetric module $S(L)$ of any pre-Lie algebra L. It turns $S(L)$ into a Hopf algebra which is isomorphic to the envelopping algebra of $L_{\mathrm{Lie}}$. Then we prove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees. To cite this article: J.-M. Oudom, D. Guin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Applications aux sommes elliptiques d'Apostol–Dedekind–Zagier

In this Note, we give two applications to our work [Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004); DOI: 10.1016/j.crma.2004.07.018] concerning multiple elliptic Apostol–Dedekind–Zagier sums. These elliptic sums are defined by means of certain Jacobi modular forms of two variables $D_{\tau }(z\text{;}\phi )$. When $\text{Im}(\tau )\to \infty$, these elliptic sums give the classical Apostol–Dedekind–Zagier multiple sums [Apostol, Duke Math. J. 17 (1950) 147–157, Pacific. J. Math 2 (1952) 1–9; Zagier, Math. Ann, 202 (1973) 149–172]. We give a reciprocity law for these sums. To cite this article: A. Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Spectres pour l'approximation d'un nombre réel et de son carré

Let $\alpha (\xi )$ be the exponent that measures how a non-quadratic real number ξ and its square can be simultaneously approximated by rational numbers with the same denominator. Davenport and Schmidt have proved that $\alpha (\xi )$ is always between the golden ratio γ and 2. Roy, and after him Bugeaud and Laurent, have constructed numbers ξ such that $\alpha (\xi )<2$. Their method involves infinite words with many palindrome prefixes. In this text, we define new exponents of approximation that allow us to obtain, to some extent, a characterization of the values $\alpha (\xi )$ obtained by these authors. To cite this article: S. Fischler, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Solutions globales d'énergie infinie de l'équation de Navier–Stokes 2D

We study in this Note the solutions of the 2D Navier–Stokes equations with initial data in ?BMO. For $u{|}_{t=0}$ in the closure of the Schwartz class, we obtain the existence and uniqueness of a global solution, and besides an estimate on its norm in ?BMO. To cite this article: P. Germain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Cohérence des faisceaux d'idéaux multiplicateurs avec estimations

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a bounded pseudoconvex open set and let ? be a plurisubharmonic function on $\text{Ω.}$ For every positive integer m, we consider the multiplier ideal sheaf $\text{I}\text{(m?)}$ and the Hilbert space $\text{H}{}_{\mathrm{\Omega }}\text{(m?)}$ of holomorphic functions f on $\text{Ω}$ such that $\text{|f|}{}^2\text{e}{}^{\mathrm{?2m?}}$ is integrable on $\text{Ω.}$ We give an effective version, with estimates, of Nadel's result stating that the sheaf $\text{I}\text{(m?)}$ is coherent and generated by an arbitrary orthonormal basis of $\text{H}{}_{\mathrm{\Omega }}\text{(m?).}$ This result is expected to play a major part in the context of current regularizations with estimates of the Monge–Ampère masses. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Estimation spatio-temporelle d'un modèle de système de particules

Let X be a discrete time contact process (CP) on ${\mathbb{Z}}^2$ as defined by Durrett and Levin (1994). We study the estimation of the model based on space–time evolution of X, that is, $T+1$ successive observations of X on a finite subset S of sites. We consider the maximum marginal pseudo-likelihood (MPL) estimator and show that, when $T\to \infty$, this estimator is consistent and asymptotically normal for a non vanishing supercritical CP. To cite this article: X. Guyon, B. Pumo, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Estimation du nombre de dérivées d'un processus Gaussien

We consider a real Gaussian process X with unknown smoothness $r_0\in \mathbb{N}$ where the mean-square derivative $X^{(r_0)}$ is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations $X(t_1),\dots ,X(t_n)$, we study reconstruction of $X(t)$, $t\in [0,1]$, with ${\stackrel{?}{X}}_r(t)$, a piecewise polynomial interpolation of degree $r?1$. We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as $r?r_0$. Next, from an interpolation-based empirical criterion, we derive an estimator $\stackrel{?}{r}$ of $r_0$ and prove its strong consistency by giving an exponential inequality for $P(\stackrel{?}{r}\ne r_0)$. Finally, we prove the strong convergence of ${\stackrel{?}{X}}_{\stackrel{?}{r}}(t)$ toward $X(t)$ with a similar rate as in the case ‘$r_0$ known’. To cite this article: D. Blanke, C. Vial, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Processus de Bickel–Rosenblatt pondéré et tests d'ajustement

The goal of this work is to establish the limit distribution of the process

$\text{I}{}_{\mathrm{n}}\text{(W):=}\text{∫}\text{A}\text{f}{}_{\mathrm{n}}\text{(x)?Ef}{}_{\mathrm{n}}\text{(x)}{}^2\text{W(x)}\text{d}\text{x,}\text{W∈}\text{W}\text{,}$

where $\text{W}$ is a class of weight functions W, f_n is the kernel density estimator of the density f and A is a Borelian subset of $\text{R}$. We apply this result to derive new statistics to test goodness-of-fit of the density function f. Under some local alternatives, these new tests are more powerful than the usual Bickel–Rosenblatt one. To cite this article: F. Chebana, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Le nombre des diviseurs d'un entier dans les progressions arithmétiques

Let d_k,?(n) be the function number of divisors of the integer n?1, in arithmetic progressions {?+mk}, with 1???k and ?,k coprime, and let F(n;k,?) defined as follows:

$\text{F(n;k,?)=}\text{ln}\text{d}{}_{\mathrm{k,?}}\text{(n)}\text{ln}\text{(?(k)}\text{ln}\text{n)}\text{ln}\text{2}\text{ln}\text{n}\text{.}$

In this Note, we study and give the structure of d_k,?-superior, highly composite numbers, which generalize those defined by S. Ramanujan. We prove that F(n;k,?) reaches its maximum among these numbers. We give it explicitly for k=2,…,13. This generalizes the study of Nicolas and Robin, in which the case k=1 is treated. To cite this article: A. Derbal, A. Smati, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Singularité de la mesure harmonique sur le bord d'un groupe hyperbolique

This Note deals with the dimension of the harmonic measure ν associated with a random walk on the isometry group of a Gromov hyperbolic space. We establish a link of the form $\dim \nu ?h/l$ between the dimension of the harmonic measure, the asymptotic entropy h of the random walk and its rate of escape l. Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure. To cite this article: V. Le Prince, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Prolongement d'un courant négatif plurisousharmonique avec condition sur les tranches

In this Note, we prove a theorem on the extension of a negative (or positive) plurisubharmonic current T (i.e. such that $d{d}^{c}T?0$) with condition on the slices with respect to some coordinates. This theorem generalizes a result proved by El Mir–Ben Messaoud relative to d-closed positive currents with a condition on slices. The method consists first of proving a Chern–Levine–Nirenberg inequality for a positive (or negative) psh current, which is a generalization of results obtained by Bedford–Taylor, Demailly and Sibony for d-closed positive currents. Also we prove an Oka type inequality for positive psh currents, thereby generalizing former results by Ben Messaoud–El Mir concerning positive currents with a negative $d{d}^c$. To cite this article: M. Toujani, H. Ben Messaoud, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Lois conjointes du processus et de son maximum,des premier instant et position d'atteinte d'une demi-droite pour le pseudo-processus régi par l'équation ∂∂t=±∂N∂xN

In this Note, we obtain explicit formulas for the joint distribution of the pseudo-process driven by the equation $\frac{?}{?t}=\pm \frac{{?}^N}{?x^N}$ coupled together with its maximum, as well as that of the first time when this pseudo-process overshoots a fixed level coupled together with the corresponding overshooting place. To cite this article: A. Lachal, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Continuité lipschitzienne des solutions d'un problème en calcul des variations

In this Note, we describe some recent developments concerning the regularity of the minimizers u of ${\int }_{\Omega }F(\mathrm{?}u)+G(x,u)$, over the functions $u\in W^{1,1}(\Omega )$ that assume given boundary values ? on ?Ω. The classical Hilbert–Haar theory derives regularity of u from an assumption on ?, the well-known bounded slope condition. Instead of this, we impose the less restrictive lower (or upper) bounded slope condition, which is satisfied if ? is the restriction to ?Ω of a convex (or even semiconvex) function. Under this new assumption and some convexity hypotheses on F and Ω, we show that any minimizer u is locally Lipschitz in Ω. In some cases we are also able to assert that u is continuous on $\overline{\Omega }$. To cite this article: P. Bousquet, F. Clarke, C. R. Acad. Sci. Paris, Ser. I 343 (2006).