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).     

Symétrie des grandes solutions d'équations elliptiques semi linéaires

Let g be a locally Lipschitz continuous function defined on $\mathbb{R}$. We assume that g satisfies the Keller–Osserman condition and there exists a positive real number a such that g is convex on $[a,\infty )$. Then any solution u of $?\mathrm{\Delta }u+g(u)=0$ in a ball B of ${\mathbb{R}}^N$, $N?2$, which tends to infinity on ?B, is spherically symmetric. To cite this article: A. Porretta, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

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).     

Interprétation motivique de la formule d'excès d'intersection

Following Voevodsky, one can associate a Gysin triangle in the category of triangulated motives to every couple (X,Z) such that X is a smooth scheme and Z is a smooth closed subscheme of X. We study the functoriality of this triangle: if one considers a morphism $\text{f}\text{:Y→X}$ which is transversal to Z, this functoriality is simple. We are rather interested in the case where f is not quite transversal to Z. In that case, one obtains a motivic excess intersection formula and a ramification formula in the sense of equicharacteristic discrete valuation rings. To cite this article: F. Déglise, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

L'équation aux variations verticales d'un équilibre relatif comme source de nouvelles solutions périodiques du problème des N corps

We start with a rotating frame which enforces a resonance between the frequency of a relative equilibrium of the N-body problem and that of an infinitesimal variation normal to the plane of the equilibrium. Continuation then yields a remarkable class of periodic solutions. The first example is the $P_{12}$ family, discovered by Christian Marchal, which links the relative equilibrium of Lagrange to the Eight. To cite this article: A. Chenciner, J. Féjoz, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Lissité rationnelle des variétés de représentations d'un carquois

Recent works are devoted to the rational smoothness of varieties of representations for quivers, notably those of Robert Bédard, Ralf Schiffler and Philippe Caldéro. Their main result is the equivalence for a variety of representations $\text{O}$ of a quiver of Dynkin type, between

$\text{O}\text{is rationally smooth}\text{?}\text{O}\text{is smooth}\text{?}\text{O}\text{is an affine space.}$

This Note gives a different approach to the question and a strengthened version of Bédard–Schiffler–Caldéro's equivalences. To cite this article: A. Arabia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Inégalités d'incertitude associées à des fonctions homogènes

The uncertainty principle states that a nonzero function and its Fourier transform cannot be both sharply localized. It is well known that the support and the spectrum of a function cannot both have finite measure. In Shubin et al. [Geom. Funct. Anal. 8 (1998) 932–964], it is shown that they cannot be contained in ?-thin sets E and $\stackrel{?}{E}$. We give here other examples of sets E and $\stackrel{?}{E}$ having this property. The thinness of the sets is expressed in terms of a dyadic decomposition of the space, which is related to the functions ${|x_1|}^{{\alpha }_1}?{|x_d|}^{{\alpha }_d}$ on ${\mathbb{R}}^d$. These sets are not ?-thin in general. We prove that the pairs of sets we consider are strongly annihilating in the sense of Havin and Jöricke. To cite this article: B. Demange, 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).     

Borne sur l'advection et la hauteur d'eau du problème de shallow-water avec conditions aux limites de Dirichlet

Using some properties of the Hardy spaces, we give a regularity result on the advection term of the shallow-water equations and we show a $L^2$ bound holding up to the boundary on the water height with Dirichlet boundary conditions. To cite this article: F. Flori, P. Orenga, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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).     

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).     

Densité des zéros des transformés de Lévy itérés d'un mouvement brownien

The Lévy transform of a Brownian motion B is the Brownian motion $\text{B′}{}_{\mathrm{t}}\text{=∫}{}_0{}^{\mathrm{t}}\text{sgn}\text{B}{}_{\mathrm{s}}\text{d}\text{B}{}_{\mathrm{s}}$; denote by Bⁿ the Brownian motion obtained from B by iterating n times the Lévy transform. We establish that the set of all instants t such that B_tⁿ=0 for some n, is a.s. dense in the time-axis $\text{R}{}_{\mathrm{+}}$. To cite this article: M. Malric, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

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

Let the functions $d_{k\text{,}l}^{*}(n)$ and $d_{k\text{,}l}(n)$ be number of unitary divisors (see below) and number of divisors n in arithmetic progressions $\{l+mk\}$; k and l are integers relatively prime such that $1?l?k$ and let, for $n?2$

$F(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}(n))\ln (\phi (k)\ln n)}{\ln n}\text{,}{F}^{*}(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}^{*}(n))\ln (\phi (k)\ln n)}{\ln n}\text{and}$

$D^{*}(n\text{;}k\text{,}l)=\frac{\ln (d_{k\text{,}l}(n)/d_{k\text{,}l}^{*}(n))\ln (\phi (k)\ln n)}{\ln n}\text{,}$

where $\phi (k)$ is Euler's totient. The function $F(n\text{;}k\text{,}l)$ has been studied in [A. Derbal, A. Smati, C. A. Acad. Sci. Paris, Ser. I 339 (2004) 87–90]. In this Note we study the functions $F^{*}(n\text{;}k\text{,}l)$ and $D^{*}(n\text{;}k\text{,}l)$. We give explicitly their maximal orders and we compute effectively the maximum of $F^{*}(n\text{;}k\text{,}l)$ for $k=1\text{,}2\text{,}3$ and that of $D^{*}(n\text{;}k\text{,}l)$ for $k=1\text{,}3\text{,}5\text{,}7\text{,}8\text{,}9\text{,}10\text{,}11\text{,}13$. To cite this article: A. Derbal, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

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).     

Sur la topologie des fibres d'une fonction définissable dans une structure o-minimale

Let $\text{V?}\text{R}{}^{\mathrm{n}}$ be a closed, non compact C² manifold and $\text{f}\text{:V→}\text{R}$ be a C² function definable in an o-minimal structure. We prove that the flow of the gradient field of f with respect to the induced riemannian metric on V embeds a non singular asymptotic critical level of f into a typical level of f. We apply this result to complex polynomials. To cite this article: D. D'Acunto, C. R. Acad. Sci. Paris, Ser. I 337 (2003).