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 auto-similaires non radiales pour l'équation quasi-géostrophique dissipative critique

We prove the existence of self-similar solutions for the critical dissipative quasi-geostrophic equation by using the formalism of mild solutions in a space close to $L^{\infty }$. To cite this article: F. Marchand, P.G. Lemarié-Rieusset, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

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).     

Paradoxe de Klein pour l'équation de Klein–Gordon chargée : superradiance et opérateur de diffusion

We develop the scattering theory for the charged Klein–Gordon equation on ${\mathbb{R}}_t\times {\mathbb{R}}_x$, when the electrostatic potential $A(x)$ has different asymptotics $a^{\pm }$ as $x\to \pm \infty$. In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation, and we establish the existence of a Scattering Operator the symbol of which has a norm strictly larger than 1, for the frequencies in $(a^{?}\text{,}{a}^{+})$. These results can be applied to the DeSitter–Reissner–Nordstrøm metric, to justify the notion of superradiance of the charged black-holes. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D

We present a parameterized family of finite difference schemes for the exact controllability of the 1-D wave equation. These schemes differ from the usual centered ones by additional terms of order to $h^2$, where h denotes the discretization step in space. Using a discrete version of Ingham's inequality for nonharmonic Fourier series, the spectral properties and dispersion diagrams of the schemes, we determine the parameters leading to a uniform controllability property with respect to h and an optimal stability CFL condition. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (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).     

Décomposition de domaine pour un calcul hybride de l'équation de helmholtz

We prove an estimate for the Dirichlet-Neumann operator, and for the H¹ local norm for solutions of Helmholtz equation outside an obstacle without trapping rays. We give an algorithm solving Helmholtz equation outside a union of such obstacles. Convergence follows from this estimate. At each step of the resolution, only one obstacle is considered for itself; this defines a decomposition domain technique fitting this equation. One can use different numerical schemes, one at each step, adapted to the considered component of the obstacle; therefore, this algorithm is a hybrid computation. The results are given for two obstacles, and the generalization is straightforward     

Un problème d'optimisation de forme pour la contrôlabilité exacte de l'équation des ondes 2D

We consider the wave equation defined on $\Omega ?{\mathbb{R}}^2$ and $\omega ?\Omega$. We designate by $v_{\omega }$ the distributed control of minimal $L^2(\omega \times (0,T))$ norm obtained with the Hilbert Uniqueness Method which stabilizes the system at time $T>0$. This Note addresses the question of the optimal position of ω in order to minimize $J:\omega \to 6v_{\omega }{6}_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^{1,1}(\Omega )$, we express the shape derivative of J as a curvilinear integral on ?ω (independently of any adjoint solution) leading to a descent algorithm. A numerical application is given. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Un problème mixte pour l’équation de Korteweg-de Vries sur un intervalle borné

The Korteweg-de Vries equation occurs as a model for unidirectional propagation of small amplitude long waves in numerous physical systems. The aim of this work is to propose a well posed mixed initial-boundary-value problem when the spacial domain is of finite extent. More precisely, we establish local existence of solutions for arbitrary initial data in the Sobolev space H¹, and global existence for small initial data in this space.     

Valuations invariantes pour l'action des groupes de Galois différentiels

Let $(F/K\text{,}?)$ be a differential field extension with differential Galois group $G={Gal}_{?}(F/K)$. For the natural action of G on the Riemann–Zariski variety $S^{?}=S^{?}(F/K)$ of the field extension $F/K$, we study the invariant valuations $\nu \in {\left(S^{?}\right)}^G$ when they do exist. We show close relations between these invariant valuations and the elements of F holonomic over K. Next, we study the continuity of the derivation ? with respect to these ν-adic topologies. We give a geometric structure property of G-invariant valuation inspired by Zariski. Finally, we give an answer for the existence problem of invariant valuations in the context of Picard–Vessiot extension. To cite this article: G. Duval, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Vitesse de convergence en norme p-intégrale et normalité asymptotique de l'estimateur crible de l'opérateur d'un ARB(1)

The autoregressive model in a Banach space (ARB) allows to represent many continuous time processes used in practice (see, for example, D. Bosq, Linear Processes in Function Spaces: Theory and Applications, 2000, Springer, p. 150). In this Note we study an estimator of the operator in ARB(1) by the least squares method, when the operator is strictly p-integral, $p\in ]1,\infty [$, and we use Grenander's method of sieves (From U. Grenander, Abstract Inference, Wiley, 1981). We show consistency of the sieve estimator and we derive a central limit theorem for this estimator. To cite this article: F. Rachedi, C. R. Acad. Sci. Paris, Ser. I 341 (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).     

Algorithmique et calculs de complexité pour un solveur de type dissections emboîtées

Résumé On considère la méthode de dissections emboîtées basée sur des théorèmes de séparation introduite par Gilbert-Tarjan et Roman utilisée pour la résolution par élimination de Gauss de grands systèmes linéaires creux. Plus précisemment, on étudie une structure de données par blocs similaire à celle proposée par George dans le cadre des graphes en grille, et on démontre les propriétés suivantes: d'une part, pour des familles de graphes à degré borné admettant unn -théorème de séparation, 1/2<1, le stockage secondaire de la structure par blocs contenant la matrice factorisée est linéaire par rapport à la taille du système; d'autre part, en rajoutant une hypothèse non restrictive sur la manière d'effectuer la séparation, la structure peut être construite en temps linéaire par une factorisation logique par blocs. Des exemples numériques illustrent ces résultats théoriques.

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Algorithmic study and complexity bounds for a nested dissection solver

Summary We consider the nested dissection method based on separator theorems introduced by Gilbert-Tarjan and Roman used for solving large sparse systems of linear equations. More precisely, we study a block storage scheme such as proposed by George for regular square grids and we prove the following results: first, for families of graphs of bounded degree withn -separator theorem, 1/2<1, the overhead storage of the block data structure for the factored matrix is linear in system dimension; on the other hand, by adding a non restrictive assumption on the separation, this structure can be constructed by a block symbolic factorization which runs in time linear in matrix dimension. Numerical experiments illustrating these theoretical results are provided.

     

Une caractérisation de l'algèbre de Fourier pour certains groupes localement compacts

Let G be a separable locally compact group with type-I left regular representation, $\stackrel{?}{G}$ its dual and $A(G)$ its Fourier algebra. We prove an analogue of Parseval's theorem and that the mapping

$T?u(x):=\underset{\stackrel{?}{G}}{\int }Tr[T(\pi )\pi {(x)}^{?1}]{\mathrm{d}}_{\mu }(\pi )$

is an isometric isomorphism of Banach spaces from $L^1(\stackrel{?}{G})$ onto $A(G)$.     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.