Un théorème de Liouville pour l'opérateur de Schrödinger avec dérive

Let $(M,g)$ be a complete Riemannian manifold without boundary of dimension n and V be a $C^2$ vector field on M such that $r(x)|V(x)|$ is bounded. Suppose that ${\mathrm{Ric}}_g(x)??\min \{\lambda (r(x))?{\mu }_{\mathrm{?}V}(x),\beta (r(x))\}$ outside a compact set of M, where ${\mu }_{\mathrm{?}V}$ denotes the upper eigenvalue of ?V and $\lambda ,\beta$ are non-negative decreasing functions such that ${\lim }_{t\to +\infty }{t}^2\lambda (t)=0$. There exists positive numbers $b_n$ and $c_n$ which depend only on n and $6V{6}_{\infty }$ such that if h is a $C^2$ function defined on M with $\mathrm{\Delta }h??c_n{a}^2$ and ${\mathrm{lim?sup}}_{R\to \infty }{R}^{\mathrm{?2}}{\min }_{x\in B_p(3R)?B_p(R)}h(x)??b_n{a}^2$, where $0?a<{\mathrm{lim?inf}}_{j\to \infty }h(z_j)$, where $(z_j)$ is a sequence of M such that $r(z_j)\to \infty$, then the equation $\mathrm{\Delta }u(x)+V(x)?\mathrm{?}u(x)+h(x)u(x)=0$ has no positive $C^2$ solution on M. To cite this article: S. Asserda, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

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).     

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).     

Comportement semi-classique pour l'opérateur de Schrödinger à potentiel périodique

In this paper, the Schrödinger operator with a periodic potential is considered. Let V a smooth periodic function, we study the semi-classical behavior for a continuum spectrum of −h² Δ + V (h → 0). We are interested by localization and width of bands. We give the interaction matrix up to an exponentially small error, measured by Agmon's distance between the wells. A detailed investigation of the spectrum is made for the case where V has one nondegenerate minima per unit cell. We also investigate the spectral properties of −h² Δ + V + ΔV, where ΔV is a smooth positive perturbation with compact support.     

Sur la distance à l'instabilité de polynômes matriciels quadratiques

A bisection method is developed for computing the distance to instability of quadratic matrix polynomials. The computation takes rounding errors into account.     

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).     

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).     

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).     

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).     

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).     

Régularité dans une équation de Schrödinger avec potentiel singulier à distance finie et à l'infini

In this Note we study the Schrödinger equation i?_tu+Δu+V₀u+V₁u=0 on $\text{R}{}^3\text{×(0,T)}$ with initial condition $\text{u}{}_0\text{∈{v∈H}{}^2\text{(}\text{R}{}^3\text{)}$, where V₀ is a coulombian potential, singular at finite distance and V₁ is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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).     

Un algorithme dérivé de l'algorithme de Metropolis

Statistical physics and stochastic modelling in economic sciences share the same mathematical bases given by the Gibbs distribution, but system characteristics are different. For instance, an economic system can be described by a Bose–Einstein statistics with few non-degenerate states and an infinitesimal “temperature”; under such conditions, the approximation of the most probable configuration is invalid. Therefore, the calculus of the exact solution needs using a Metropolis algorithm, which estimates a Gibbs distribution. This paper presents a much more efficient algorithm. For small systems, the exact distribution on the canonical set can be computed, and then this distribution is compared to the solutions of the old and new algorithms.     

Sur la détermination du périmètre de l'ovale à huit centres

This note aims to present the history of the main researches carried out to represent an ellipse by means of an eight-centered oval and to complete them by giving a geometrical demonstration of the formulas expressing the coordinates of the centers, the radii and the central angles of these arcs of circles making up this oval. These formulas will make it possible to determine its perimeter and thus to approximate the value of that of the ellipse. This result could be useful for archaeologists, who will test the hypothesis according to which some Roman amphitheatres, such as the Coliseum, were built from eight-centered ovals.     

Estimation de la densité de probabilité améliorée par pre-testing

We consider the nonparametric problem of multidimensional probability density estimate. Using concept of minimax risk with random normalizing factor introduced by Lepski [Math. Methods Statist. 8 (1999) 441–486], by considering an independence hypothesis, we build an estimator which can be adaptive and whose accuracy, depending on the observation, is better than the minimax estimate, $n^{?\frac{\beta }{2\beta +d}}$, with prescribed confidence level. To cite this article: A.F. Yode, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Une propriété de composition dans l'espace Hs

Let us assume that $1<p?q?+\infty$, $1/p<s?[s]<1$, and $[s]?1$. If f and g are functions in the Besov space $B_p^{s\text{,}q}(\mathbb{R})$, such that g is real valued and such that $f(0)=0$, then the composed function $f○g$ belongs to $B_p^{s\text{,}q}(\mathbb{R})$. To cite this article: G. Bourdaud, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiquesLyapounov exponents for discrete,quasi-periodic Schrödinger operators

We consider quasi-periodic Schrödinger operators H on $\text{Z}$ of the form H=H_λ,x,ω=λv(x+nω)δ_n,n′+Δ where v is a non-constant real analytic function on the d-torus $\text{T}{}^{\mathrm{d}}\text{(d?1)}$ and Δ denotes the discrete lattice Laplacian on $\text{Z}$. Denote by L_ω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector $\text{ω∈}\text{T}{}^{\mathrm{d}}$. It is shown that for |λ|>λ₀(v), there is the uniform minoration $\text{L}{}_{\mathrm{\omega }}\text{(E)>}\text{1}\text{2}\text{log}\text{|λ|}$ for all E and ω. For all λ and ω, L_ω(E) is a continuous function of E. Moreover, L_ω(E) is jointly continuous in (ω,E), at any point $\text{(ω}{}_0\text{,E}{}_0\text{)∈}\text{T}{}^{\mathrm{d}}\text{×}\text{R}$ such that k·ω₀≠0 for all $\text{k∈}\text{Z}{}^{\mathrm{d}}\text{?{0}}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.