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).     

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).     

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).     

Sur la densité d'état de l'opérateur de Schrödinger quasi-périodique unidimensionnel

We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrödinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Hölder-$\frac{1}{2}$ function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. To cite this article: S. Hadj Amor, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On Saint Venant's compatibility conditions and Poincaré's lemma

Saint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in $L^2$ by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in $H^{\mathrm{?1}}$. Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On Poincaré's and J.L. Lions' lemmas

Let Ω be a bounded, connected and simply connected open subset of ${\mathbb{R}}^N$ with a Lipschitz continuous boundary. It is shown that an irrotational vector field whose components are in $H^{\mathrm{?1}}(\Omega )$ is the gradient of a function in $L^2(\Omega )$. It is also shown that this generalization of a classical lemma of Poincaré is equivalent to a well-known lemma of J.L. Lions. To cite this article: S. Kesavan, 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).     

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).     

A variant of Poincaré's inequality

We show that if $\text{Ω?}\text{R}{}^{\mathrm{N}}\text{,}\text{N?2}$, is a bounded Lipschitz domain and $\text{(ρ}{}_{\mathrm{n}}\text{)?L}{}^1\text{(}\text{R}{}^{\mathrm{N}}\text{)}$ is a sequence of nonnegative radial functions weakly converging to δ₀ then there exist C>0 and n₀?1 such that

$\text{∫}\text{Ω}\text{f??}\text{∫}\text{Ω}\text{f}{}^{\mathrm{p}}\text{?C}\text{∫}\text{Ω}\text{∫}\text{Ω}\text{|f(x)?f(y)|}{}^{\mathrm{p}}\text{|x?y|}{}^{\mathrm{p}}\text{ρ}{}_{\mathrm{n}}\text{(|x?y|)}\text{d}\text{x}\text{d}\text{y}\text{?f∈L}{}^{\mathrm{p}}\text{(Ω)}\text{?n?n}{}_0\text{.}$

The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu (in: Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). As n→∞ in (1) we recover Poincaré's inequality. We also extend a compactness result of Bourgain, Brezis and Mironescu. To cite this article: A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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).     

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).     

Sur les bornes inférieures de l'écart des variétés invariantes

Let m be an integer ?3, set $?(r)=\frac{1}{2}(r_1^2+?+r_{m?1}^2)+r_m$ for $r\in {\mathbb{R}}^m$, and consider a badly approximable vector ${\overline{\omega }}^0\in {\mathbb{R}}^{m?2}$. Fix $\alpha >1$, $L>0$ and $R>1+6{\overline{\omega }}^0{6}_{\infty }$. We construct a sequence $(H_N)$ of $\text{Gevrey}\text{-}(\alpha ,L)$ Hamiltonian functions of ${\mathbb{T}}^m\times {\overline{B}}_{\infty }(0,R)$, which converges to ? when $N\to \infty$, such that for each N the system generated by $H_N$ possesses a $(m?1)$-dimensional hyperbolic invariant torus with fixed frequency vector $({\overline{\omega }}^0,1)$, which admits a homoclinic point with splitting matrix of the form $\mathrm{diag}(0,{\nu }_N,\dots ,{\nu }_N,0)\in M_m(\mathbb{R})$, with ${\nu }_N?\exp (?c{(\frac{1}{{?}_N})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2(\alpha ?1)(m?2)$}\right.})$, where ${?}_N:=6H_N??6_{\alpha ,L}$ and $c>0$. To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Test d'hétéroscédasticité quand les covariables sont fonctionnelles

We present in this paper a consistent nonparametric test for heteroscedasticity when data are of functional kind. The latter is constructed by evaluating the difference between the conditional and unconditional variances. We show the asymptotic normality of the statistical test under the null hypothesis. In addition, we prove that this test is consistent against all deviations from homoscedasticity condition.     

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).     

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.     

L'électron habillé non relativiste dans un champ magnétique

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the $x_3$-axis and with a quantized electromagnetic field. Because of the translation invariance along the $x_3$-axis, we consider the reduced Hamiltonian associated with the total momentum along the $x_3$-axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the $x_3$-axis are sufficiently small. Finally, we determine the absolutely continuous spectrum of the reduced Hamiltonian and we prove that the renormalized mass of the electron is greater than its bare one. To cite this article: L. Amour et al., 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).     

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).