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).     

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

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).     

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).     

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).     

Répartition des densités des sous-suites d'une suite d'entiers

Let $\text{A}$ denote a strictly increasing sequence of integers; for any integer n, define A(n) to be the number of positive elements of $\text{A}$ not exceeding n. The upper and lower asymptotic densities of $\text{A}$ are defined by

We describe the set of pairs (d$\text{B}$, $\text{d}$$\text{B}$), where $\text{B}$ runs over all subsequences of $\text{A}$, as being a closed convex region of the plane. The converse statement is also proved.     

<Emphasis Type="Italic">F</Emphasis>-isocristaux surconvergents et surcohérence différentielle

Résumé Soient un anneau de valuation discrète complet d’inégales caractéristiques, de corps résiduel parfait k, un -schéma formel propre et lisse, T un diviseur de la fibre spéciale P de , U l’ouvert de P complémentaire de T, Y un sous-k-schéma fermé lisse de U. Nous prouvons que la catégorie des F-isocristaux surconvergents sur Y est équivalente à celle des F-isocristaux surcohérents sur Y (voir [Car, 6.2.1 et 6.4.3.a)]). Plus généralement, nous établissons par recollement une telle équivalence pour tout k-schéma séparé lisse Y. Nous vérifions de plus que les F-complexes de -modules à cohomologie bornée et -surcohérente se dévissent en F-isocristaux surconvergents.     

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.     

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 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).     

Sous-groupes canoniques et cycles évanescents <Emphasis Type="Italic">p</Emphasis>-adiques pour les variétés abéliennes

Sans résumé     

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).     

Schémas d'approximation associés à une équation différentielle dirigée par une fonction höldérienne ; cas du mouvement brownien fractionnaire

We study here classical approximation schemes (Euler, Milshtein) associated with a differential equation of the type $\mathrm{d}{x}_t=\sigma (x_t)\mathrm{d}{g}_t+b(x_t)\mathrm{d}t$, $x_t\in \mathbb{R}$, where g is a function, supposed Hölderian of order α somewhere in $(0,1]$. When $g=B^H$ is the trajectory of fractional Brownian movement, we deduce probability properties to refine the results. To cite this article: I. Nourdin, 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).     

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).