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).     

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).     

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).     

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).     

Turán问题37—39的完全解

本文给出Turán的关于Birkhoff插值的问题37—39的完全解。     

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).     

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).     

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).     

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).     

Minimal submanifolds in ℝ<Superscript>4</Superscript> with a G.C.K. structure

In this paper we obtain all invariant, anti-invariant and CR submanifolds in (?⁴, g, J) endowed with a globally conformal Kähler structure which are minimal and tangent or normal to the Lee vector field of the g.c.K. structure.     

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).     

Lah numbers and Lindström's lemma

We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a consequence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.     

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 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).     

Généralisation d'un théorème de R. Perez-Marco au cas p-adique

In this Note, we prove, in the p-adic case, a 0–1 law about the linearization of families ${(f_t:{\mathbb{C}}_p^m\to {\mathbb{C}}_p^m)}_{t\in {\mathbb{Q}}_p}$ under a polynomial perturbation in t. This gives new examples of non linearizable functions in the p-adic case. To cite this article: D. Vieugué, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

本刊英文版Vol.39(2023),No.6论文摘要（英文）

<正>On the Termination of Flips for Log Canonical Generalized Pairs Guo Du CHEN Nikolaos TSAKANIKAS Abstract We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4.As main ingredients,we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs,and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of...     

本刊英文版Vol.39(2023),No.5论文摘要（英文）

<正>Monge-Ampère Type Equations on Almost Hermitian Manifolds Jiao Gen ZHANG Abstract In this paper we consider the Monge-Ampere type equations on compact almost Hermitian manifolds.We derive C^∞ a priori estimates under the existence of an admissible C-subsolution.Finally,we obtain an existence result if there exists an admissible supersolution.     

本刊英文版Vol.39(2023),No.4论文摘要（英文）

<正>Variable Besov-type Spaces Zouheyr ZEGHAD Douadi DRIHEM Abstract In this paper we introduce Besov-type spaces with variable smoothness and integrability.We show that these spaces are characterized by theφ-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces.Moreover the Sobolev embeddings for these function spaces are obtained.     

本刊英文版Vol.39(2023),No.3论文摘要（英文）

<正>Maps on Positive Cones of C^*-algebras Ming Chu GAO Gui Mei AN Abstract We prove that a surjective map (on the positive cones of unital C^*-algebras)preserves the minimum spectrum values of harmonic means if and only if it has a Jordan*-isomorphism extension to the whole algebra.We represent weighted geometric mean preserving bijective maps on the positive cones of prime C^*-algebras in terms of Jordan*-isomorphisms of the algebras.We also characterize orde...