Stockage diophantien et hypothèse abc généralisée

Let (M, L, φ) be a triple consisting of a first order language L and a set M that admits an interpretationof L. We define what it means for a second such triple (M′, L′, φ′) to be positive existential in (M, L, φ), which generalizes the notion of positive existential set. If the cartesian product (M × M, L,φ×φ ) is positive existential in (M, L, φ), we say it admits positive existential storing. This generalizes the notion of a storing function, known from recursion theory. We then prove that Z and certain function fields admit positive existential storing.     

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 généralisation du problème de Dirichlet

Sans résumé     

Sur la généralisation du problème de Dirichlet

Sans résumé     

Une généralisation du théorème de Kobayashi-Ochiai

We establish relations between the growth of non-degenerate holomorphic maps \(\varphi\) from \({\mathbb{C}}^n\) to X, an n-dimensional compact Kähler manifold, and the positivity of the canonical bundle of X. The general principle is that the growth increases with this positivity. In the extreme case where X is of general type, such maps do not exist, a result of Kobayashi-Ochiai. In the other extreme, if the growth is sufficiently slow (see theorem 1), we show that X is uniruled if projective. K. Kodaira obtained the weaker property that X has no nonzero pluricanonical forms. Our results interpolate between, and include, these two extreme cases.     

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).     

Généralisation de la notion de l'enveloppe

Sans résumé Présenté par G. Hajós     

Intégrales orbitales semi-simples et centre de l'algèbre enveloppante

In a previous Note, the first author has established an explicit local formula for semi-simple orbital integrals associated with the Casimir. In this Note, we extend the formula to all elements of the center of the Lie algebra.     

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 $A(G)$.     

Un critère de Darboux d'existence d'intégrale première pour les 1-formes différentielles analytiques

Resumé On considère une 1-forme analytique définie sur un ouvert borné de ⁿ et qui s'étend au voisinage de l'adhérence de cet ouvert. On montre que si elle possède une infinité de feuilles contenues dans des hypersurfaces algébriques différentes et de degrés bornés alors elle admet une intégrale première méromorphe. Si de plus l'origine est dans l'adhérence de chacune de ces feuilles alors la 1-forme admet une intégrale première méromorphe de graphe algébrique. Des exemples illustrent l'optimalité de ces résultats.

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Consider a germ of analytic differential one form and suppose there exists infinitely separatrix in algebraic surfaces of bounded degree. The conclusion is that there exists an algebraic first integral. We give a list of examples wich explain the necessity of the hypotheses.

Financements: CNRS et réseau européen TMR Sing. Eq. Diff. et Feuilletages     

Généralisation d’un théorème de Greenberg

We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic \(\mathbb {Z}_\ell \)-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the classical conjectures of Leopoldt and of Gross-Kuz’min. We so extend a result of semi-simplicity of Greenberg and, by the way, an isomorphism of Kuz’min.     

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).     

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).     

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).     

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.     

Borne sur l'advection et la hauteur d'eau du problème de shallow-water avec conditions aux limites de Dirichlet

Using some properties of the Hardy spaces, we give a regularity result on the advection term of the shallow-water equations and we show a $L^2$ bound holding up to the boundary on the water height with Dirichlet boundary conditions. To cite this article: F. Flori, P. Orenga, C. R. Acad. Sci. Paris, Ser. I 340 (2005).