Multiplication de fréquence et stabilité en phase

Summary A frequency multiplying system which delivers an exact multiple of the frequency of a 100 kc/s standard in the microwave region has been developed. High spectral purity of the signal has been obtained by servo-controlling the phase of a high frequency quartz crystal oscillator through an integrating filter with long time-constant.     

Solutions en dualité pour les gaz sans pression

We introduce the notion of duality solution for the system of pressureless gases, which ensures existence for the Cauchy problem, and uniqueness under optimal conditions on initial data.     

Visibilité dans le modèle Booléen

The aim of this Note is to give an estimate on the tail probability of the visibility function in a Boolean model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle. Convergence results are added, when the size of obstacles goes to zero and when the distance between the origin and the closest obstacle goes to infinity. To cite this article: P. Calka et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Une identité remarquable en théorie des partitions

We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials. The proof given is using -ring techniques. It would be interesting to obtain a bijective proof.

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Received January 18, 2000 / Revised January 23, 2000 / Published online July 20, 2000     

Inégalité d’Ahlfors en dimension supérieure

We prove an Ahlfors’ like inequality for the holomorphic curves with boundary of a complex compact Kobayashi-hyperbolic manifold.     

Bandes phononiques interdites en élasticité linéarisée

In this Note we rigorously justify the existence of elastic band gaps in three-dimensional periodic composite materials with strong heterogeneities. In particular, we show how to compute these bands. To cite this article: A. Ávila et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Ordre de Nevanlinna des fonctions sousharmoniques continues en une singularité isolée

Dans ce travail, nous étudions l"ordre de Nevanlinna des solutions de l'équation deSchrödinger u = u , où est une mesure de Kato, positive et radiale. Nous montrons que si ne vérifie pas le principe de Picard, alors toute solution non bornée de u = u est d'ordre infini. En utilisant la théorie de perturbation, l'effilement minimal de M. Brelot et la notion d'ensembles essentiels, nous donnons des conditions necessaires et des conditions suffisantes pour que l'ordre des solutions, non bornées au voisinage d'une singularité, soit infini. Enfin nous prouvons que le comportement des solutions d'une telle équation ne permet pas de caractériser la validité ou non du principe de Picard.     

Non-annulation effective et positivité locale des fibrés en droites amples adjoints

We show how to use effective non-vanishing to prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. We prove the effective non-vanishing conjecture of Ionescu–Kawamata in dimension 3 in the case of line bundles of “high” volume.     

Sur le discriminant et la forme trace d'une involution orthogonale en degré 4

Résumé. let A be a central simple algebra of degree 4 over a fieldF of caracteristic not 2 and an orthogonal involution on A. Denotes the trace form of . In this note we show that , ie. is similar to a 4-Pfister form (over F) if and only if is a sum of two squares in F. It follows, in the split case , that if then, the hermitian level of is finite if and only if is weakly isotropic. A similar result holds with ordinary squares and trace form for any central simple algebra (see [5] and [7]).

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Received July 15, 1998 / in final from November 3, 1999 / Published online July 20, 2000     

L’inégalité de Melin-Hörmander en Caractéristiques Multiples

We prove here the Melin-Hörmander inequality with the optimal loss of k/2 - 1/4 derivatives. We then prove that if k ≥ 2, the inequality with k/2-1/4 loss is false in general. We correct and improve the work of [13], which has well posed the question.     

Dualité de Spanier–Whitehead en géométrie algébrique

In this Note, we study finitely presented objects in the stable ${\mathbb{A}}^1$-homotopy category introduced by Morel and Voevodsky and we use a theorem by Voevodsky and Ayoub on the four functors to get Spanier–Whitehead duality in ${\mathbb{A}}^1$-homotopy theory. To cite this article: J. Riou, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Estimation de la densité invariante de systèmes dynamiques en dimension 1

In this Note we obtain a central limit theorem for standard kernel invariant density estimates of one-dimensional dynamical systems. The two main steps in the proof of this theorem are the following: the study of speed of convergence for the variance of the estimator and then a variation on the Lindeberg–Rio method [6].