Insensitizing controls for a large-scale ocean circulation model

We consider here a linear quasi-geostrophic ocean model. We look for controls insensitizing (resp. ε-insensitizing) an observation function of the state. The existence of such controls is equivalent to a null controllability property (resp. an approximate controllability property) for a cascade Stokes-like system. Under reasonable assumptions on the spatial domains where the observation and the control are performed, we are able to prove these properties. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Méthode d'éléments finis avec hybridisation frontière pour les problèmes de contact avec frottementFinite element method with Lagrange multipliers for contact problems with friction

In this Note, we propose a finite element method with Lagrange multipliers in order to approximate contact problems with friction. The discretized normal and tangential constraints at the candidate contact interface are expressed by using continuous piecewise linear Lagrange multipliers in the saddle-point formulation. An optimal error estimate is established and several numerical studies corresponding to this choice of the discretized normal and tangential constraints are achieved. To cite this article: L. Baillet, T. Sassi, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 917–922.     

Local nonglobal minima for solving large-scale extended trust-region subproblems

We study large-scale extended trust-region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust-region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS  and that there are efficient algorithms for solving large-scale TRS  problems. It is also known that there can exist at most one local non-global minimizer (LNGM) for TRS. We combine this with known characterizations for strong duality for eTRS  and, in particular, connect this with the so-called hard case for TRS. We begin with a recent characterization of the minimum for the TRS  via a generalized eigenvalue problem and extend this result to the LNGM. We then use this to derive an efficient algorithm that finds the global minimum for eTRS  by solving at most three generalized eigenvalue problems.     

Braids on surfaces and finite type invariants

We prove that there is no functorial universal finite type invariant for braids in Σ×I if the genus of Σ is positive. To cite this article: P. Bellingeri, L. Funar, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Orthogonal polynomials and a generalized Szegő condition

Asymptotical properties of orthogonal polynomials from the so-called Szeg? class are very well-studied. We obtain asymptotics of orthogonal polynomials from a considerably larger class and we apply this information to the study of their spectral behavior. To cite this article: S. Denisov, S. Kupin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Lemme du mouvement rigide infinitésimal en coordonnées lipschitziennes et application aux coques de régularité minimale

We establish a version of the infinitesimal rigid displacement lemma in curvilinear Lipschitz coordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz. To cite this article: S. Anicic et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Actions moyennables et fonctions harmoniquesAmeanable actions and harmonic functions

We prove that the action of a countable discrete group on a locally compact invariant space of minimal harmonic functions is ameanable. To cite this article: P. Biane, E. Germain, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 355–358.     

Explicit Plancherel formula for the p-adic group GL(n)

We provide an explicit Plancherel formula for the p-adic group GL(n). We determine explicitly the Bernstein decomposition of Plancherel measure, including all numerical constants. We also prove a transfer-of-measure formula for GL(n). To cite this article: A.-M. Aubert, R. Plymen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A level-set formulation of immersed boundary methods for fluid–structure interaction problems

We propose a level set formulation of the immersed boundary method for fluid–structure problems in two and three dimensions. We prove that the resulting model verifies an energy estimate. To cite this article: G.-H. Cottet, E. Maitre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Archimedean zeta integrals on GL n × GL m and SO2n+1 × GL m

In this paper, we evaluate archimedean zeta integrals for automorphic L-functions on GL _n × GL _n-1+? and on SO_2n+1 × GL _n+? , for ? = ?1, 0, and 1. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals. When ? = 0 or ? = 1, the archimedean zeta integrals amount to integrals over the full torus. We show that, as has been predicted by Bump for such domains of integration, these zeta integrals are equal to the corresponding local L-factors—which are simple rational combinations of Gamma functions. (In the cases of GL _n × GL _n-1 and GL _n × GL _n this has, in large part, been shown previously by the second author of the present work, though the results here are more general in that they do not require the assumption of trivial central characters. Our techniques here are also quite different. New formulas for GL(n, R) Whittaker functions, obtained recently by the authors of this work, allow for substantially simplified computations). In the case ? = ?1, we express our archimedean zeta integrals explicitly in terms of Gamma functions and certain Barnes-type integrals. These evaluations rely on new recursive formulas, derived herein, for GL(n, R) Whittaker functions. Finally, we indicate an approach to certain unramified calculations, on SO_2n+1 × GL _n and SO_2n+1 × GL _n+1, that parallels our method herein for the corresponding archimedean situation. While the unramified theory has already been treated using more direct methods, we hope that the connections evoked herein might facilitate future archimedean computations.     

Sur la complexité des nombres algébriques

Let b?2 be an integer. We prove that real numbers whose b-ary expansion satisfies some given, simple, combinatorial condition are transcendental. This implies that the b-ary expansion of any algebraic irrational number cannot be generated by a finite automaton. To cite this article: B. Adamczewski et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Comment établir des conditions nécessaires d'optimalité dans les problèmes de contrôle dont certains arguments sont déviés ?

We give a general idea to state necessary optimality conditions for problems of variational calculus or optimal control problems with deviating arguments. To cite this article: L. Samassi, R. Tahraoui, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Counting moves in knight's tours

A knight's tour contains eight types of elementary moves. We prove that the only asymptotic constraints on the numbers of moves of each type are the trivial ones: for all proportions compatible with these constraints, there exists a sequence of tours asymptotically achieving these proportions. We deduce a positive answer to the question asked by A. Grigis in C. R. Acad. Sci. Paris, Ser. I 335 989–992 about the existence of tours with an arbitrarily large index. To cite this article: P. Dehornoy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Factorialité de l'anneau des séries de Dirichlet analytiques

We study arithmetical properties of the ring of analytic Dirichlet series. In particular, we prove a theorem of division by several series and we deduce from it that the ring is factorial. To cite this article: F. Bayart, A. Mouze, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the contact boundaries of normal surface singularities

The abstract boundary M of a normal complex-analytic surface singularity is canonically equipped with a contact structure. We show that if M is a rational homology sphere, then this contact structure is uniquely determined by the topological type of M. An essential tool is the notion of open book carrying a contact structure, defined by E. Giroux. To cite this article: C. Caubel, P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Limiting laws associated with Brownian motion perturbated by normalized exponential weights

We perturb Brownian motion on the time interval [0,t] by an exponential weight; we show that for a large class of these weights the corresponding probability laws converge weakly as t→∞. To cite this article: B. Roynette et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Utilisation des modules de Drinfeld en cryptologie

We present in this Note a new and efficient public-key cryptosystem based on Drinfeld modules. The details will appear elsewhere. To cite this article: R. Gillard et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).