Multiplicité des solutions d'équilibre d'une coque de Marguerre–von Kármán membranaireMultiplicity of solutions to the Marguerre–von Kármán membrane equations

We build explicitly an infinite number of equilibrium solutions of unloaded Marguerre–von Kármán membrane shells. This construction is based upon the existence of three elementary solutions, together with the solution of a Monge–Ampère equation associated with a partition of the reference configuration of the shell. To cite this article: A. Léger, B. Miara, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 649–654.     

Using local search to speed up filtering algorithms for?some NP-hard constraints

This paper proposes to use local search inside filtering algorithms of combinatorial structures for which achieving a desired level of consistency is too computationally expensive. Local search quickly provides supports for many variable-value pairs, thus reducing the effort required to check and potentially filter the rest of them. The idea is demonstrated on the SomeDifferent constraint, a graph coloring substructure. An experimental evaluation confirms its significant computational gain in many cases.     

On the spectrum of the sum of generators for a finitely generated group

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π. For the universal representationπ _un of Γ, we prove in particular the following results. First, the spectrum Sp(π _un (h)) contains the complex numberz of modulus one iff Sp(π _un (h)) is invariant under multiplication byz, iff there exists a character such that η(S)={z}. Second, forS ⁻¹=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π _un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.      

Central limit theorems for the zeros of Lamé polynomials

In a recent paper, it was shown that the zeros of Lamé polynomials satisfy a strong law of large numbers. In this paper, we show that the zeros also satisfy two central limit theorems.     

A Newton type rational interpolation formula

We give a Newton type rational interpolation formula (Theorem 2.2). It contains as a special case the original Newton interpolation, as well as the interpolation formula of Liu, which allows to recover many important classical q-series identities. We show in particular that some bibasic identities are a consequence of our formula.     

Shrinkage structure in biased regression

Biased regression is an alternative to ordinary least squares (OLS) regression, especially when explanatory variables are highly correlated. In this paper, we examine the geometrical structure of the shrinkage factors of biased estimators. We show that, in most cases, shrinkage factors cannot belong to [0,1] in all directions. We also compare the shrinkage factors of ridge regression (RR), principal component regression (PCR) and partial least-squares regression (PLSR) in the orthogonal directions obtained by the signal-to-noise ratio (SNR) algorithm. In these directions, we find that PLSR and RR behave well, whereas shrinkage factors of PCR have an erratic behaviour.