On the Colin de Verdière number of graphs

Let μ(G) and ω(G) be the Colin de Verdière and clique numbers of a graph G, respectively. It is well-known that μ(G)?ω(G)-1 for all graphs. Our main results include μ(G)?ω(G) for all chordal graphs; μ(G)?tw(G)+1 for all graphs (where tw is the tree-width), and a characterization of those split (⊆ chordal) graphs for which μ(G)=ω(G). The bound μ(G)?tw(G)+1 improves a result of Colin de Verdière by a factor of 2.     

Boundary value problems for mixed type equations and applications

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations which arise in applications to differential geometry.     

Interior gradient estimates for solutions to the linearized Monge–Ampère equation

Let ? be a convex function on a convex domain Ω⊂Rn, n?1. The corresponding linearized Monge–Ampère equation istrace(ΦD²u)=f, where is the matrix of cofactors of D²?. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to Lp(Ω) for some p>n. The function ? is assumed to be such that with ?=0 on ∂Ω and the Monge–Ampère measure is given by a density g∈C(Ω) which is bounded away from zero and infinity.     

Calcul à la rupture en présence d'un écoulement : formulation cinématique avec un champ de pression approchéYield design formulation for porous media subjected to flow,using approximate pressure field

We attempt here to use the kinematic method of yield design in the case of a porous medium subjected to flow (with or without free surface), without looking for the exact solution of the pressure field. The method proposed here is based on the use of approximate pressure fields. In this paper, we show how, under different conditions concerning the yield criterion and the velocity field, the use of such approximate fields allows one to obtain a necessary condition for stability without having to find the real pressure field. To cite this article: A. Corfdir, C. R. Mecanique 334 (2006).     

Generalized Polak-Ribière algorithm

A new generalized Polak-Ribière conjugate gradient algorithm is proposed for unconstrained optimization, and its numerical and theoretical properties are discussed. The new method is, in fact, a particular type of two-dimensional Newton method and is based on a finite-difference approximation to the product of a Hessian and a vector.     

A globally convergent version of the Polak-Ribière conjugate gradient method

In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribière conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that with this line search every limit point produced by the Polak-Ribière iteration is a stationary point of the objective function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the step-size acceptance rules proposed here will retain the same good features of the Polak-Ribière method, while avoiding pathological situations. This research was supported by Agenzia Spaziale Italiana, Rome, Italy.     

Analyse de films minces et ultra-minces: Perspectives nouvelles en spectrométrie d'émission des rayons X

This paper illustrates the capabilities of “low-energy electron induced X-ray spectrometry” (LEEIXS) for surface and thin, or ultra-thin, film analysis. The work deals with anodic films and shows that it is possible to measure oxide film thicknesses in a range between 0 and a few hundred angströms. In addition, it is demonstrated that this technique provides some information about the distribution of impurities in the interior of the films.     

Thermodynamic behaviour of some glycol—water mixtures. Excess and partial volumes

Densities of water—glycol (mono- di-, tri- and tetraethyleneglycol) mixtures have been measured over the entire composition range at 298. 15 K. Mixtures involving monoethyleneglycol (MEG) have also been studied at different temperatures from 308.15 to 288.15 K. Deviations, V^E, from ideal volumes of mixing have been calculated: negative values are observed for all systems. They increase with the number of ether functions present in the organic molecule; a slight dependence of V^E towards temperature has been shown in the water—MEG mixtures. Partial molal volumes have also been calculated for both components of each system; they exhibit an extremum in the water-rich region.     

Mechanism of a secondary reaction during the coulometric determination of uranium in a mixture with iron

Summary The mechanism of a photochemical secondary reaction which takes place between iron and uranium during the coulometric determination of uranium is explained. As a result of this interfering effect lower uranium values are found. The appropriate working conditions to obtain reliable uranium measurements are indicated.We wish to thank Mr. H. L. Eschbach and Mr. V. Verdingh for their helpful comments when drafting this paper.