Relation de Rankine Hugoniot faible pour les lois de conservation avec contraintes primitives convexes

We describe a Weak Rankine Hugoniot relation for constrained discontinuous solutions of conservation laws. The constraint is convex in the primitive variable. Various examples show the potential interest of this approach. To cite this article: B. Després, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Twin Paradox and the Logical Foundation of Relativity Theory

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND.     

Dynamics of kinks in biological membranes

kinks created in a biological membrane by the interaction with a movable bead. We arrive at the evolution equations for both the bead and the membrane, whence we conclude that the force exerted on the bead by a fixed membrane points in the direction along which the curvature of the membrane is more concentrated. This is the first step towards understanding the basic mechanism behind the dynamics of protein aggregation which takes place on biological membranes. Received November 6, 2001 / Published online February 4, 2002     

An attempt on the calculation of relativistic potential energy curves: Noble gas difluorides

Total energy SCF calculations were performed for noble gas difluorides in a relativistic procedure and compared with analogous non-relativistic calculations. The discrete variational method with numerical basis functions was used. Rather smooth potential energy curves could be obtained. The theoretical Kr-F and Xe-F bond distances were calculated to be 3.5 a.u. and 3.6 a.u. which should be compared with the experimental values of 3.54 a.u. and 3.7 a.u. Although the dissociation energies are off by a factor of about five it was found that ArF₂ may be a stable molecule. Theoretical ionization energies for the outer levels reproduce the experimental values for KrF₂ and XeF₂ to within 2 eV.     

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.     

On the Behavior of Some Second Order Nonlinear Differential Equations

Qualitative behavior of second order nonlinear differential equations of type y″+p(t)y+q(t)f(y)=0, is studied. It includes properties such as positivity, number of zeros, oscillating nature, boundedness and monotonicity of the solutions.     

Extremal positive and self-adjoint extensions of suboperators

This work is supported in part by the fund OTKA (No. 5-134).