Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions

For generic families of vector fields or transformations, normally hyperbolic invariant products of spheres appear near partially elliptic rest points. To cite this article: M. Kammerer-Colin de Verdière, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A Multidimensional Branch-and-Prune Method for Interval Global Optimization

In this paper a new multidimensional extension of the recently developed one-dimensional enclosure method called kite is given for interval global optimization. A more sophisticated version of the pruning technique based on the kite method is introduced. By the new componentwise approach all the one-dimensional theoretical results and procedures can be used in the higher-dimensional case. The possibilities in the implementation of the new algorithm together with numerical results on 40 standard test problems are presented.     

Trois formulations d'un modèle de plasma quasi-neutre avec courant non nul

In this Note, we propose three formulations of a model describing a quasi-neutral plasma with non-vanishing current. In order to study and compare the numerical efficiency of each formulation, two test-problems are implemented in one dimension. The first is a periodic perturbation of a uniform stationary plasma. The second is a case of plasma expansion in vacuum between two electrodes. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On Certain Perturbations of the Erdös–Renyi Random Graph

We study perturbations of the Erdös–Renyi model for which the statistical weight of a graph depends on the abundance of certain geometrical patterns. Using the formal correspondance with an exactly solvable effective model, we show the existence of a percolation transition in the thermodynamical limit and derive perturbatively the expression of the threshold. The free energy and the moments of the degree distribution are also computed perturbatively in that limit and the percolation criterion is compared with the Molloy–Reed criterion.     

Weak density of smooth maps for the Dirichlet energy between manifolds

We prove that given a simply connected compact manifold M and a closed manifold N, any map in the Sobolev space W ^1,2(M,N) can be approximated weakly by smooth maps between M and N. Submitted: September 2002, Final version: November 2002.     

Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations

In this work we introduce a class of balanced numerical schemes, up to second order, for the solution of general non-homogeneous hyperbolic systems of conservation laws. We give a general technique to build such schemes. We also prove that they balance up to second order a large class of steady solutions in the whole domain but some subset whose measure tends to zero as the grid size decreases to zero. We finally present an application to Shallow Water equations that exhibit the good performances of some of the schemes introduced. To cite this article: T. Chacón Rebollo et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)}$ for $\text{σ}\text{→}\text{∈Σ}{}_{\mathrm{N}}\text{={0,1}}{}^{\mathrm{<mtext>N</mtext>}}\text{.}$ We assume that the family $\text{(H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{))}$ is jointly Gaussian centered and that for $\text{σ}\text{→}{}^1\text{,}\text{σ}\text{→}{}^2\text{∈Σ}{}_{\mathrm{N}}\text{,}$$\text{N}{}^{\mathrm{?1}}\text{EH}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^1\text{)H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^2\text{)}$ =ξ(N^?1∑_i?Nσ¹_iσ²_i) for a certain function ξ on $\text{R}$. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)+h∑}{}_{\mathrm{i?N}}\text{σ}{}_{\mathrm{i}}$ is bounded below by the free energy of the Parisi solution provided that ξ is convex on $\text{R}$. We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on $\text{R}{}^{\mathrm{+}}$. This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On covering bridged plane triangulations with balls

We show that every plane graph of diameter 2r in which all inner faces are triangles and all inner vertices have degree larger than 5 can be covered with two balls of radius r. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 65–80, 2003     

A theoretical analysis of the TiO2/Sn doped (1 1 0) surface properties

We have used the periodic quantum-mechanical method with density functional theory at the B3LYP level in order to study TiO₂/Sn doped (1 1 0) surfaces and have investigated the structural, electronic and energy band properties of these oxides. Our calculated relaxation directions for TiO₂ is the experimental one and is also in agreement with other theoretical results. We also observe for the doped systems relaxation of lattice positions of the atoms. Modification of Sn, O and Ti charges depend on the planes and positions of the substituted atoms. Doping can modify the Fermi levels, energy gaps as well as the localization and composition of both valence and conduction band main components. Doping can also modify the chemical, electronic and optical properties of these oxides surfaces increasing their suitability for use as gas sensors and optoelectronic devices.