An optimization problem related to the modeling of atmospheric organic aerosols

A mathematical model for the computation of the phase equilibrium related to atmospheric organic aerosols is proposed. The equilibrium is given by the minimum of the Gibbs free energy and is characterized using the notion of phase simplex of its convex hull. A primal-dual interior-point method solving the Karush–Kuhn–Tucker conditions is detailed. Numerical results show the efficiency of our algorithm. To cite this article: N.R. Amundson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Primal-Dual Active-Set Algorithm for Chemical Equilibrium Problems Related to the Modeling of Atmospheric Inorganic Aerosols

A general equilibrium model for multiphase multicomponent inorganic atmospheric aerosols is proposed. The thermodynamic equilibrium is given by the minimum of the Gibbs free energy for a system involving an aqueous phase, a gas phase, and solid salts. A primal-dual algorithm solving the Karush-Kuhn-Tucker conditions is detailed. An active set/Newton method permits to compute the minimum of the energy and tracks the presence or not of solid salts at the equilibrium. Numerical results show the efficiency of our algorithm for the prediction of multiphase multireaction chemical equilibria.Communicated by R. GlowinskiThis work has been partially supported by the United States Environmental Protection Agency through Cooperative Agreement X-83234201 to the University of Houston. The second author was supported by the Swiss National Science Foundation, Grant PBEL2-103152.     

Équilibre d'un solide élastique en grandes déformations

We investigate equilibrium positions of a solid in large deformations. We take the party that a solid can flatten in a solid with a lower dimension. A structure flattened by a power hammer is an example of such a situation. Moreover, we take into account the spacial variations of rotation matrix. We prove that under reasonable assumptions, there exist equilibrium positions which may be non-unique. To cite this article: M. Frémond, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A reduced basis method applied to the Restricted Hartree–Fock equations

In this Note, we describe a reduced basis approximation method for the computation of some electronic structure in quantum chemistry, based on the Restricted Hartree–Fock equations. Numerical results are presented to show that this approach allows for reducing the complexity and potentially the computational costs. To cite this article: Y. Maday, U. Razafison, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A particle model for fluid–structure interactionUne méthode particulaire pour le couplage fluide–structure

We propose a particle method to handle fluid–structure interactions on a 1D model problem. Interactions between fluid and solid particles implicitly enforce the continuity of stresses on the interface. Comparisons with results obtained by ALE methods allow one to evaluate the robustness and accuracy of the method. To cite this article: G.-H. Cottet, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 833–838.     

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.     

The multiconfiguration methods in quantum chemistry: Palais–Smale condition and existence of minimizersLes méthodes de multiconfiguration en chimie quantique : condition de Palais–Smale et existence de minima

In this Note, we propose a new proof for the existence of a minimum in the multiconfiguration methods in Quantum Chemistry. We use a Palais–Smale condition with Morse-type information, whose proof is based on the Euler–Lagrange equations, written in a simple and useful way. To cite this article: M. Lewin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 299–304.     

A formal computation of the splitting for the Klein–Gordon operator

We study the semi-classical Klein–Gordon operator in the one dimensional case, for a double-well potential. We obtain a formal computation of the splitting in cases that were not yet studied. To cite this article: E. Servat, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Une nouvelle méthode de relaxation pour les équations de Navier–Stokes compressibles

We consider the compressible Navier–Stokes equations for gas flows endowed with general pressure and temperature laws as long as they are compatible with the existence of an entropy and Gibbs relations. We extend the relaxation method introduced for the Euler equations by Coquel and Perthame. Keeping the same “sub-characteristic” conditions for the hyperbolic fluxes and using a consistent splitting of the diffusive fluxes based on a global temperature, we prove the stability of the relaxation system via the sign of the production of a suitable entropy. A first order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method. To cite this article: E. Bongiovanni et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structureAnalyse de perturbation singulière pour la réduction de la chimie complexe des mélanges gazeux avec structure entropique

In this Note, we investigate the reduction of complex chemistry in gaseous mixtures. We consider an arbitrarily complex network of reversible reactions, the equilibrium constant of which are compatible with thermodynamics, thus providing an entropic structure. We assume that a subset of the reactions is consituted of fast reactions and define a constant and linear projection onto the partial equilibrium manifold compatible with the entropy production. This reduction step is used for the study of a homogeneous reactor at constant density and internal energy where the temperature can encounter strong variations. We prove the global existence of a smooth solution and of an asymptotically stable equilibrium state for both the reduced system and the complete one. A global in time singular perturbation analysis proves that the reduced system on the partial equilibrium manifold approximates the full chemistry system. To cite this article: M. Massot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 93–98.     

Un algorithme probabiliste de calcul d'approximations polynômiales sur un hypercube

We describe a Monte Carlo method which enables an iterative computation of the L² approximation of a function on any orthonormal basis. We use it for the approximation of smooth functions on an hypercube with the help of multidimensional orthogonal polynomial basis containing only few terms. The algorithm is both a tool for approximation and numerical integration. To cite this article: S. Maire, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Caractérisation des moments d'inertie d'un système matériel

We characterize those real-valued functions on the set of subspaces of a euclidian affine space that represent the moments of inertia of some suitable solid body. Moreover, we find the lowest number of point masses that can be used to obtain such a solid body. To cite this article: P. Barbaroux, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1067–1070.     

Girth in graphs

It is shown that a graph of large girth and minimum degree at least 3 share many properties with a graph of large minimum degree. For example, it has a contraction containing a large complete graph, it contains a subgraph of large cyclic vertex-connectivity (a property which guarantees, e.g., that many prescribed independent edges are in a common cycle), it contains cycles of all even lengths modulo a prescribed natural number, and it contains many disjoint cycles of the same length. The analogous results for graphs of large minimum degree are due to Mader (Math. Ann.194 (1971), 295–312; Abh. Math. Sem. Univ. Hamburg37 (1972), 86–97), Woodall (J. Combin. Theory Ser. B22 (1977), 274–278), Bollobás (Bull. London Math. Soc.9 (1977), 97–98) and Häggkvist (Equicardinal disjoint cycles in sparse graphs, to appear). Also, a graph of large girth and minimum degree at least 3 has a cycle with many chords. An analogous result for graphs of chromatic number at least 4 has been announced by Voss (J. Combin. Theory Ser. B32 (1982), 264–285).     

Primal-Dual Interior-Point Method for an Optimization Problem Related to the Modeling of Atmospheric Organic Aerosols

A mathematical model for the computation of the phase equilibrium related to atmospheric organic aerosols is presented. The phase equilibrium is given by the global minimum of the Gibbs free energy for a system that involves water and organic components. This minimization problem is equivalent to the determination of the convex hull of the corresponding molar Gibbs free energy function. A geometrical notion of phase simplex related to the convex hull is introduced to characterize mathematically the phases at equilibrium. A primal-dual interior-point algorithm for the efficient solution of the phase equilibrium problem is presented. A novel initialization of the algorithm, based on the properties of the phase simplex, is proposed to ensure the convergence to a global minimum of the Gibbs free energy. For a finite termination of the interior-point method, an active phase identification procedure is incorporated. Numerical results show the robustness and efficiency of the approach for the prediction of liquid-liquid equilibrium in multicomponent mixtures.Communicated by R. GlowinskiThis work was supported by US Environmental Protection Grant X-83234201. The second author was partially supported by Swiss National Science Foundation Grant PBEL2-103152.     

Consistency of a simple multidimensional scheme for Hamilton–Jacobi–Bellman equations

This Note presents an approximation scheme for second-order Hamilton–Jacobi–Bellman equations arising in stochastic optimal control. The scheme is based on a Markov chain approximation method. It is easy to implement in any dimension. The consistency of the scheme is proved, which guarantees its convergence. To cite this article: R. Munos, H. Zidani, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Étude d'une forme volume naturelle sur l'espace de représentations du groupe d'un nœud dans SU(2)

For a knot K in S³, we construct in the line of Casson – or more precisely taking into account Lin's (J. Differential Geom. 35 (1992) 337–357) and Heusener's (Topology Appl. 127 (2003) 175–197) further works – a 1-volume form on the SU(2)-representation space of the group of K and we show how to interpret this volume form as a Reidemeister torsion. In the last part of this Note, we give an explicit computation of this volume form for torus knots. To cite this article: J. Dubois, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Homologies d'algèbres Artin–Schelter régulières cubiques

Hochschild homology of cubic Artin–Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by Van den Bergh (K-Theory 8 (1994) 213–230) in the quadratic case, by considering these algebras as deformations of a polynomial algebra, with remarkable Poisson brackets. A new quasi-isomorphism is introduced. De Rham cohomology, cyclic and periodic cyclic homologies, and Hochschild cohomology are also computed. To cite this article: N. Marconnet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Hauteur des sous-schémas toriques et dualité de Legendre–Fenchel

We announce a formula for the height of a complete toric variety defined over a number field. The exact expression rests on tools from convex analysis and in particular, on the Legendre–Fenchel dual of the logarithm of the local norms of a natural section of the relevant metrized line bundle. We apply this formula to the computation of the height of projective toric curves and of toric bundles. To cite this article: J.I. Burgos Gil et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

L'équation aux variations verticales d'un équilibre relatif comme source de nouvelles solutions périodiques du problème des N corps

We start with a rotating frame which enforces a resonance between the frequency of a relative equilibrium of the N-body problem and that of an infinitesimal variation normal to the plane of the equilibrium. Continuation then yields a remarkable class of periodic solutions. The first example is the $P_{12}$ family, discovered by Christian Marchal, which links the relative equilibrium of Lagrange to the Eight. To cite this article: A. Chenciner, J. Féjoz, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Weights computation for simplicial Whitney forms of degree one

We investigate some simple techniques of computation of the weights (‘moments’) of simplicial Whitney p-forms of first polynomial degree. The classical metric-dependent computation of weights is shown to be equivalent to an affine one, more suitable in the context of differential forms. To cite this article: F. Rapetti, C. R. Acad. Sci. Paris, Ser. I 341 (2005).