On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

Spacetime Causality in the Study of the Hankel Transform

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving Cauchy-Goursat problems of hyperbolic type. Communicated by Vincent Rivasseau Submitted: October 28, 2005; Accepted: February 17, 2006     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

Curvature Conditions for Immersions of Submanifolds and Applications

Some curvature conditions about the geodesics emanating from a submanifold are obtained. These conditions are used to to study the topological and geometric properties of the ambient spaces which admit some minimal submanifolds.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

Fonctions constructibles et intégration motivique II

We extend the formalism of an earlier Note to a global setting for which a theorem on fiber integrals and a Fubini theorem are obtained. We compare our formalism to the previous constructions given by Denef and Loeser. To cite this article: R. Cluckers, F. Loeser, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Des équations de Dirac et de Schrödinger pour la transformation de Fourier

Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrödinger equation on the half-axis and has used, in tandem, the Gel'fand–Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms. To cite this article: J.-F. Burnol, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Speed of adaptation in structured populations

Adaptation of populations takes place with the occurrence and subsequent fixation of mutations that confer some selective advantage to the individuals which acquire it. For this reason, the study of the process of fixation of advantageous mutations has a long history in the population genetics literature. Particularly, the previous investigations aimed to find out the main evolutionary forces affecting the strength of natural selection in the populations. In the current work, we investigate the dynamics of fixation of beneficial mutations in a subdivided population. The subpopulations (demes) can exchange migrants among their neighbors, in a migration network which is assumed to have either a random graph or a scale-free topology. We have observed that the migration rate drastically affects the dynamics of mutation fixation, despite of the fact that the probability of fixation is invariant on the migration rate, accordingly to Maruyama's conjecture. In addition, we have noticed a topological dependence of the adaptive evolution of the population when clonal interference becomes effective.     

Diagramme polythermique du systeme ternaire H2O—Al(NO3)3—Mg(NO3)2

The solid—liquid equilibria of the ternary system H₂O—Al(NO₃)₃—Mg(NO₃)₂ were studied at –30, –20, –10 and 0°C by using a synthetic method which allows to detemine all the characteristic points of isothermal sections. The stable solid phases which appear are respectively: ice, Al(NO₃)₃·9H₂O, Mg(NO₃)₂·9H₂O and Mg(NO₃)₂·6H₂O. Neither double salts nor mixed crystals are observed in the temperature and composition field studied. Polytherm diagram layout show two invariant transformations correspond with an eutectic point and a peritectic point.This revised version was published online in November 2005 with corrections to the Cover Date.