We study the effect of the so-called half-hound states on the Titchmarsh-Weyl M(λ)· function and the S-matrix for a one dimensional Dirac system. For short range potentials with finite first (absolute) moments, we gave an M(λ) characterization of half bound states and, as a corollary, we deduce the behavior of the spectral function near the spectral gap endpoints. Further, we establish community of the S-matrix in momentum space and prove the Levinson theorem as a corollary to this analysis. We also obtain explicit asymptotics of the S-matrix for power-law potentials / Ph. D.

<p>In this thesis, we propose and analyze a fast method for computing the solution of the Helmholtz equation in a bounded domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is first split into one--way wave equations which are then solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the solutions of the one--way wave equations from the previous iteration. The solution of the Helmholtz equation is then approximated by the sum of the one--way solutions at every iteration. To improve the computational cost, the source functions are thresholded and in the domain where they are equal to zero, the one--way wave equations are solved with GO with a computational cost independent of the frequency. Elsewhere, the equations are fully resolved with a Runge-Kutta method. We have been able to show rigorously in one dimension that the algorithm is convergent and that for fixed accuracy, the computational cost is just O(ω^1/p) for a p-th order Runge-Kutta method. This has been confirmed by numerical experiments.</p>

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We enlarge the class of Beltrami equations by developping a stability theory for the sheaf of solutions of an overdetermined elliptic system of first order homogeneous partial differential equations with constant coefficients in the Euclidean space.

As increasing emphases are placed on the importance of a muscular male physique in Westernized culture, more men are experiencing eating, exercise, and body image (EEBI) disturbances. Clinician-researchers have identified a syndrome, termed muscle dysmorphia (MD), in which individuals, usually men, are pathologically preoccupied with their perceived lack of muscularity. The current study tested a modified version of an extant theoretical model of MD symptomatology as well as an alternative model of MD symptomatology. Over 700 adult men completed a demographic questionnaire, a symptom inventory, a self-esteem questionnaire, a measure of perfectionism, a measure of the media’s influence on EEBI disturbances, and measures of body dissatisfaction and MD symptoms. Structural equation modeling (SEM) was used to examine the goodness of fit of the proposed models. Overall, the first model demonstrated poor fit with the data. Conversely, the alternative model fit the data adequately. The alternative model was cross validated with a second sample, and also fit this data adequately.

Le travail de cette thèse a porté sur l'étude mathématique et numérique de quelques modèles multi-échelles issus de la physique des matériaux. La première partie de ce travail est consacrée à l'homogénéisation mathématique d'un problème elliptique avec une petite échelle. Nous étudions le cas particulier d'un matériau présentant une structure périodique avec un défaut. En adaptant la théorie classique d'Avellaneda et Lin pour les milieux périodiques, on démontre qu'on peut approximer finement la solution d'un tel problème, notamment à l'échelle microscopique. Nous obtenons des taux de convergence dépendant de l'étalement du défaut. On démontre aussi quelques propriétés des fonctions de Green d'un problème elliptique périodique avec conditions de bord périodiques. Les dislocations sont des lignes de défaut de la matière responsables du phénomène de plasticité. Les deuxième et troisième parties de ce mémoire portent sur la simulation de dislocations, d'abord en régime stationnaire puis en régime dynamique. Nous utilisons le modèle de Peierls, qui couple échelle atomique et échelle mésoscopique. Dans le cadre stationnaire, on obtient une équation intégrodifférentielle non-linéaire avec un laplacien fractionnaire: l'équation de Weertman. Nous en étudions les propriétés mathématiques et proposons un schéma numérique pour en approximer la solution. Dans le cadre dynamique, on obtient une équation intégrodifférentielle à la fois en temps et en espace. Nous en faisons une brève étude mathématique, et comparons différents algorithmes pour la simuler. Enfin, dans la quatrième partie, nous étudions la limite macroscopique d'une chaîne d'atomes soumis à la loi de Newton. Des arguments formels suggèrent que celle-ci devrait être décrite par une équation des ondes non-linéaires. Or, nous démontrons --sous certaines hypothèses-- qu'il n'en est rien lorsque des chocs apparaissent / In this thesis we study mathematically and numerically some multi-scale models from materials science. First, we investigate an homogenization problem for an oscillating elliptic equation. The material under consideration is described by a periodic structure with a defect at the microscopic scale. By adapting Avellaneda and Lin's theory for periodic structures, we prove that the solution of the oscillating equation can be approximated at a fine scale. The rates of convergence depend upon the integrability of the defect. We also study some properties of the Green function of periodic materials with periodic boundary conditions. Dislocations are lines of defects inside materials, which induce plasticity. The second part and the third part of this manuscript are concerned with simulation of dislocations, first in the stationnary regime then in the dynamical regime. We use the Peierls model, which couples atomistic and mesoscopic scales and involves integrodifferential equations. In the stationary regime, dislocations are described by the so-called Weertman equation, which is nonlinear and involves a fractional Laplacian. We study some mathematical properties of this equation and propose a numerical scheme for approximating its solution. In the dynamical regime, dislocations are described by an equation which is integrodifferential in time and space. We compare some numerical methods for recovering its solution. In the last chapter, we investigate the macroscopic limit of a simple chain of atoms governed by the Newton equation. Surprisingly enough, under technical assumptions, we show that it is not described by a nonlinear wave equation when shocks occur