In this dissertation we study The Refractor Problem and its analytic formulation which leads to Monge-Ampere type equation. This problem can be described as follows: Suppose that A and B are two domains of the unit sphere in n dimensions and g and f are two positive functions integrable on A and B respectively. Consider two homogeneous, isotropic media; medium I and medium II, which have different optical densities and assume that from a point O inside medium I, light emanates with intensity g(x); where x is in A. When an incident ray of light hits an interface between two media with different indices of refraction, it splits into two rays a reflected ray that propagates back into medium I and a refracted ray that proceeds into medium II. Consequently, the incident ray loses some of its energy as it proceeds into medium II. By using Fresnel equations, which are consequences of Maxwell's Equations, one can determine precisely how much of the energy is lost due to internal reflection. The problem is to take into account this loss and construct a surface such that all rays emitted from a point O in the first medium, with directions in A are refracted by the surface into media II with directions in B and the prescribed illumination intensity received in the direction m, where m is in B is f(m). We propose a model to this problem. We introduce weak solutions for the problem and prove their existence by using approximation by ellipsoids or hyperboloids depending on whether n1 is less than n2 or n1 is greater than n2. We will also prove that a solution of the problem satisfies a Monge-Ampere type of PDE. / Mathematics

We introduce two models to design near field reflectors in R^3 that solve an inverse problem in radiometry, taking into account the inverse square law of irradiance. The problem leads to a Monge-Ampere type inequality. The surfaces in the first model are strictly convex and require to be far from the source to avoid obstruction. In the second model, the reflectors are neither convex nor concave and do not block the rays even if they are close to the source. / Mathematics

The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\`ere type arises here. / Mathematics

[pt] O presente trabalho propõe técnicas numéricas para a síntese de antenas refletoras que utilizando seções de cônicas ou superfícies quádricas confocais. Para tal, utilizando os princípios da Óptica Geométrica, foram desenvolvidos algoritmos capazes de sintetizar as superfícies refletoras desejadas. São analisadas duas geometrias distintas: a antena duplo-refletora com cobertura omnidirecional e a antena refletora offset com um único refletor. No primeiro problema, é apresentado um método alternativo para a síntese geométrica de antenas duplo-refletoras com cobertura omnidirecional e diagrama de radiação arbitrário no plano de elevação. O subrefletor é um corpo de revolução gerado por uma única seção cônica e o refletor principal modelado é gerado por uma série de seções cônicas locais sequencialmente concatenadas. Para ilustrar o método, duas configurações axialmente simétrica são sintetizadas para proporcionar diagramas de radiação uniforme ou cossecante ao quadrado no plano de elevação. Os resultados são validados por uma técnica híbrida baseada em Casamento de Modos e o Método de Momentos. No segundo problema, é investigado um procedimento numérico alternativo para a síntese geométrica de antenas refletoras offset com diagrama de radiação arbitrário na região de campo distante. O método usa superfícies quádricas confocais com eixos deslocados para representar localmente a superfície modelada. Nesta abordagem, um operador não linear deve ser resolvido como um problema de contorno. Para ilustrar o método, são apresentadas antenas modeladas para prover diagrama de radiação Gaussiano em contornos de cobertura circular, elíptico e super-elíptico. / [en] This work proposes numerical techniques for synthesis of reflector antennas, using conic sections or confocal quadric surfaces. Under Geometrical Optics principles, algorithms to shape desired reflective surfaces have been developed. Two different geometries have been considered: omnidirectional dual-reflector antenna and single offset reflector antennas. In the first problem, it was presented an alternative method for synthesis of omnidirectional dual-reflector antennas with an arbitrary radiation pattern in elevation plane. The body-of-revolution subreflector is generated by a single conic section, while the shaped main reflector is generated by a series of local conic sections, sequentially consecutively concatenated. In order to illustrate the method, omnidirectional axisdisplaced ellipse (OADE) and Cassegrain (OADC) configurations are synthesized to provide uniform or cosecant squared radiation pattern in the elevation plane. The GO shaping results are validated by a hybrid technique based on Mode Matching and Method of Moments. In the second problem, an alternative numerical procedure was investigated for the geometrical synthesis of offset reflector antennas with an arbitrary radiation pattern in the far-field region, according to geometrical optics. The method uses local axis-displaced confocal quadric surfaces to describe the shaped reflector. In this approach, a nonlinear operator must be solved as a boundary value problem. To illustrate the method, we have chosen several offset configurations with circular, elliptical and super-elliptical contour coverage and Gaussian power density. The results were validated by the physical optics approximation.

Ce travail de thèse porte sur différentes caractérisations des modèles multivariés de stables-Tweedie multiples dans le cadre des familles exponentielles naturelles sous la propriété de "steepness". Ces modèles parus en 2014 dans la littérature ont été d’abord introduits et décrits sous une forme restreinte des stables-Tweedie normaux avant les extensions aux cas multiples. Ils sont composés d’un mélange d’une loi unidimensionnelle stable-Tweedie de variable réelle positive fixée, et des lois stables-Tweedie de variables réelles indépendantes conditionnées par la première fixée, de même variance égale à la valeur de la variable fixée. Les modèles stables-Tweedie normaux correspondants sont ceux du mélange d’une loi unidimensionnelle stable-Tweedie positive fixé et les autres toutes gaussiennes indépendantes. A travers des cas particuliers tels que normal, Poisson, gamma, inverse gaussienne, les modèles stables-Tweedie multiples sont très fréquents dans les études de statistique et probabilités appliquées. D’abord, nous avons caractérisé les modèles stables-Tweedie normaux à travers leurs fonctions variances ou matrices de covariance exprimées en fonction de leurs vecteurs moyens. La nature des polynômes associés à ces modèles est déduite selon les valeurs de la puissance variance à l’aide des propriétés de quasi orthogonalité, des systèmes de Lévy-Sheffer, et des relations de récurrence polynomiale. Ensuite, ces premiers résultats nous ont permis de caractériser à l’aide de la fonction variance la plus grande classe des stables-Tweedie multiples. Ce qui a conduit à une nouvelle classification laquelle rend la famille beaucoup plus compréhensible. Enfin, une extension de caractérisation des stables-Tweedie normaux par fonction variance généralisée ou déterminant de la fonction variance a été établie via leur propriété d’indéfinie divisibilité et en passant par les équations de Monge-Ampère correspondantes. Exprimées sous la forme de produit des composantes du vecteur moyen aux puissances multiples, la caractérisationde tous les modèles multivariés stables-Tweedie multiples par fonction variance généralisée reste un problème ouvert. / In the framework of natural exponential families, this thesis proposes differents characterizations of multivariate multiple stables-Tweedie under "steepness" property. These models appeared in 2014 in the literature were first introduced and described in a restricted form of the normal stables-Tweedie models before extensions to multiple cases. They are composed by a fixed univariate stable-Tweedie variable having a positive domain, and the remaining random variables given the fixed one are reals independent stables-Tweedie variables, possibly different, with the same dispersion parameter equal to the fixed component. The corresponding normal stables-Tweedie models have a fixed univariate stable-Tweedie and all the others are reals Gaussian variables. Through special cases such that normal, Poisson, gamma, inverse Gaussian, multiple stables-Tweedie models are very common in applied probability and statistical studies. We first characterized the normal stable-Tweedie through their variances function or covariance matrices expressed in terms of their means vector. According to the power variance parameter values, the nature of polynomials associated with these models is deduced with the properties of the quasi orthogonal, Levy-Sheffer systems, and polynomial recurrence relations. Then, these results allowed us to characterize by function variance the largest class of multiple stables-Tweedie. Which led to a new classification, which makes more understandable the family. Finally, a extension characterization of normal stable-Tweedie by generalized variance function or determinant of variance function have been established via their infinite divisibility property and through the corresponding Monge-Ampere equations. Expressed as product of the components of the mean vector with multiple powers parameters reals, the characterization of all multivariate multiple stable- Tweedie models by generalized variance function remains an open problem.