<p> In this thesis we study the representation theorems for evenly positive definite functions on Euclidean spaces. A generalization of the concept of evenness on R^n to a concept of symmetry on any locally compact abelian group is given. In addition, a result analogous to the Weil-Povzner-Raikov Theorem is obtained for the representation of symmetrically positive definite functions on locally compact abelian groups.</p> / Thesis / Master of Science (MSc)

Let $\Omega\subset\mathbb{R}^n$ be an open and connected subset of $\mathbb{R}^n$. We say that a function $F\colon \Omega-\Omega\to\mathbb{C}$, where $\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, is positive definite if for any $x_1,\ldots,x_m\in\Omega$ and any $c_1,\ldots,c_m\in \mathbb{C}$ we have that $\sum_{j,k=1}^m F(x_j-x_k)c_j\overline{c_k}\geq 0$. Let $F\colon\Omega-\Omega\to\mathbb{C}$ be a continuous positive definite function. We give necessary and sufficient conditions for $F$ to have an extension to a continuous and positive definite function defined on the entire Euclidean space $\mathbb{R}^n$. The conditions are formulated in terms of strong commutativity of some certain selfadjoint operators defined on a Hilbert space associated to our positive definite function.

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Apresentamos uma caracterização das funções positivas definidas em esferas complexas, generalizando assim, um resultado de Schoenberg ([41]). Como no caso real, uma classe importante dessas funções é aquela composta pelas funções estritamente positivas definidas de uma certa ordem; estas podem ser utilizadas para resolver certos problemas de interpolação de dados arbitrários associados a pontos distintos distribuídos nas esferas. Com esse objetivo, obtivemos algumas condições necessárias e suficientes (separadamente) para que funções positivas definidas sejam estritamente positivas definidas. Os resultados apresentados fornecem uma caracterização final elementar para funções estritamente positivas definidas de todas as ordens em quase todas as esferas complexas. Funções estritamente positivas definidas de ordem 2 são caracterizadas em todas as esferas complexas. Analisamos também a relação entre funções estritamente positivas definidas em esferas complexas e funções estritamente positivas definidas em esferas reais. / We characterize positive definite functions on complex spheres, generalizing a famous result due to I. J. Schoenberg ([41]). As in the real case, we study the so-called strictly positive definite functions. They can be used to perform interpolation of scattered data on those spheres. We present (separated) necessary and sufficient conditions for a positive definite function to be strictly positive definite of a certain order. These conditions produce a final characterization for those positive definite functions which are strictly positive definite of all orders, on almost all spheres. Strictly positive definite functions of order 2 are identified. Finally, we study a connection between strictly positive definite functions on real spheres and strictly positive definite functions on complex spheres.

Apresentamos uma caracterização das funções positivas definidas em esferas complexas, generalizando assim, um resultado de Schoenberg ([41]). Como no caso real, uma classe importante dessas funções é aquela composta pelas funções estritamente positivas definidas de uma certa ordem; estas podem ser utilizadas para resolver certos problemas de interpolação de dados arbitrários associados a pontos distintos distribuídos nas esferas. Com esse objetivo, obtivemos algumas condições necessárias e suficientes (separadamente) para que funções positivas definidas sejam estritamente positivas definidas. Os resultados apresentados fornecem uma caracterização final elementar para funções estritamente positivas definidas de todas as ordens em quase todas as esferas complexas. Funções estritamente positivas definidas de ordem 2 são caracterizadas em todas as esferas complexas. Analisamos também a relação entre funções estritamente positivas definidas em esferas complexas e funções estritamente positivas definidas em esferas reais. / We characterize positive definite functions on complex spheres, generalizing a famous result due to I. J. Schoenberg ([41]). As in the real case, we study the so-called strictly positive definite functions. They can be used to perform interpolation of scattered data on those spheres. We present (separated) necessary and sufficient conditions for a positive definite function to be strictly positive definite of a certain order. These conditions produce a final characterization for those positive definite functions which are strictly positive definite of all orders, on almost all spheres. Strictly positive definite functions of order 2 are identified. Finally, we study a connection between strictly positive definite functions on real spheres and strictly positive definite functions on complex spheres.

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex numbers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagonalizable matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingular matrices into Involutions. Chapter 5 studies factorization of a complex matrix into Positive-(semi)definite matrices, emphasizing the least number of such factors required. / Mathematical Sciences / M. Sc. (Mathematics)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex numbers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagonalizable matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingular matrices into Involutions. Chapter 5 studies factorization of a complex matrix into Positive-(semi)definite matrices, emphasizing the least number of such factors required. / Mathematical Sciences / M. Sc. (Mathematics)

Cette thèse traite de l'évitement de collision entre un engin spatial opérationnel, appelé objet primaire, et un débris orbital, dit secondaire. Ces travaux concernent aussi bien la question de l'estimation du risque pour une paire d'objets sphériques que celle du calcul d'un plan de manoeuvres d'évitement pour le primaire. Pour ce qui est du premier point, sous certaines hypothèses, la probabilité de collision s'exprime comme l'intégrale d'une fonction gaussienne sur une boule euclidienne, en dimension deux ou trois. On en propose ici une nouvelle méthode de calcul, basée sur les théories de la transformée de Laplace et des fonctions holonomes. En ce qui concerne le calcul de manoeuvres de propulsion, différentes méthodes sont développées en fonction du modèle considéré. En toute généralité, le problème peut être formulé dans le cadre de l'optimisation sous contrainte probabiliste et s'avère difficile à résoudre. Dans le cas d'un mouvement considéré comme relatif rectiligne, l'approche par scénarios se prête bien au problème et permet d'obtenir des solutions admissibles. Concernant les rapprochements lents, une linéarisation de la dynamique des objets et un recouvrement polyédral de l'objet combiné sont à la base de la construction d'un problème de substitution. Deux approches sont proposées pour sa résolution : une première directe et une seconde par sélection du risque. Enfin, la question du calcul de manoeuvres de proximité en consommation optimale et temps fixé, sans contrainte d'évitement, est abordée. Par l'intermédiaire de la théorie du vecteur efficacité, la solution analytique est obtenue pour la partie hors-plan de la dynamique képlérienne linéarisée. / This thesis is about collision avoidance for a pair of spherical orbiting objects. The primary object - the operational satellite - is active in the sense that it can use its thrusters to change its trajectory, while the secondary object is a space debris that cannot be controlled in any way. Onground radars or other means allow to foresee a conjunction involving an operational space craft,leading in the production of a collision alert. The latter contains statistical data on the position and velocity of the two objects, enabling for the construction of a probabilistic collision model.The work is divided in two parts : the computation of collision probabilities and the design of maneuvers to lower the collision risk. In the first part, two kinds of probabilities - that can be written as integrals of a Gaussian distribution over an Euclidean ball in 2 and 3 dimensions -are expanded in convergent power series with positive terms. It is done using the theories of Laplace transform and Definite functions. In the second part, the question of collision avoidance is formulated as a chance-constrained optimization problem. Depending on the collision model, namely short or long-term encounters, it is respectively tackled via the scenario approach or relaxed using polyhedral collision sets. For the latter, two methods are proposed. The first one directly tackles the joint chance constraints while the second uses another relaxation called risk selection to obtain a mixed-integer program. Additionaly, the solution to the problem of fixed-time fuel minimizing out-of-plane proximity maneuvers is derived. This optimal control problem is solved via the primer vector theory.