<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)

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This thesis presents a method to correct for non-positive-definiteness in linear viscoelastic material functions. Viscoelastic material functions for anisotropic materials need to be interconverted in a matrix coefficient prony series form, with a requirement of positive definiteness. Fitting is usually done as a uniaxial prony series, resulting in scalar coefficients. When these uniaxial coefficients are placed in a coefficient matrix, the required positive definiteness cannot be guaranteed. For those matrices that do not meet this requirement, finding the nearest symmetric semi-positive definite form of the matrix results in a viable prony series matrix coefficient with the required positive definiteness. These corrected prony series coefficients allow for material functions to be interconverted with minimal changes to experimental data.

An integral quadratic form is said to be strictly regular if it primitively represents all integers that are primitively represented by its genus. The goal of this dissertation is to extend the systematic investigation of the positive definite ternary primitive integral quadratic forms and lattices that are candidates for strict regularity. An integer that is primitively represented by a genus, but not by some specific form in that genus, is called a primitive exception for that form. So, the strictly regular forms are those forms for which there are no primitive exceptions. Our computations of primitive exceptions for each of the 119 positive definite regular ternary forms which lie in multiple-class genera, and of the companion forms in their genera, show that there are 45 inequivalent such forms that are candidates for strict regularity. We provide a proof of the strict regularity of one of these candidates, bringing the total number of forms for which such proofs are known to 15, and prove partial results on the integers primitively represented by the other form in its genus. The theory of primitive spinor exceptional integers is used to analyze the primitive exceptions for the forms in two other genera known to contain a regular ternary form. In these cases, results are obtained relating the primitive representation of certain integers c by a given form in one of these genera to the primitive representation of the integers 4c and 9c by the forms in the genus.

Este trabalho é composto de duas partes distintas, ambas dentro de um mesmo tema: núcleos positivos definidos sobre variedades. Na primeira delas fornecemos uma caracterização para os núcleos contínuos, isotrópicos e positivos definidos a valores matriciais sobre um espaço compacto 2-homogêneo. Utilizando-a, investigamos a positividade definida estrita destes núcleos, apresentando inicialmente algumas condições suficientes para garantir tal propriedade. No caso em que o espaço 2-homogêneo não é uma esfera, descrevemos uma caracterização definitiva para a positividade definida estrita do núcleo. Neste mesmo caso, para núcleos a valores no espaço das matrizes de ordem 2, apresentamos uma caraterização alternativa para a positividade definida estrita do núcleo via os dois elementos na diagonal principal da representação matricial do núcleo. Na segunda parte, nos restringimos a núcleos positivos definidos escalares sobre os mesmos espaços e determinamos condições necessárias e suficientes para a positividade definida estrita de um produto de núcleos positivos definidos sobre um mesmo espaço compacto 2-homogêneo. Apresentamos ainda uma extensão deste resultado para núcleos positivos definidos sobre o produto cartesiano de um grupo localmente compacto com uma esfera de dimensão alta, mantendo-se a isotropia na componente esférica. / In this work we present a characterization for the continuous, isotropic and positive definite matrix-valued kernels on a compact two-point homogeneous space. After that, we consider the strict positive definiteness of the kernels, describing some independent sufficient conditions for that property to hold. In the case the space is not a sphere, one of the conditions becomes necessary and sufficient for the strict positive definiteness of the kernel. Further, for 22- matrix-valued kernels on a compact two-point homogeneous space which is not a sphere, we present a characterization for the strict positive definiteness of the kernels based upon the main diagonal elements in its matrix representation. In the last part of this work, we restrict ourselves to scalar kernels and determine necessary and sufficient conditions in order that the product of two continuous, isotropic and positive definite kernels on a compact two-point homogeneous space be strictly positive definite. We also discuss the extension of this result for kernels defined on a product of a locally compact group and a high dimensional sphere.

Este trabalho é composto de duas partes distintas, ambas dentro de um mesmo tema: núcleos positivos definidos sobre variedades. Na primeira delas fornecemos uma caracterização para os núcleos contínuos, isotrópicos e positivos definidos a valores matriciais sobre um espaço compacto 2-homogêneo. Utilizando-a, investigamos a positividade definida estrita destes núcleos, apresentando inicialmente algumas condições suficientes para garantir tal propriedade. No caso em que o espaço 2-homogêneo não é uma esfera, descrevemos uma caracterização definitiva para a positividade definida estrita do núcleo. Neste mesmo caso, para núcleos a valores no espaço das matrizes de ordem 2, apresentamos uma caraterização alternativa para a positividade definida estrita do núcleo via os dois elementos na diagonal principal da representação matricial do núcleo. Na segunda parte, nos restringimos a núcleos positivos definidos escalares sobre os mesmos espaços e determinamos condições necessárias e suficientes para a positividade definida estrita de um produto de núcleos positivos definidos sobre um mesmo espaço compacto 2-homogêneo. Apresentamos ainda uma extensão deste resultado para núcleos positivos definidos sobre o produto cartesiano de um grupo localmente compacto com uma esfera de dimensão alta, mantendo-se a isotropia na componente esférica. / In this work we present a characterization for the continuous, isotropic and positive definite matrix-valued kernels on a compact two-point homogeneous space. After that, we consider the strict positive definiteness of the kernels, describing some independent sufficient conditions for that property to hold. In the case the space is not a sphere, one of the conditions becomes necessary and sufficient for the strict positive definiteness of the kernel. Further, for 22- matrix-valued kernels on a compact two-point homogeneous space which is not a sphere, we present a characterization for the strict positive definiteness of the kernels based upon the main diagonal elements in its matrix representation. In the last part of this work, we restrict ourselves to scalar kernels and determine necessary and sufficient conditions in order that the product of two continuous, isotropic and positive definite kernels on a compact two-point homogeneous space be strictly positive definite. We also discuss the extension of this result for kernels defined on a product of a locally compact group and a high dimensional sphere.

A partial matrix, is a matrix for which some entries are specified and some unspecified. In general completion problems ask whether a given partial matrix, may be completed to a matrix where all the entries are specified, such that this completion admits a specific structure. The positive definite completion problem asks whether a partial Hermitian matrix admits a completion such that the completed matrix is positive semidefinite. The minimum solution criterion, is that every fully specified principal submatrix is nonnegative. Then the set of partial Hermitian matrices, which admit a positive semidefinite completion, forms a convex cone, and its dual cone can be identified as the set of positive semidefinite Hermitian matrices with zeros in the entries that correspond to non-edges in the graph G: Furthermore, the set of partial Hermitian matrices, with non-negative fully specified principal minors, also forms a convex cone, and its dual cone can be identified as the set of positive semidefinite Hermitian matrices which can be written as the sum of rank one matrices, with underlying graph G. Consequently, the problem reduces to determining when these cones are equal. Indeed, we find that this happens if and only if the underlying graph is chordal. It then follows that the extreme rays of the cone of positive semidefinite Hermitian matrices with zeros in the entries that correspond to non-edges in the graph G is generated by rank one matrices. The question that arises, is what happens if the underlying graph is not chordal. In particular, what can be said about the extreme rays of the cone of positive semidefinite matrices with some non-chordal pattern. This gives rise to the notion of the sparsity order of a graph G; that is, the maximum rank of matrices lying on extreme rays of the cone of positive semidefinite Hermitian matrices with zeros in the entries that correspond to non-edges in the graph G: We will see that those graphs having sparsity order less than or equal to 2 can be fully characterized. Moreover, one can determine in polynomial time whether a graph has sparsity order less than or equal to 2, using a clique-sum decomposition. We also show that one can determine whether a graph has sparsity order less than or equal to 2, by considering the characteristic polynomial of the adjacency matrix of certain forbidden induced subgraphs and comparing it with the characteristic polynomial of principal submatrices of appropriate size. / MSc (Mathematics), North-West University, Potchefstroom Campus, 2015

A partial matrix, is a matrix for which some entries are specified and some unspecified. In general completion problems ask whether a given partial matrix, may be completed to a matrix where all the entries are specified, such that this completion admits a specific structure. The positive definite completion problem asks whether a partial Hermitian matrix admits a completion such that the completed matrix is positive semidefinite. The minimum solution criterion, is that every fully specified principal submatrix is nonnegative. Then the set of partial Hermitian matrices, which admit a positive semidefinite completion, forms a convex cone, and its dual cone can be identified as the set of positive semidefinite Hermitian matrices with zeros in the entries that correspond to non-edges in the graph G: Furthermore, the set of partial Hermitian matrices, with non-negative fully specified principal minors, also forms a convex cone, and its dual cone can be identified as the set of positive semidefinite Hermitian matrices which can be written as the sum of rank one matrices, with underlying graph G. Consequently, the problem reduces to determining when these cones are equal. Indeed, we find that this happens if and only if the underlying graph is chordal. It then follows that the extreme rays of the cone of positive semidefinite Hermitian matrices with zeros in the entries that correspond to non-edges in the graph G is generated by rank one matrices. The question that arises, is what happens if the underlying graph is not chordal. In particular, what can be said about the extreme rays of the cone of positive semidefinite matrices with some non-chordal pattern. This gives rise to the notion of the sparsity order of a graph G; that is, the maximum rank of matrices lying on extreme rays of the cone of positive semidefinite Hermitian matrices with zeros in the entries that correspond to non-edges in the graph G: We will see that those graphs having sparsity order less than or equal to 2 can be fully characterized. Moreover, one can determine in polynomial time whether a graph has sparsity order less than or equal to 2, using a clique-sum decomposition. We also show that one can determine whether a graph has sparsity order less than or equal to 2, by considering the characteristic polynomial of the adjacency matrix of certain forbidden induced subgraphs and comparing it with the characteristic polynomial of principal submatrices of appropriate size. / MSc (Mathematics), North-West University, Potchefstroom Campus, 2015

Neste trabalho apresentamos uma demonstração detalhada para um conhecido teorema de I. J. Schoenberg que caracteriza certas funções positivas definidas em esferas. Analisamos ainda um critério para a obtenção de positividade definida de uma função a partir de condições de suavidade e convexidade dela, em uma tentativa de ratificar alguns resultados da literatura conhecidos como critérios de Pólya. / In this work we present a proof for a famous theorem of Schoenberg on positive definite functions on spheres. We analyze some results that deduce positive definiteness from diferentiability and convexity assumption on the function, an attempt to ratify some Pólya type conjectures found in the literature.