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.

In this work, we generalize three famous results obtained by Schoenberg: I) the characterization of the continuous positive definite isotropic kernels defined on a real sphere; II) the characterization of the continuous positive definite radial kernels defined on an Euclidean space; III) the characterization of the continuous conditionally negative radial kernels defined on an Euclidean space. From this new approach, we reobtain several results in the literature and obtain some new ones as well. With the exception of S1 and R , we obtain necessary and sufficient conditions in order that these kernels be strictly positive definite and strictly conditionally negative definite. / Neste trabalho, nós generalizamos três resultados famosos obtidos por Schoenberg: I) a caracterização dos núcleos contínuos isotrópicos positivos definidos em esferas reais; II) a caracterização dos núcleos contínuos radiais positivos definidos em espaços Euclidianos; III) a caracterização dos núcleos contínuos radiais condicionalmente negativos definidos em espaços Euclidianos. A partir destas novas abordagens, reobtemos vários resultados da literatura assim como obtemos novos. Com a exceção de S1 e R, obtemos condições necessárias e suficientes para que estes núcleos sejam estritamente positivos definidos e estritamente condicionalmente negativos definidos.

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.

Objectives: The inverse relationship between QT interval variability (QTV) and T wave amplitude potentially confounds QT variability assessment. We quantified the influence of the T wave amplitude on QTV in a comprehensive dataset and devised a correction formula. Methods: Three ECG datasets of healthy subjects were analyzed to model the relationship between T wave amplitude and QTV. To derive a generally valid correction formula, linear regression analysis was used. The proposed correction formula was applied to patients enrolled in the Evaluation of Defibrillator in Non-Ischemic Cardiomyopathy Treatment Evaluation trial (DEFINITE) to assess the prognostic significance of QTV for all-cause mortality in patients with non-ischemic dilated cardiomyopathy. Results: A strong inverse relationship between T wave amplitude and QTV was demonstrated, both in healthy subjects (R2 = 0.68, p < 0.001) and DEFINITE patients (R2 = 0.20, p < 0.001). Applying the T wave amplitude correction to QTV achieved 2.5-times better group discrimination between patients enrolled in the DEFINITE study and healthy subjects. Kaplan-Meier estimator analysis showed that T wave amplitude corrected QTVi is inversely related to survival (p < 0.01) and a significant predictor of all-cause mortality. Conclusion: We have proposed a simple correction formula for improved QTV assessment. Using this correction, predictive value of QTV for all-cause mortality in patients with non-ischemic cardiomyopathy has been demonstrated.

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

Esta dissertação apresenta uma análise semântica, no paradigma da Semântica Formal, da (i) restrição de domínio no DP e da (ii) expressão kar, que veicula a ideia de totalidade, no dialeto paranaense da língua Kaingang, do tronco Macro-Jê, família Jê, falada nas regiões sul e sudeste do Brasil. Num primeiro momento, o artigo definido parece ter, no Kaingang Paranaense, uma distribuição não trivial: aplica-se primeiro a um determinante quantificacional e restringe, via essa combinação, o domínio do quantificador, tal como no Basco, Grego e Státimcets (Giannakidou 2003, Etxeberria 2005 e Etxeberria & Giannakidou 2009), línguas muito parecidas com o Kaingang no domínio nominal. Alguns dados, contudo, apontaram a análise em outra direção. Entendemos que certos padrões de comportamento semântico e sintático apresentado por kar nas sentenças, tal como sua neutralidade quanto à propriedade de distributividade e a possibilidade da conjunção de duas sequências de [NP+kar] sob um mesmo artigo definido (ao contrário do Basco, no qual a mesma estrutura é agramatical, sugerindo que os quantificadores universais em Basco criam um QP), são algumas das evidências que, no conjunto, dão suporte para a hipótese, defendida nesta dissertação, de que kar, no Kaingang Paranaense, parece ser um modificador - à la Lasersohn (1999) -, não tendo, por isso, uma força quantificacional própria. A função semântica de kar seria de controlar o quanto de desvio da verdade é pragmaticamente permissível. Como consequência desta análise, no Kaingang Paranaense o artigo definido não operaria sobre um determinante quantificacional, mas sim sobre um NP. O que aponta no sentido de que os artigos definidos ag/fag (os/as) no Kaingang Paranaense, em contextos em que eles co-ocorrem com kar, não perderiam a sua função max, i.e., de formadores de indivíduo a partir de um conjunto, para funcionar meramente como um operador preservador de tipo e restritor de domínio adjungido ao determinante quantificacional, como proposto por Giannakidou 2003, Etxeberria (2005) e Etxeberria & Giannakidou (2009) para o Basco e o Grego. Ag/fag continuariam sendo artigos definidos clássicos ocupando o núcleo de uma projeção DP e kar um modificador. Também discutimos brevemente algumas das vantagens e problemas de se tentar estender esta análise para o Basco, Grego e Státimcets. E, por último, investigamos o comportamento de kar na sentença, as possíveis leituras quando da sua interação com indefinidos, numerais, tipos de predicados e o operador distributivo introduzido via reduplicação verbal. Com base nos dados, nossa proposta é de que - em função de a leitura distributiva, na maioria dos exemplos, ser permitida pelos informantes somente quando houve reduplicação verbal - kar é neutro quanto à propriedade da distributividade e que o operador distributivo introduzido por reduplicação verbal tem escopo sobre todo o VP. / This dissertation presents, in the paradigm of formal semantics, a semantic analysis of both (i) the phenomenon of domain restriction in the DP and (ii) the expression kar, which conveys the idea of totality, in a dialect of the Kaingang language, a Brazilian language from the Macro-Jê Stock, Jê family, spoken in southern and southeastern Brazil. Although, at first, the definite article in Kaingang seems to have a non-trivial distribution: it applies first to a quantificational expression, and via such combination restricts the domain of quantifier, such as in Basque, Greek and Státimcets (Giannakidou 2003, Etxeberria 2005 and Etxeberria & Giannakidou 2009), some data pointed the analysis in another direction. Patterns of semantic and syntactic bevavior presented by kar in some sentences, such as its neutralite regarding the property of distributivity and the possibility of conjoining two [NP + kar] sequences under the same definite article (unlike Basque, which does not allow such structure, suggesting that in Basque the universal quantifier creats a QP), seem to be as a whole evidences for the hypothesis, advocated in this thesis, that the expression kar may be a modifier - à la Lasersohn (1999) - and as such would not have a quantificational force of its own. The semantic function of kar would be to control how much deviation from the truth conditions of the sentences is pragmatically allowed. As a result of this analysis, the definite articles ag/fag in Kaingang do not operate on a quantificational expression, as in Basque and Greek, but on the NP. Such fact suggests that the definite articles in Kaingang, in contexts they co-occur with kar, do not lose their max function in order to work merely as a type preserver and a domain restrictor combined with a quantificational expression, as proposed by Giannakidou 2003, Etxeberria (2005) e Etxeberria & Giannakidou (2009) for Basque and Greek. Ag/fag would still be a classical definite article occupying the head of a DP projection and kar a modifier, instead of a universal quantifier. We also discuss briefly some of the advantages and problems of trying to extend this analysis to the Basque, Greek and Státimcets languages. And lastly, we investigated the behavior of kar in the sentence, its interaction with indefinites, numerals, types of predicates and the distributive operator introduced via verbal reduplication. Based on such data, once distributive readings were permitted by the informants only via verbal reduplication, our proposal is that kar is neutral regarding the property of distributivity and that the distributive operator introduced via verbal reduplication has scope over the VP.

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.

Inicialmente, estudamos alguns resultados clássicos da teoria dos núcleos positivos definidos e alguns resultados pertinentes. Estudamos em seguida, o Teorema de Mercer e algumas de suas generalizações e conseqüências, incluindo a caracterização da transformada de Fourier de um núcleo positivo definido com domínio Rm£Rm, m ¸ 1. O trabalho traz um enfoque especial nos núcleos cujo domínio é um subconjunto não-compacto de Rm £ Rm, uma vez que os demais casos são considerados de maneira extensiva na literatura. Aplicamos esses estudos na análise do decaimento dos autovalores de operadores integrais gerados por núcleos positivos definidos / Firstly, we study some classical results from the theory of positive definite kernels along with some related results. Secondly, we focus on generalizations of Mercer\'s theorem and some of their implications. Special attention is given to the cases where the domain of the kernel is not compact, once the other cases are considered consistently in the literature. We include a characterization for the Fourier transform of a positive definite kernel on Rm£Rm, m ¸ 1. Finally, we apply the previous study in the analysis of decay rates for eigenvalues of integral operators generated by positive definite kernels