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)

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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)

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.

Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(-1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix.

The thesis deals with the use of articles in German, English and Czech. In the theoretical part the characteristics of German and English article, its varieties and historical development is introduced, and one chapter is dedicated to the existence of potential article in Czech. Further the usage of the articles in German and English is described, primarily with the abstracta. In the practical part ten selected abstracta will be analysed. For the analyses the cases in which there are different articles will be crucial. The occurrence and function of Czech demonstrative and indefinite pronouns that evince certain similarities of an article will be also examined.

By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised.

Esta tese tem por objetivo discutir algumas das questões semântico- pragmáticas relacionadas com as descrições demonstrativas, especificamente aquelas ligadas aos usos dêiticos dessas expressões, os quais englobam, ao nosso ver, os usos dêiticos canônicos, os uso descritivos e os usos com referência deferida (NUNBERG, 1993, 2004; ELBOURNE, 2005, 2008). Para realizar nossas análises, utilizamos a semântica de situações (KRATZER, 1989) e as noções de situação default e situa_c~ao não-default (WOLTER, 2006). Para iniciar a investigação, no Capítulo 1, retomamos conceitos da abordagem de Kaplan (1989), uma das pioneiras em relação ao es- tudo dos demonstrativos. Além disso, também expomos os principais aspectos da teoria de Wolter (2006) { uma abordagem semântico- pragmática elegante que tenta explicar diferentes usos das descrições demonstrativas através da semântica de situações. Utilizamos a teoria de Wolter (2006) como parâmetro de comparação pela sua importância e simplicidade em relação as outras abordagens. Por isso, a partir da abordagem da autora levantamos algumas questões no primeiro capítulo, as quais nortearam nossa investigação ao longo da tese. No Capítulo 2, detemo-nos no papel da demonstração nos usos dêiticos das descrições demonstrativas. Nossa investigação procurava entender se apontamentos (gestos físicos que acompanham e são importantes para a determinação do valor semântico das expressões dêiticas) são elemento\essenciais", no sentido de serem convencionalmente ligados às descrições demonstrativas. Concluímos que demonstrações e saliência são fatores ”não-essenciais" da semântica das descrições demonstrativas, porque eles podem ser substituídos por outros elementos dado um contexto adequado, ponto de vista que se aproxima da abordagem de Wolter (2006). No Capítulo 3, buscamos elucidar a influência e/ou a função semântica do conteúdo descritivo adicional na interpretação das descrições demonstrativas dêiticas. Tendo como pano de fundo a abordagem de Wolter (2006), suas noções de situação default e de situação não- default e suas análises dos \usos descritivos W" ou NDNS (KING, 2001), verificamos que a ideia da autora de que o material adicional introduz uma variável de situação extra na forma lógica _e uma assunção desnecessária. Afirmamos isso, com base na constatação de que o conteúdo descritivo adicional é o ultimo e o mais poderoso meio de restrição do domínio de referentes das descrições demonstrativas, i.e., ele é o elemento que assegura a unicidade quando outros elementos { apontamento, saliência, informação espacial do determinante não são suficientes. No Capítulo 4, analisamos os usos descritivos e os usos com referência deferida das descrições demonstrativas. Buscamos verificar a natureza e a importância do componente relacional, proposto por Nunberg (1993) e formalizado por Elbourne (2005), na semântica/pragmática dessas expressões. Concluímos, após análises empíricas, que o componente relacional é resultado de muitos processos pragmáticos, o que torna sua representação demasiadamente complexa e, muitas vezes, não completamente fiel aos processos envolvidos. Os usos descritivos e com referência deferida também serviram para elucidarmos o nível de semelhança/diferença semântico-pragmática das descrições demonstrativas e das descrições definidas. Chegamos à conclusão que devido às diferenças nos processos de interpretação desses dois grupos de expressões, aspecto evidenciado pelos usos descritivos e com referência deferida, elas não devem ser consideradas expressões similares do ponto de vista semântico-pragmático, apesar de alguns autores afirmarem o contrário (WOLTER, 2006; ELBOURNE, 2005, 2008, entre outros). / This dissertation aims to discuss some of the main points about seman- tics and pragmatics of demonstratives descriptions, speci cally those related to deictic uses of these expressions, which in our point of view include canonical deictic uses, descriptive uses and, uses with deferred reference (NUNBERG, 1993, 2004; ELBOURNE, 2005, 2008). We use the situations semantics (KRATZER, 1989) and conceptions of default situation and non-default situation (WOLTER, 2006) to work in our analyzes. To begin the investigation, in Chapter 1, we recall concepts from Kaplan's approach (1989), one of the pioneers in relation to the study of demonstratives. In addition, we also discuss the main aspects of Wolter's theory (2006); an elegant semantic-pragmatic approach that tries to explain different uses of demonstrative descriptions through situation semantics. Wolter (2006) is our parameter of comparison for its importance and simplicity. Because of this, in this chapter we also raise some questions related to Wolter's theory which guide our investigation throughout the thesis. The purpose of Chapter 2 is analyzing the role of demonstration in deictic uses of demonstrative descriptions. Our investigation tryes to explain whether pointings (physical gestures associated to deictic ex- pressions and important to semantic value determination) are \essential elements" in the sense of being conventionally related to demonstrative descriptions. We conclude that demonstrations and salience are \non- essential" elements to the semantics of the demonstrative descriptions, mainly because they can be replaced by other elements given a suitable context. In Chapter 3, our focus is to elucidate the semantic in uence and/or the semantic function of additional descriptive content in the interpretation of deictic demonstrative descriptions. Our investigation is based on Wolter's approach (2006), her concepts of default and non-default situations and her analysis of \W descriptive uses" or NDNS's uses (KING, 2001) { these uses are distintc from deictic uses, but they also have additional descriptive content associated with demonstrative descriptions. We conclude that Wolter's conception, in which the additional material introduces an extra situation variable in logical form, is an unnecessary assumption. We affirm this, based on examples where the additional descriptive content is the last resource and the most powerful means of restricting the domain of reference of demons- trative descriptions, i.e., it is the element that ensures the uniqueness when other elements, such as pointing, contextual salience and, spatial information of the determiner, are not enough to do this. In Chapter 4, we analyze descriptive uses and uses with deferred reference of demonstrative descriptions. We verify the nature and the importance of the relational component, proposed by Nunberg (1993) and formalized by Elbourne (2005), in the semantics / pragmatics of these expressions. We conclude, after empirical analysis, that the relational component is the result of many pragmatic processes. This complexity makes its formal representation too complicated and often not completely faithful to all the processes involved in interpretation. The descriptive uses and uses with deferred reference also contributed to elucidate the semantics / pragmatics level of similarity / difference between demonstrative descriptions and de nite descriptions. We conclude that the distinct processes of interpretation of these two groups of expressions are relevant, so they should not be considered similar expressions from the semantic-pragmatic point of view, although some authors affirm the opposite (WOLTER, 2006; ELBOURNE, 2005, 2008, among others).

A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\mathscr{H}$ of functions with the property that the values $f(x)$ for $f \in \mathscr{H}$ are reproduced from the inner product in $\mathscr{H}$. Recent applications are found in stochastic processes (Ito Calculus), harmonic analysis, complex analysis, learning theory, and machine learning algorithms. This research began with the study of RKHSs to areas such as learning theory, sampling theory, and harmonic analysis. From the Moore-Aronszajn theorem, we have an explicit correspondence between reproducing kernel Hilbert spaces (RKHS) and reproducing kernel functions—also called positive definite kernels or positive definite functions. The focus here is on the duality between positive definite functions and their boundary spaces; these boundary spaces often lead to the study of Gaussian processes or Brownian motion. It is known that every reproducing kernel Hilbert space has an associated generalized boundary probability space. The Arveson (reproducing) kernel is $K(z,w) = \frac{1}{1-_{\C^d}}, z,w \in \B_d$, and Arveson showed, \cite{Arveson}, that the Arveson kernel does not follow the boundary analysis we were finding in other RKHS. Thus, we were led to define a new reproducing kernel on the unit ball in complex $n$-space, and naturally this lead to the study of a new reproducing kernel Hilbert space. This reproducing kernel Hilbert space stems from boundary analysis of the Arveson kernel. The construction of the new RKHS resolves the problem we faced while researching “natural” boundary spaces (for the Drury-Arveson RKHS) that yield boundary factorizations: \[K(z,w) = \int_{\mathcal{B}} K^{\mathcal{B}}_z(b)\overline{K^{\mathcal{B}}_w(b)}d\mu(b), \;\;\; z,w \in \B_d \text{ and } b \in \mathcal{B} \tag*{\it{(Factorization of} $K$).}\] Results from classical harmonic analysis on the disk (the Hardy space) are generalized and extended to the new RKHS. Particularly, our main theorem proves that, relaxing the criteria to the contractive property, we can do the generalization that Arveson's paper showed (criteria being an isometry) is not possible.