Nós consideramos decomposições de estatísticas $U$ para obter testes para componentes de variância. As distribuições assintóticas das estatísticas de testes sob a hipótese nula são obtidas supondo apenas a existência do quarto momento do erro condicional e do segundo momento dos efeitos aleatórios. Isso permite sua utilização em uma classe bastante ampla de distribuições. Sob a suposição adicional de existência do quarto momento dos efeitos aleatórios, obtemos também a distribuição assintótica das estatísticas sob uma seqüência de hipóteses alternativas locais. Comparamos a eficiência dos testes propostos com aqueles dos testes clássicos, obtidos sob suposição de normalidade, por meio de estudos de simu-lação. Os testes propostos se mostram mais adequados nas situações em que a amostra é de tamanho moderado ou grande, independentemente da distribuição das fontes de variação, e nas situações em que existe fortes afastamentos da normalidade. / We consider decompositions of U-statistics to obtain tests for null variance components in linear mixed models. Their asymptotic distributions under the null hypothesis are obtained only assuming the existence of the first four moments of the conditional error distribution and the existence of the first two moments of the random effects distribution. Thus, the proposed U-tests may be employed in a large class of models. Under the additional assumption of the existence of the fourth moment of the distribution of the random effects, we also obtain the asymptotic distribution of the U-tests under a sequence of local hypothesis. We compare their efficiency with that of classical tests derived under the assumption of normality, through simulation studies. The proposed tests are more efficient in situations where the sample size is moderate or large, independently of the distribution of the sources of variation; they also perform better in situations where the underlying distributions are far from normal.

Nós consideramos decomposições de estatísticas $U$ para obter testes para componentes de variância. As distribuições assintóticas das estatísticas de testes sob a hipótese nula são obtidas supondo apenas a existência do quarto momento do erro condicional e do segundo momento dos efeitos aleatórios. Isso permite sua utilização em uma classe bastante ampla de distribuições. Sob a suposição adicional de existência do quarto momento dos efeitos aleatórios, obtemos também a distribuição assintótica das estatísticas sob uma seqüência de hipóteses alternativas locais. Comparamos a eficiência dos testes propostos com aqueles dos testes clássicos, obtidos sob suposição de normalidade, por meio de estudos de simu-lação. Os testes propostos se mostram mais adequados nas situações em que a amostra é de tamanho moderado ou grande, independentemente da distribuição das fontes de variação, e nas situações em que existe fortes afastamentos da normalidade. / We consider decompositions of U-statistics to obtain tests for null variance components in linear mixed models. Their asymptotic distributions under the null hypothesis are obtained only assuming the existence of the first four moments of the conditional error distribution and the existence of the first two moments of the random effects distribution. Thus, the proposed U-tests may be employed in a large class of models. Under the additional assumption of the existence of the fourth moment of the distribution of the random effects, we also obtain the asymptotic distribution of the U-tests under a sequence of local hypothesis. We compare their efficiency with that of classical tests derived under the assumption of normality, through simulation studies. The proposed tests are more efficient in situations where the sample size is moderate or large, independently of the distribution of the sources of variation; they also perform better in situations where the underlying distributions are far from normal.

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Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Durant la dernière décennie, les développements technologiques en radiothérapie ont transformé considérablement les techniques de traitement. Les nouveaux faisceaux non standard améliorent la conformité de la dose aux volumes cibles, mais également complexifient les procédures dosimétriques. Puisque des études récentes ont démontré l’invalidité de ces protocoles actuels avec les faisceaux non standard, un nouveau protocole applicable à la dosimétrie de référence de ces faisceaux est en préparation par l’IAEA-AAPM. Le but premier de cette étude est de caractériser les facteurs responsables des corrections non unitaires en dosimétrie des faisceaux non standard, et ainsi fournir des solutions conceptuelles afin de minimiser l’ordre de grandeur des corrections proposées dans le nouveau formalisme de l’IAEA-AAPM. Le deuxième but de l’étude est de construire des méthodes servant à estimer les incertitudes d’une manière exacte en dosimétrie non standard, et d’évaluer les niveaux d’incertitudes réalistes pouvant être obtenus dans des situations cliniques. Les résultats de l’étude démontrent que de rapporter la dose au volume sensible de la chambre remplie d’eau réduit la correction d’environ la moitié sous de hauts gradients de dose. Une relation théorique entre le facteur de correction de champs non standard idéaux et le facteur de gradient du champ de référence est obtenue. En dosimétrie par film radiochromique, des niveaux d’incertitude de l’ordre de 0.3% sont obtenus par l’application d’une procédure stricte, ce qui démontre un intérêt potentiel pour les mesures de faisceaux non standard. Les résultats suggèrent également que les incertitudes expérimentales des faisceaux non standard doivent être considérées sérieusement, que ce soit durant les procédures quotidiennes de vérification ou durant les procédures de calibration. De plus, ces incertitudes pourraient être un facteur limitatif dans la nouvelle génération de protocoles. / During the past decade, technological developments in radiation therapy have considerably transformed treatment techniques. Novel nonstandard beams improve target dose conformity, but increase the complexity of dosimetry procedures. As recent studies demonstrated the invalidity of these protocols to nonstandard beams, a new protocol applicable to nonstandard beam reference dosimetry is in preparation by the IAEA-AAPM. The first goal of the study is to characterize the factors responsible for non-unity corrections in nonstandard beam dosimetry, and provide conceptual solutions to minimize the magnitude of the corrections. The second goal is to provide methods to estimate uncertainties accurately in nonstandard beam dosimetry, and estimate uncertainty levels achievable in typical clinical situations. Results of this study show that reporting dose to the sensitive volume of the chamber filled with water reduces the correction factor approximately by half under high gradients. A theoretical expression of correction factor is obtained for ideal nonstandard reference fields. In radiochromic film dosimetry, levels of uncertainty of the order of 0.3% are achieved with strict procedures and show great potential for nonstandard beam measurements. Results also suggest that experimental uncertainties in nonstandard beam are an important issue to consider both during daily QA routine and reference dosimetry, and could be a limiting factor in the new generation of protocols.

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

This diploma thesis presents the perspective of a future teacher on mathematical contests and their impact on the development of crucial competencies, with special focus on the contest Mathematical kangaroo. The thesis deals with test questions from categories Benjamin and Kadet from 2015. Furthermore, a set of worksheets expounding similar examples that can be found in this contest, is included in the thesis. These exercises are established to serve pupils with inferior results as a preparation for the contest, or for the enlivening of math classes. Solution is illustrated via pictures, sketches, geometrical constructions, tables and verbal comments that should help pupils to better understand the topics.

In this thesis, a mathematical model for HIV and TB co-infection with TB treatment among populations of Ethiopia is developed and analyzed. The TB model includes an age of infection. We compute the basic reproduction numbers RTB and RH for TB and HIV respectively, and the overall repro- duction number R for the system. We find that if R < 1 and R > 1; then the disease-free and the endemic equilibria are locally asymptotically stable, respectively. Otherwise these equilibria are unstable. The TB-only endemic equilibrium is locally asymptotically stable if RTB > 1, and RH < 1. How- ever, the symmetric condition, RTB < 1 and RH > 1, does not necessarily guarantee the stability of the HIV-only equilibrium, but it is possible that TB can coexist with HIV when RH > 1: As a result, we assess the impact of TB treatment on the prevalence of TB and HIV co-infection. To derive and formulate the nonlinear differential equations models for HIV and TB co-infection that accounts for treatment, we formulate and analyze the HIV only sub models, the TB-only sub models and the full models of HIV and TB combined. The TB-only sub model includes both ODEs and PDEs in order to describe the variable infectiousness and e ect of TB treatment during the infectious period. To analyse and solve the three models, we construct robust methods, namely the numerical nonstandard definite difference methods (NSFDMs). Moreover, we improve the order of convergence of these methods in their applications to solve the model of HIV and TB co-infection with TB treatment at the population level in Ethiopia. The methods developed in this thesis work and show convergence, especially for individuals with small tolerance either to the disease free or the endemic equilibria for first order mixed ODE and PDE as we observed in our models. / Mathematical Sciences / Ph. D. (Applied Mathematics)

This essay deals with the use of linguistic stereotypes in Hollywood movies. It investigates whether attitudes towards English dialects found in studies on perceptual dialectology are reflected in the selected movies and discusses the notion of linguistic identity and how standard and nonstandard speech, respectively, are used symbolically to emphasize features of characters in eleven movies from three different movie series, namely The Lord of the Rings, Star Wars and Transformers, with a main focus on syntactic and phonological dimensions. The essay finds a correlation between standard speech and features of competence and wisdom, and nonstandard speech and features of solidarity, sociability and traits of stupidity and humor. Moreover, very specific perceptions of certain varieties of English are probably utilized as amplifiers of equally specific characteristics of some characters. The use of dialects and accents in these movies is probably intentional and not coincidental.

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature. / South Africa