M.Phil. (Energy Studies) / Nodal diffusion methods are often used to calculate the distribution of neutrons in a nuclear reactor core. They require few-group homogenized neutron cross sections for every heterogeneous sub-region of the core. The homogenized cross sections are pre-calculated at various reactor states and represented in a way that facilitates the reconstruction of cross sections at other possible states. In this study a number of such representations were built for the homogenized cross sections of a MOX (mixed oxide) fuel assembly via hierarchical Lagrange interpolation on Clenshaw-Curtis sparse grids. These cross sections were represented as a function of various thermal hydraulic and material composition parameters of a pressurized water reactor core (i.e. burnup, soluble boron concentration, fuel temperature, moderator temperature and moderator density), which are generally referred to as state parameters. Representations were produced for the homogenized cross sections of a number of individual isotopes, as well as the e ective (lumped) cross section of all the materials in the assembly. This was done for both two and six energy groups. Additionally, two sets of state parameter intervals were considered for each of the group structures. The first set of intervals was chosen to correspond to conditions that may be encountered during day-to-day reactor operations. The second set of intervals was chosen to be applicable to the simulation of accident scenarios and therefore have wider ranges for fuel temperature, moderator temperature and moderator density.

MA, Clinical Psychology, Faculty of Humanities, University of the Witwatersrand

A definir / A definir

Este trabalho procura contribuir para uma melhor compreensão dos fenômenos relacionados ao ensino e à aprendizagem de Matemática, tendo como foco a exploração e apreensão de um objeto matemático por meio de suas diversas representações semióticas. Em nossa pesquisa exploratória, nos apoiamos nos estudos do psicólogo francês Raymond Duval, que trata do uso dos registros de representação semiótica na Matemática, para compreender o contexto e as variáveis envolvidas em tais fenômenos. Buscamos estender o alcance das observações de Duval com um estudo prévio das teorias relacionadas à Semiótica, na forma concebida por Charles Sanders Peirce, e com estudos filosóficos que buscaram entender como o homem percebe a realidade que o cerca, principalmente por meio dos escritos de Ernst Cassirer. Utilizamos, nesse trabalho, a teoria das matrizes introduzida aqui com alguns detalhes de sua intricada evolução histórica como exemplo para mostrar a variedade de representações que um objeto matemático pode carregar, e avaliamos, sob este aspecto, uma amostra de livros didáticos representativa das décadas de 1980, 1990 e 2000. / This study is aimed to make a contribution for a better understanding of the phenomena related to teaching and learning of Mathematics. Its focus is the exploration and apprehension of a mathematical object by way of its several semiotic representations. In our exploratory research, we got support from the studies of the French psychologist Raymond Duval, which has dealt with the use of semiotics representations in Mathematics to understand the context and aspects related to such phenomena. We aimed to reinforce Duvals observations by adding a study on the theories that surround the field of Semiotics, from the first concepts of Charles Sanders Peirce and including several philosophical studies, like Ernst Cassirers works, that have offered different understandings on the perception of reality that surrounds the Man. In this work we will find the Theory of Matrices as field of application; we included some historical background on this subject and we also showed the variety of representations that such Mathematical object can support and we evaluated, on such aspect, a representative sample of textbooks published on 1980, 1990 and 2000 decades.

O teorema de Hilbert-Burch fornece uma boa descrição de variedades determinantais de codi- mensão dois e de suas deformações em termos da matriz de representação. Neste trabalho, usamos esta correspondência para estudar propriedades de tais variedades usando métodos da teoria de singularidades. Na primeira parte da tese, estabelecemos a teoria de singularidades de matrizes n X p, generalizando os resultados obtidos por J. W. Bruce and F. Tari em [5], para ma- trizes quadradas, e por A. Frühbis-Krüger em [16], para matrizes n X (n+1). Na segunda parte, nos concentramos em variedades determinantais de codimensão 2, com singularidade isolada na origem. Para estas variedades, podemos mostrar a existência e a unicidade de suavizações, o que possibilita definir seu número de Milnor como o número de Betti na dimensão média de sua fibra genérica. Para superfícies em \'C POT. 4\', obtemos uma fórmula Lê-Greuel expressando o número de Milnor da superfície em termos da segunda multiplicidade polar e do número de Milnor de uma seção genérica / The theorem of Hilbert- Burch provides a good description of codimension two determinantal varieties and their deformations in terms of their presentation matrices. In this work we use this correspondence to study properties of determinantal varieties, based on methods of singularity theory of their presentation matrices. In the first part of the thesis we establish the theory of singularities for n X p matrices extending previous results of J. W. Bruce and F. Tari in [5], for classes of square matrices, and A. Frühbis-Krüger for n X (n+1) matrices in [16]. In the second part we concentrate on codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in \'C POT. 4\' , we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of the generic section

Les méthodes de projection sur des espaces de Krylov ont été employées avec grand succès pour diverses tâches en calcul scientifique, par exemple la résolution de grands systèmes d’équations linéaires, le calcul approché de valeurs propres, ou encore le calcul approché des fonctions de matrices fois un vecteur. L’objectif majeur de cette thèse est d’étudier et d’expliquer la convergence superlinéaire des méthodes de Krylov. La plupart des résultats existants sont asymptotiques avec passage à la racine n-ième et considèrent des suites de matrices. Dans un premier temps, nous généralisons une formule de Ipsen et al. concernant la convergence superlinéaire des méthodes MR valable pour des disques, à l’aide des opérateurs de Hankel et de la théorie AAK. Notre analyse permet aussi d’obtenir des bornes supérieures pour des ensembles convexes en utilisant la transformée de Faber. Ensuite nous énonçons notre principal résultat qui est un théorème d’optimalité en théorie du potentiel logarithmique. Nous montrons, à l’aide d’une nouvelle technique de discrétisation d’un potentiel, que l’inégalité de Bernstein-Walsh à poids sur un intervalle réel est optimale, à un facteur universel près, dans le cas où le champs extérieur est un potentiel d’une mesure à support réel à gauche de l’intervalle, ce qui inclut le cas des poids polynômiaux. Via un lien avec un problème sous contrainte, l’inégalité précédente s’applique à l’analyse de la convergence des méthodes de Krylov, et permet de prédire analytiquement un taux de convergence superlinéaire de la méthode du gradient conjugué et des approximations de Rayleigh-Ritz pour des fonctions de Markov, à chaque étape et pour une seule matrice. / Projection methods on Krylov spaces were used with great success for various tasks in scientific computing, for example the resolution of large systems of linear equations, the approximate computation of eigenvalues, or the approximate computation of matrix functions times a vector. The main goal in this thesis is to study and explain superlinear convergence of Krylov methods. Most of the existing formulas provide asymptotic results for the n-th root considering an increasing sequence of matrices. Firstly, we generalize a formula of Ipsen et al. concerning superlinear convergence of MR methods valid for disks using Hankel operators and AAK theory, our analysis also allows to obtain upper bounds for convex sets using the Faber transform. Then we state our main theorem which is a sharpness result in logarithmic potential theory using a new technique of discretization of a logarithmic potential. We prove that the weighted Bernstein-Walsh inequality on a real interval is sharp up to some universal constant, when the external field is given by a potential of a real measure supported at the left of the interval. As a special case this result includes the case of weights given by polynomials. Via a link with a constrained extremal problem our inequality applies to the analysis of the convergence of Krylov methods, and allows us to predict analytically the superlinear convergence of the conjugate gradient method and of the error for Rayleigh-Ritz approximations for Markov functions. Our results apply to a simple matrix, without taking the limit and without n-th root.

Esta dissertação tem como objetivo salientar a utilidade e importância de cálculos matriciais no ensino médio. Para tanto, foram estudados alguns tópicos que descrevem situações que necessitam de recursos gerados por operações matriciais. Foi observado que esses tópicos apresentam situações que evidenciam a utilidade da multiplicação de matrizes não somente no desenvolvimento teórico, mas também nas aplicações de matrizes, e têm potencial para serem abordados no ensino médio / The aim is this work is to stress on the use of algebraic operations with matrices in the mathematics teaching for secondary school students. For this purpose, we studied some topics that require algebraic operations with matrices. It was observed that these topics reveal circumstances in which the matrix multiplication is not only useful in the theoretical development but also in the applications. In addition, the studied showed that these themes have potential to be considered in the secondary school

by Lo Tai-yan, Milton. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 112-114). / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- The Noncentral Multivariate F distribution --- p.2 / Chapter 1.1.2 --- The Central Problems and the Approach --- p.4 / Chapter 1.2 --- Concepts and Terminology --- p.7 / Chapter 1.3 --- Choice of Estimates --- p.10 / Chapter 1.4 --- Related Work --- p.11 / Chapter 2 --- Estimation of the noncentrality parameter of a Noncentral Mul- tivariate F distribution --- p.19 / Chapter 2.1 --- Unbiased and Linear Estimators --- p.19 / Chapter 2.1.1 --- The unbiased estimate --- p.20 / Chapter 2.1.2 --- The Class of Linear Estimates --- p.24 / Chapter 2.2 --- Optimal Linear Estimate --- p.32 / Chapter 2.3 --- Nonlinear Estimate --- p.34 / Chapter 2.4 --- Monte Carlo Simulation Study --- p.41 / Chapter 2.5 --- Evaluation and Further Investigation --- p.42 / Chapter 3 --- Estimation of Canonical Correlation Coefficients --- p.73 / Chapter 3.1 --- Preliminary --- p.73 / Chapter 3.2 --- The Estimation Problem --- p.76 / Chapter 3.3 --- Orthogonally Invariant Estimates --- p.77 / Chapter 3.3.1 --- The Unbiased Estimate --- p.78 / Chapter 3.3.2 --- The Class of Linear Estimates --- p.78 / Chapter 3.3.3 --- The Class of Nonlinear Estimates --- p.80 / Chapter 3.4 --- Monte Carlo Simulation Study --- p.87 / Chapter 3.5 --- Evaluation and Further Investigation --- p.89 / Chapter A --- p.104 / Chapter A.1 --- Lemma 3.2 --- p.104 / Chapter A.2 --- Theorem 3.3 Leung(1992) --- p.105 / Chapter A.3 --- The Noncentral F Identity --- p.106 / Chapter B --- Bibliography --- p.111

Leung Kit Ying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 86-88). / Abstracts in English and Chinese. / Declaration --- p.i / Acknowledgement --- p.ii / Chapter 1 --- INTRODUCTION --- p.1 / Chapter 2 --- IMPROVED ESTIMATION OF THE NORMAL PRECISION MATRIX USING THE L1 AND L2 LOSS FUNCTIONS --- p.7 / Chapter 2.1 --- Previous Work --- p.9 / Chapter 2.2 --- Important Lemmas --- p.13 / Chapter 2.3 --- Improved Estimation of Σ-1 under L1 Loss Function --- p.20 / Chapter 2.4 --- Improved Estimation of Σ-1 under L2 Loss Function --- p.26 / Chapter 2.5 --- Simulation Study --- p.31 / Chapter 2.6 --- Comparison with Krishnammorthy and Gupta's result --- p.38 / Chapter 3 --- IMPROVED ESTIMATION OF THE NORMAL PRECISION MATRIX USING THE L3 AND L4 LOSS FUNCTIONS --- p.43 / Chapter 3.1 --- Justification of the Loss Functions --- p.46 / Chapter 3.2 --- Important Lemmas for Calculating Risks --- p.48 / Chapter 3.3 --- Improved Estimation of Σ-1 under L3 Loss Function --- p.55 / Chapter 3.4 --- Improved Estimation of Σ-1 under L4 Loss Function --- p.62 / Chapter 3.5 --- Simulation Study --- p.69 / Appendix --- p.77 / Reference --- p.35

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