In this thesis we shall examine the role of measurerability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bounded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration. / Mathematical Sciences / M. Sc. (Mathematics)

No description available.

In this thesis we shall examine the role of measurerability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bounded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration. / Mathematical Sciences / M. Sc. (Mathematics)

No description available.

This thesis explores the theoretical representation of localised electrons in magnetic systems, using Majorana fermions. A motivation is provided for the Majorana fermion representation, which is then developed and applied as a mean-field theory and in the path-integral formalism to the Ising model in transversal-field (TFIM) in one, two and three dimensions, on an orthonormal lattice. In one dimension the development of domain walls precludes long-range order in discrete systems; this is as free energy savings due to entropy outweigh the energetic cost of a domain wall. An argument due to Peierls exists in 2D which allows the formation of domains of ordered spins amidst a disordered background, however, which may be extended to 3D. The forms of the couplings to the bosons used in the Random Phase Analysis (RPA) are considered and an explanation for the non-existence of the phases calculated in this thesis is discussed, in terms of spare degrees of freedom in the Majorana representation. This thesis contains the first known application of Majorana fermions at the RPA level.

Dans cette thèse, nous étudions des transformées de type Radon sur certains espaces symétriques. Une transformée de type Radon associe à toute fonction continue à support compact sur une variété $M$ ses intégrales sur une classe $Xi$ de sous-variétés de $M$. Le problème sur lequel nous nous concentrons est l'inversion d'une telle transformée, c'est-à-dire déterminer la fonction à partir de ses intégrales sur les sous-variétés dans $Xi$. Nous présentons d'abord la solution de ce problème inverse due à Sigurdur Helgason et François Rouvière, entre autres, lorsque $M$ est un espace symétrique riemannien isotrope et $Xi$ une certaine orbite de sous-variétés totalement géodésiques de $M$ sous l'action d'un groupe de transformations de Lie de $M$. La transformée de Radon associée est qualifiée de totalement géodésique.Sur les espaces symétriques pseudo-riemanniens semisimples, nous considérons une autre transformée de type Radon, qui associe à toute fonction continue à support compact ses intégrales orbitales, c'est-à-dire ses intégrales sur les orbites du sous-groupe d'isotropie du groupe des transvections. L'inversion des intégrales orbitales, qui est donnée par une formule-limite, a été obtenue par Sigurdur Helgason sur les espaces symétriques lorentziens à courbure sectionnelle constante et par Jeremy Orloff sur tout espace symétrique pseudo-riemannien semisimple de rang un. Nous résolvons le problème d'inversion des intégrales orbitales sur les espaces de Cahen-Wallach, qui sont les modèles d'espaces symétriques lorentziens indécomposables résolubles.Pour finir, nous nous intéressons aux transformées de type Radon sur les espaces symétriques symplectiques à courbure de type Ricci. L'inversion des orbitales intégrales sur ces espaces lorsqu'ils sont semisimples a déjà été obtenue par Jeremy Orloff. En revanche, lorsque ces espaces ne sont pas semisimples, la transformée donnée par les intégrales orbitales n’est pas inversible. Ensuite, nous déterminons les orbites de sous-variétés totalement géodésiques symplectiques ou lagrangiennes sous l'action d'un groupe de transformations de Lie de l'espace de départ. Dans ce contexte, la méthode d'inversion développée par Sigurdur Helgason et François Rouvière, entre autres, ne fonctionne que pour les transformées de Radon totalement géodésiques symplectiques sur les espaces symétriques kählériens à courbure holomorphe constante. Les formules d'inversion de ces transformées sur les espaces hyperboliques complexes sont dues à François Rouvière. Nous calculons les formules d'inversion de ces transformées sur les espaces projectifs complexes. / In this thesis, we study Radon-type transforms on some symmetric spaces. A Radon-type transform associates to any compactly supported continuous function on a manifold $M$ its integrals over a class $Xi$ of submanifolds of $M$. The problem we address is the inversion of such a transform, that is determining the function in terms of its integrals over the submanifolds in $Xi$. We first present the solution to this inverse problem which is due to Sigurdur Helgason and François Rouvière, amongst others, when $M$ is an isotropic Riemannian symmetric space and $Xi$ a particular orbit of totally geodesic submanifolds of $M$ under the action of a Lie transformation group of $M$. The associated Radon transform is qualified as totally geodesic.On semisimple pseudo-Riemannian symmetric spaces, we consider an other Radon-type transform, which associates to any compactly supported continuous function its orbital integrals, that is its integrals over the orbits of the isotropy subgroup of the transvection group. The inversion of orbital integrals, which is given by a limit-formula, has been obtained by Sigurdur Helgason on Lorentzian symmetric spaces with constant sectional curvature and by Jeremy Orloff on any rank-one semisimple pseudo-Riemannian symmetric space. We solve the inverse problem for orbital integrals on Cahen-Wallach spaces, which are model spaces of solvable indecomposable Lorentzian symmetric spaces.In the last part of the thesis, we are interested in Radon-type transforms on symplectic symmetric spaces with Ricci-type curvature. The inversion of orbital integrals on these spaces when they are semisimple has already been obtained by Jeremy Orloff. However, when these spaces are not semisimple, the orbital integral operator is not invertible. Next, we determine the orbits of symplectic or Lagrangian totally geodesic submanifolds under the action of a Lie transformation group of the starting space. In this context, the technique of inversion that has been developed by Sigurdur Helgason and François Rouvière, amongst others, only works for symplectic totally geodesic Radon transforms on Kählerian symmetric spaces with constant holomorphic curvature. The inversion formulas for these transforms on complex hyperbolic spaces are due to François Rouvière. We compute the inversion formulas for these transforms on complex projective spaces. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Die Auswertung von Mehrschleifen-Feynman-Integralen ist eine der größten Herausforderungen bei der Berechnung präziser theoretischer Vorhersagen für die am LHC gemessenen Wirkungsquerschnitte. In den vergangenen Jahren hat sich die Nutzung von Differentialgleichungen bei der Berechnung von Feynman-Integralen als sehr erfolgreich erwiesen. Es wurde dabei beobachtet, dass die von den Feynman-Integralen erfüllte Differentialgleichung oftmals in eine sogenannte kanonische Form transformiert werden kann, welche die Integration der Differentialgleichung mittels iterierter Integrale wesentlich vereinfacht. Das zentrale Ergebnis der vorliegenden Arbeit ist ein Algorithmus zur Berechnung rationaler Transformationen von Differentialgleichungen von Feynman-Integralen in eine kanonische Form. Neben der Existenz einer solchen rationalen Transformation stellt der Algorithmus keinerlei weitere Bedingungen an die Differentialgleichung. Insbesondere ist der Algorithmus auf Mehrskalenprobleme anwendbar und erlaubt eine rationale Abhängigkeit der Differentialgleichung vom dimensionalen Regulator. Bei der Anwendung des Algorithmus wird zunächst das Transformationsgesetz im dimensionalen Regulator entwickelt, um Differentialgleichungen für die Koeffizienten in der Entwicklung der Transformation herzuleiten. Diese Differentialgleichungen werden dann mit einem rationalen Ansatz für die gesuchte Transformation gelöst. Es wird zudem eine Implementation des Algorithmus in dem Mathematica Paket CANONICA vorgestellt, welches das erste veröffentlichte Programm dieser Art ist, das auf Mehrskalenprobleme anwendbar ist. CANONICAs Potential für moderne Mehrschleifenrechnungen wird anhand mehrerer nicht trivialer Mehrschleifen-Integraltopologien demonstriert. Die gezeigten Topologien hängen von bis zu drei Variablen ab und umfassen auch vormals ungelöste Topologien, die zu Korrekturen höherer Ordnung zum Wirkungsquerschnitt der Produktion einzelner Top-Quarks am LHC beitragen. / The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven to be a powerful tool for the computation of Feynman integrals. It has been observed that the differential equation of Feynman integrals can in many instances be transformed into a so-called canonical form, which significantly simplifies its integration in terms of iterated integrals. The main result of this thesis is an algorithm to compute rational transformations of differential equations of Feynman integrals into a canonical form. Apart from requiring the existence of such a rational transformation, the algorithm needs no further assumptions about the differential equation. In particular, it is applicable to problems depending on multiple kinematic variables and also allows for a rational dependence on the dimensional regulator. First, the transformation law is expanded in the dimensional regulator to derive differential equations for the coefficients of the transformation. Using an ansatz in terms of rational functions, these differential equations are then solved to determine the transformation. This thesis also presents an implementation of the algorithm in the Mathematica package CANONICA, which is the first publicly available program to compute transformations to a canonical form for differential equations depending on multiple variables. The main functionality and its usage are illustrated with some simple examples. Furthermore, the package is applied to state-of-the-art integral topologies appearing in recent multi-loop calculations. These topologies depend on up to three variables and include previously unknown topologies contributing to higher-order corrections to the cross section of single top-quark production at the LHC.

Neste trabalho os problemas de determinação dos Fatores de Intensidade de Tensão KI e KII estáticos e dinâmicos são tratados numericamente utilizando uma formulação alternativa do Método dos Elementos de Contorno (MEC) com solução fundamental de Kelvin e matriz de massa para os problemas dinâmicos. A trinca é suposta retangular inicialmente, com suas faces não-coincidentes. Tanto as faces da trinca, quanto o contorno externo são discretizados em elementos de contorno reto com variação de forças de deslocamentos quadráticas, não havendo, portanto distinção entre elementos de trinca e de contorno externo. Integrais analíticas também são obtidas para o elemento linear isoparamétrico. As células de domínio apresentam formato triangular e suas integrais são solucionadas semi-analiticamente. Quanto às integrais de contorno, essas são obtidas analiticamente segundo eixos de referência locais, procedendo-se em seguida a rotação pra eixos globais. O algoritmo de Houbolt é empregado como integrador temporal. Exemplos numéricos da determinação desses Fatores de Intensidade de Tensão são mostrados e comparados com resultados analíticos e resultados numéricos disponíveis na literatura. / In this work the stress intensity factors KI and KII for static and dynamic two-dimensional problem are obtained numerically by an alternative mass matrix boundary element formulation. The crack is considered a rectangular hole inside the domain and its faces are not coincident. Both crack faces and boundary are discretized by straight boundary elements with quadratic approximation. Domain cells are triangular with linear approximation and their integrals are developed semi-analytically. Boundary integrals are analytically performed, for linear and quadratic approximations. They are performed at local co-ordinate axes and transformed to global co-ordinate axes. The Houbolt algorithm is used to integrate the matrix time differential equation along time. Numerical examples are shown in order to compare the results obtained by the proposed formulation and the ones presents in literature.

La mise en œuvre de la méthode des éléments finis de frontière nécessite l'évaluation d'intégrales comportant un intégrand singulier. Un calcul fiable et précis de ces intégrales peut dans certains cas se révéler à la fois crucial et difficile. La méthode que nous proposons consiste en une réduction récursive de la dimension du domaine d'intégration et aboutit à une représentation de l'intégrale sous la forme d'une combinaison linéaire d'intégrales mono-dimensionnelles dont l'intégrand est régulier et qui peuvent s'évaluer numériquement mais aussi explicitement. L'équation de Helmholtz 3-D sert d'équation modèle mais ces résultats peuvent être utilisés pour les équations de Laplace et de Maxwell 3-D. L'intégrand est décomposé en une partie homogène et une partie régulière ; cette dernière peut être traitée par les méthodes usuelles d'intégration numérique. Pour la discrétisation du domaine, des triangles plans sont utilisés ; par conséquent, nous évaluons des intégrales sur le produit de deux triangles. La technique que nous avons développée nécessite de distinguer entre diverses configurations géométriques ; c'est pourquoi nous traitons séparément le cas de triangles coplanaires, dans des plans sécants ou parallèles. Divers prolongements significatifs de la méthode sont présentés : son extension à l'électromagnétisme, l'évaluation de l'intégrale du noyau de Green complet pour les coefficients d'auto-influence, et le calcul de la partie finie d'intégrales hypersingulières. / The implementation of the boundary element method requires the evaluation of integrals with a singular integrand. A reliable and accurate calculation of these integrals can in some cases be crucial and difficult. The proposed method is a recursive reduction of the dimension of the integration domain and leads to a representation of the integral as a linear combination of one-dimensional integrals whose integrand is regular and that can be evaluated numerically and even explicitly. The 3-D Helmholtz equation is used as a model equation, but these results can be used for the Laplace and the Maxwell equations in 3-D. The integrand is decomposed into a homogeneous part and a regular part, the latter can be treated by conventional numerical integration methods. For the discretization of the domain we use planar triangles, so we evaluate integrals over the product of two triangles. The technique we have developped requires to distinguish between several geometric configurations, that's why we treat separately the case of triangles in the same plane, in secant planes and in parallel planes.

Este trabalho é dedicado à aplicação de integrais de trajetória de Grassmann para o cálculo de operadores relevantes aos problemas da teoria quântica relativística. Primeiramente uma visão geral detalhada do método é fornecida. Então concentramos nas definições e aplicações das integrais de trajetória sobre as variáveis de Grassmann. Discutimos, em detalhe, um importante papel das integrais de trajetória de Grassmann na representação de propagadores de partículas relativísticas. Derivamos o chamado fatores de spin para tais representações, fazendo as integrações Grasmannianas. Uma contribuição completamente original foi feita aplicando tais integrais ao cálculo de operadores. Derivamos, desta forma, um conjunto de fórmulas novas para as funções de operadores das matrizes y. A aplicações de tais fórmulas são apresentadas. / This work is devoted to an application of Grassmann path integrals to operator calculus relevant to problems of relativistic quantum theory. A detailed survey of path integral method is given first. Then we concentrate ourselves on definitions and applications of path integrals over Grassmann variables. We discuss in detail an important role of Grassmann path integrals in representations of relativistic particle propagators. We derive the so called spin factors for such representations doing Grassmann integrations. A completely original contribution was made in application of such integrals to operator calculus. We have derived in such a way a set of new formulas for operator functions of y-matrices. Applications of such formulas are presented.