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.

In dieser Dissertation geht es um Schwinger-parametrische Feynmanintegrale in der Quantenelektrodynamik. Mittels einer Vielzahl von Methoden aus der Kombinatorik und Graphentheorie wird eine signifikante Vereinfachung des Integranden erreicht. Nach einer größtenteils in sich geschlossenen Einführung zu Feynmangraphen und -integralen wird die Herleitung der Schwinger-parametrischen Darstellung aus den klassischen Impulsraumintegralen ausführlich erläutert, sowohl für skalare Theorien als auch Quantenelektrodynamik. Es stellt sich heraus, dass die Ableitungen, die benötigt werden um Integrale aus der Quantenelektrodynamik in ihrer parametrischen Version zu formulieren, neue Graphpolynome enthalten, die auf Zykeln und minimalen Schnitten (engl. "bonds") basieren. Danach wird die Tensorstruktur der Quantenelektrodynamik, bestehend aus Dirac-Matrizen und ihren Spuren, durch eine diagrammatische Interpretation ihrer Kontraktion zu ganzzahligen Faktoren reduziert. Dabei werden insbesondere gefärbte Sehnendiagramme benutzt. Dies liefert einen parametrischen Integranden, der über bestimmte Teilmengen solcher Diagramme summierte Produkte von Zykel- und Bondpolynomen enthält. Weitere Untersuchungen der im Integranden auftauchenden Polynome decken Verbindungen zu Dodgson- und Spannwaldpolynomen auf. Dies wird benutzt um eine Identität zu beweisen, mit der sehr große Summen von Sehnendiagrammen in einer kurzen Form ausgedrückt werden können. Insbesondere führt dies zu Aufhebungen, die den Integranden massiv vereinfachen. / This thesis is concerned with the study of Schwinger parametric Feynman integrals in quantum electrodynamics. Using a variety of tools from combinatorics and graph theory, significant simplification of the integrand is achieved. After a largely self-contained introduction to Feynman graphs and integrals, the derivation of the Schwinger parametric representation from the standard momentum space integrals is reviewed in full detail for both scalar theories and quantum electrodynamics. The derivatives needed to express Feynman integrals in quantum electrodynamics in their parametric version are found to contain new types of graph polynomials based on cycle and bond subgraphs. Then the tensor structure of quantum electrodynamics, products of Dirac matrices and their traces, is reduced to integer factors with a diagrammatic interpretation of their contraction. Specifically, chord diagrams with a particular colouring are used. This results in a parametric integrand that contains sums of products of cycle and bond polynomials over certain subsets of such chord diagrams. Further study of the polynomials occurring in the integrand reveals connections to other well-known graph polynomials, the Dodgson and spanning forest polynomials. This is used to prove an identity that expresses some of the very large sums over chord diagrams in a very concise form. In particular, this leads to cancellations that massively simplify the integrand.

La premième partie a pour but l’établissement d’un développement asymptotique pour la solution du problème de Stokes avec une petite perturbation du domaine. Dans ce travail, nous avons appliqué la théorie du potentiel. On a écrit les solutions du problème non-perturbé et du problème perturbé sous forme des opérateurs intégraux. En calculant la différence, et en utilisant des propriétés liées aux noyaux des opérateurs on a établi un développement asymptotiquede la solution.L’objectif principal de la deuxième partie de ce rapport est de déterminer les termes d’ordre élevé de l’expansion asymptotique des valeurs propres et fonctions propres pour l’opérateur de Stokes dues aux changements d’interface de l’inclusion. Dans la troisième partie, nous proposons une méthode pour l’évaluation des integrales singulières provenant de la mise en oeuvre de la méthode des éléments finis de frontière en électromagnetisme. La méthodeque nous adoptons consiste en une réduction récursive de la dimension du domained’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. Pour la discrétisation du domaine, destriangles 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. / This thesis contains three main parts. The first part concerns the derivation of an asymptotic expansion for the solution of Stokes resolvent problem with a small perturbation of the domain. Firstly, we verify the continuity of the solution with respect to the small perturbation via the stability of the density function. Secondly, we derive the asymptotic expansion ofthe solution, after deriving the expansion of the density function. The procedure is based on potential theory for Stokes problem in connection with boundary integral equation method, and geometric properties of the perturbed boundary. The main objective of the second part on this report, is to present a schematic way to derive high-order asymptotic expansions for both eigenvalues and eigenfunctions for the Stokes operator caused by small perturbationsof the boundary. Also, we rigorously derive an asymptotic formula which is in some sense dual to the leading-order term in the asymptotic expansion of the perturbations in the Stokes eigenvalues due to interface changes of the inclusion. The implementation of the boundary element method requires the evaluation of integrals with a singular integrand. A reliable andaccurate calculation of these integrals can in some cases be crucial and difficult. In the third part of this report we propose a method of evaluation of singular integrals based on recursive reductions of the dimension of the integration domain. It leads to a representation of the integralas a linear combination of one-dimensional integrals whose integrand is regular and that can be evaluated numerically and even explicitly. The Maxwell equation is used as a model equation, but these results can be used for the Laplace and the Helmholtz equations in 3-D.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.

Wir geben eine rigorose Konstruktion von analytisch-regularisierten Feynman-Integralen im D-dimensionalen Minkowski-Raum als meromorphe Distributionen in den externen Impulsen, sowohl in der Impuls- als auch in der parametrischen Darstellung. Wir zeigen, dass ihre Pole durch die üblichen Power-counting Formeln gegeben sind, und dass ihr singulärer Träger in mikrolokalen Verallgemeinerungen der (+alpha)-Landauflächen enthalten ist. Als weitere Anwendungen geben wir eine Konstruktion von dimensional regularisierten Integralen im Minkowski-Raum und beweisen Diskontinuitätsformeln für parametrische Amplituden. / We give a rigorous construction of analytically regularized Feynman integrals in D-dimensional Minkowski space as meromorphic distributions in the external momenta, both in the momentum and parametric representation. We show that their pole structure is given by the usual power-counting formula and that their singular support is contained in a microlocal generalization of the alpha-Landau surfaces. As further applications, we give a construction of dimensionally regularized integrals in Minkowski space and prove discontinuity formula for parametric amplitudes.

Diabetes can affect the structure and function of the foot, resulting in severe limitation of mobility and reduction of life expectancy. Early warning signs include limited joint mobility (LJM), soft tissue changes, high plantar pressure (HPP), high pressure time integrals (P/TI) and plantar callus. These abnormalities were examined in 216 young people with diabetes and 57 controls. The fingers, toes, ankle subtalar and first metatarsophalangeal joints shows reduced motion and the plantar aponeurosis was thicker in diabetic subjects. Skin thickness was the same for diabetic and control subjects. LJM in the feet was more common in males and older subjects. Subtalar and finger LJM was associated with early sensory nerve changes and finger LJM was associated with retinopathy and higher HbAtc. Thicker plantar aponeurosis was associated with male gander and larger feet. High peak pressure, high P/TI and callus were no more common in diabetic subjects than controls. However, high P/TI and callus were associated with early sensory nerve changes in young people with diabetes. Diabetic subjects with callus were significantly older than those without callus. Those with HPP had higher body mass index and less motion at the first MTP joints than those without HPP. Although plantar callus, HPP and high P/TI were no more common in young people with diabetes these abnormailities may be complicated by diabetes. Cushioning, custom orthoses or both in combination significantly reduced peak pressure and P/TI in diabetic subjects. / Doctor of Philosophy (PhD)

The thesis provides answers, in one case partial and in the other final, to two conjectures in the area of weighted inequalities for Singular Integral Operators. We study the mapping properties of these operators in weighted Lebesgue spaces with weight w. The novelty of this thesis resides in proving sharp dependence of the operator norm on the Muckenhoupt constant associated to the weigth w for a rich class of Singular Integral operators. The thesis also addresses the end point case p=1, providing counterexamples for the dyadic and continuous settings.

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

This thesis presents the benchmarking and development of a method to study ground state properties of hydrogen clusters using molecular dynamics. Benchmark studies are performed on our Path Integral Molecular Dynamics code using the Langevin equation for finite temperature studies and our Langevin equation Path Integral Ground State code to study systems in the zero-temperature limit when all particles occupy their nuclear ground state. A simulation is run on the first 'real' system using this method, a parahydrogen molecule interacting with a fixed water molecule using a trivial unity trial wavefunction. We further develop a systematic method of optimizing the necessary parameters required for our ground state simulations and introduce more complex trial wavefunctions to study parahydrogen clusters and their isotopologues orthodeuterium and paratritium. The effect of energy convergence with parameters is observed using the trivial unity trial wavefunction, a Jastrow-type wavefunction that represents a liquid-like system, and a normal mode wavefunction that represents a solid-like system. Using a unity wavefunction gives slower energy convergence and is inefficient compared to the other two. Using the Lindemann criterion, the normal mode wavefunction acting on floppy systems introduces an ergodicity problem in our simulation, while the Jastrow does not. However, even for the most solid-like clusters, the Jastrow and the normal mode wavefunctions are equally efficient, therefore we choose the Jastrow trial wavefunction to look at properties of a range of cluster sizes. The energetic and structural properties obtained for parahydrogen and orthodeuterium clusters are consistent with previous studies, but to our knowledge, we may be the first to predict these properties for neutral paratritium clusters. The results of our ground state simulations of parahydrogen clusters, namely the distribution of pair distances, are used to calculate Raman vibrational shifts and compare to experiment. We investigate the accuracy of four interaction potentials over a range of cluster sizes and determine that, for the most part, the ab initio derived interaction potentials predict shifts more accurately than the empirically based potentials for cluster sizes smaller than the first solvation shell and the trend is reversed as the cluster size increases. This work can serve as a guide to simulate any system in the nuclear ground state using any trial wavefunction, in addition to providing several applications in using this ground state method.

Sensor networks are an evolving technology that promise numerous applications. The random and dynamic structure of sensor networks has motivated the suggestion of greedy data-routing algorithms. / In this thesis stochastic models are developed to study the advancement of messages under greedy routing in sensor networks. A model framework that is based on homogeneous spatial Poisson processes is formulated and examined to give a better understanding of the stochastic dependencies arising in the system. The effects of the model assumptions and the inherent dependencies are discussed and analyzed. A simple power-saving sleep scheme is included, and its effects on the local node density are addressed to reveal that it reduces one of the dependencies in the model. / Single hop expressions describing the advancement of messages are derived, and asymptotic expressions for the hop length moments are obtained. Expressions for the distribution of the multihop advancement of messages are derived. These expressions involve high-dimensional integrals, which are evaluated with quasi-Monte Carlo integration methods. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The subsequent results agree extremely well with those obtained via routing simulations. A renewal process model is proposed to model multihop advancements, and is justified under certain assumptions. / The model framework is extended by incorporating a spatially dependent density, which is inversely proportional to the sink distance. The aim of this extension is to demonstrate that an inhomogeneous Poisson process can be used to model a sensor network with spatially dependent node density. Elliptic integrals and asymptotic approximations are used to describe the random behaviour of hops. The final model extension entails including random transmission radii, the effects of which are discussed and analyzed. The thesis is concluded by giving future research tasks and directions.

Sensor networks are an evolving technology that promise numerous applications. The random and dynamic structure of sensor networks has motivated the suggestion of greedy data-routing algorithms. / In this thesis stochastic models are developed to study the advancement of messages under greedy routing in sensor networks. A model framework that is based on homogeneous spatial Poisson processes is formulated and examined to give a better understanding of the stochastic dependencies arising in the system. The effects of the model assumptions and the inherent dependencies are discussed and analyzed. A simple power-saving sleep scheme is included, and its effects on the local node density are addressed to reveal that it reduces one of the dependencies in the model. / Single hop expressions describing the advancement of messages are derived, and asymptotic expressions for the hop length moments are obtained. Expressions for the distribution of the multihop advancement of messages are derived. These expressions involve high-dimensional integrals, which are evaluated with quasi-Monte Carlo integration methods. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The subsequent results agree extremely well with those obtained via routing simulations. A renewal process model is proposed to model multihop advancements, and is justified under certain assumptions. / The model framework is extended by incorporating a spatially dependent density, which is inversely proportional to the sink distance. The aim of this extension is to demonstrate that an inhomogeneous Poisson process can be used to model a sensor network with spatially dependent node density. Elliptic integrals and asymptotic approximations are used to describe the random behaviour of hops. The final model extension entails including random transmission radii, the effects of which are discussed and analyzed. The thesis is concluded by giving future research tasks and directions.