We have directed much attention towards developing quantum trajectory methods which can accurately predict the transmission probabilities for a variety of quantum mechanical barrier scattering processes. One promising method involves solving the complex quantum Hamilton-Jacobi equation with the Derivative Propagation Method (DPM). We present this method, termed complex valued DPM (CVDPM(n)). CVDPM(n) has been successfully employed in the Lagrangian frame to accurately compute transmission probabilities on 'thick' one dimensional Eckart and Gaussian potential surfaces. CVDPM(n) is able to reproduce accurate results with a much lower order of approximation than is required by real valued quantum trajectory methods, from initial wave packet energies ranging from the tunneling case (E[subscript o]=0) to high energy cases (twice the barrier height). We successfully extended CVDPM(n) to two-dimensional problems (one translational degree of freedom representing an Eckart or Gaussian barrier coupled to a vibrational degree of freedom) in the Lagrangian framework with great success. CVDPM helps to explain why barrier scattering from "thick" barriers is a much more well posed problem than barrier scattering from "thin" barriers. Though results in these two cases are in very good agreement with grid methods, the search for an appropriate set of initial conditions (termed an 'isochrone) from which to launch the trajectories leads to a time-consuming search problem that is reminiscent of the rootsearching problem from semi-classical dynamics. In order to circumvent the isochrone problem, we present CVDPM(n) equations of motion which are derived and implemented in the arbitrary Lagrangian-Eulerian frame for a metastable potential as well as the Eckart and Gaussian surfaces. In this way, the isochrone problem can be circumvented but at the cost of introducing other computational difficulties. In order to understand why CVDPM may give better transmission probabilities than real valued counterparts, much attention we have been studying and applying numerical analytic continuation techniques to visualize complex-extended wave packets as well as the complex-extended quantum potential. Numerical analytic continuation techniques have also been used to analytically continue a discrete real-valued potential into the complex plane for CVDPM with very promising results.

The focus of this thesis is optimal path planning. The path planning problem is posed as an optimal control problem, for which the viscosity solution to the static Hamilton-Jacobi-Bellman (HJB) equation is used to determine the optimal path. The Ordered Upwind Method (OUM) has been previously used to numerically approximate the viscosity solution of the static HJB equation for direction-dependent weights. The contributions of this thesis include an analytical bound on the convergence rate of the OUM for the boundary value problem to the viscosity solution of the HJB equation. The convergence result provided in this thesis is to our knowledge the tightest existing bound on the convergence order of OUM solutions to the viscosity solution of the static HJB equation. Only convergence without any guarantee of rate has been previously shown. Navigation functions are often used to provide controls to robots. These functions can suffer from local minima that are not also global minima, which correspond to the inability to find a path at those minima. Provided the weight function is positive, the viscosity solution to the static HJB equation cannot have local minima. Though this has been discussed in literature, a proof has not yet appeared. The solution of the HJB equation is shown in this work to have no local minima that is not also global. A path can be found using this method. Though finding the shortest path is often considered in optimal path planning, safe and energy efficient paths are required for rover path planning. Reducing instability risk based on tip-over axes and maximizing solar exposure are important to consider in achieving these goals. In addition to obstacle avoidance, soil risk and path length on terrain are considered. In particular, the tip-over instability risk is a direction-dependent criteria, for which accurate approximate solutions to the static HJB equation cannot be found using the simpler Fast Marching Method. An extension of the OUM to include a bi-directional search for the source-point path planning problem is also presented. The solution is found on a smaller region of the environment, containing the optimal path. Savings in computational time are observed. A comparison is made in the path planning problem in both timing and performance between a genetic algorithm rover path planner and OUM. A comparison in timing and number of updates required is made between OUM and several other algorithms that approximate the same static HJB equation. Finally, the OUM algorithm solving the boundary value problem is shown to converge numerically with the rate of the proven theoretical bound.

Les travaux de recherche de cette thèse concernent l'encadrement de la constante de Markov Bernstein pour la norme L2 associée aux mesures de Jacobi et Gegenbauer généralisée. Ce travail est composé de deux parties : dans la première partie, nous avons développé une généralisation de l'algorithme qd pour les matrices symétriques définies positives à largeur de bande $\ell$ et nous avons construit l'algorithme qd pour les matrices de Jacobi par blocs. Ensuite, nous l'avons généralisé aux cas des matrices par bloc à largeur de bande $\ell$. Ces algorithmes nous permettent de trouver un majorant de la constante. Enfin, nous avons développé le déterminant caractéristique d'une matrice symétrique définie positive pentadiagonale, ce qui nous permet d'obtenir un minorant de la constante en utilisant la méthode de Newton. La deuxième partie est consacrée à l'application de tous les outils développés à l'encadrement de la constante de Markov Bernstein pour la norme L2 associée à la mesure de Gegenbauer généralisée.

WKB procedure is used in the study of asymptotic solutions of differential equations of the hypergeometric type. Hence asymptotic forms of classical orthogonal polynomials associated with the names Jacobi, Laguerre and Hermite have been derived. In particular, the asymptotic expansion of the Jacobi polynomials $P^{(alpha, beta)}_n(x)$ as $n$ tends to infinity is emphasized.

No description available.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (p. 87-91).

Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Aug. 6, 2009). Vita. Includes bibliographical references.

Diss.--Literaturwissenschaft--Duisburg Universität, 1990. / Bibliogr. p. 243-246.

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Orientador: Bruto Max Pimentel Escobar / Banca: Dmitry Vasilevich / Banca: José Abdalla Helaÿel-Neto / Banca: Álvaro de Souza Dutra / Banca: Roldão da Rocha Júnior / Resumo: Neste trabalho apresentaremos o formalismo de Hamilton-Jacobi para sistemas singulares em teorias de campos, com foco em teorias com derivadas de ordem superior. Iniciaremos com uma análise preliminar do cálculo variacional para esses sistemas, que envolve as condições para a extremização de uma integral fundamental múltipla e a análise dos teoremas de Noether. Buscaremos seguir este caminho na construção do formalismo de Hamilton- Jacobi em forma covariante, em que nos utilizaremos da clássica abordagem de Carathéodory adaptada a teorias de campos. No terceiro capítulo, mostraremos como o formalismo pode ser construído dada a escolha de uma dinâmica relativística específica e como esta escolha nos permite tratar de sistemas singulares de forma natural. No quarto capítulo abordaremos o problema das condições de integrabilidade, análise que garantirá um método autoconsistente de análise de vínculos. Nesta análise, seremos capazes de relacionar um conjunto de geradores a simetrias da integral fundamental e um segundo tipo a uma modificação da dinâmica com a introdução de parênteses generalizados. Nos dois últimos capítulos apresentaremos aplicações deste método / Abstract: In this work we will develop the Hamilton-Jacobi formalism to singular and higher-order derivative field theories. We will begin with a preliminary approach to the variational problem concerning the search for extrema of a given fundamental integral, and the analysis of the Noether's theorems. Next, we will present a covariant Hamilton-Jacobi theory using the classical approach of Carathéodory applied to field theories. In the third chapter we will show how this formalism can be derived given a choice of relativistic dynamics, and how this choice allows us to deal with singular systems. In the fourth chapter we will address the problem of integrability conditions. This analysis will be the basic tool for a self consistent constraint analysis. We will see that we can relate a certain set of generators to symmetries of the action, as well as a second type of generators to a modification of the dynamics by means of generalized brackets. The two last chapters will be used for applications / Doutor