Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Clint N. Dawson. Vita. Includes bibliographical references.

Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 06/21/2010). Advisor and committee chair names found in the thesis' metadata record in the digital repository. Includes bibliographical references.

Thesis (Ph. D.)--University of Sydney, 2008. / Title from title screen (viewed December 12, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliographical references. Also available in print form.

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2019 / We will study the symmetry, invariance properties and conservation laws of partial dif ferential equations (pdes) that arise in a number of situations in mathematical physics. These will be range from Image Processing and noise removal algorithms to Timoshenko beam systems. Furthermore, we will study the invariance properties and approximate conservation laws of some nonlinear Schro¨dinger equation with PT-symmetric potentials with inhomogeneous nonlinearity and some nonlinear Schro¨dinger equation involving a spatially extended system consisting of two coupled elements. / TL (2019)

Introduction. Statement of problem: The primary purpose of this study is to indicate the manner in which a Boundary Value problem in Physics leads to the solution in generalized Fourier Series. The conditions to be met in problems of this sort are generally the Partial Differential Equation and several unique physical conditions which are imposed on the distribution sought after. The problem is solved when a mathematical solution of the Differential Equation is found which satisfies all of the restrictions levied by the physical considerations. The secondary purpose of this study is to obtain a view of the generalized problem which leads ultimately to the Sturm-Liouville theory.

Random functions is the central component in many statistical and probabilistic problems. This dissertation presents theoretical analysis and computation for random functions and its applications in statistics. This dissertation consists of two parts. The first part is on the topic of classic continuous random fields. We present asymptotic analysis and computation for three non-linear functionals of random fields. In Chapter 1, we propose an efficient Monte Carlo algorithm for computing P{sup_T f(t)>b} when b is large, and f is a Gaussian random field living on a compact subset T. For each pre-specified relative error ɛ, the proposed algorithm runs in a constant time for an arbitrarily large $b$ and computes the probability with the relative error ɛ. In Chapter 2, we present the asymptotic analysis for the tail probability of ∫_T e^{σf(t)+μ(t)}dt under the asymptotic regime that σ tends to zero. In Chapter 3, we consider partial differential equations (PDE) with random coefficients, and we develop an unbiased Monte Carlo estimator with finite variance for computing expectations of the solution to random PDEs. Moreover, the expected computational cost of generating one such estimator is finite. In this analysis, we employ a quadratic approximation to solve random PDEs and perform precise error analysis of this numerical solver. The second part of this dissertation focuses on topics in statistics. The random functions of interest are likelihood functions, whose maximum plays a key role in statistical inference. We present asymptotic analysis for likelihood based hypothesis tests and sequential analysis. In Chapter 4, we derive an analytical form for the exponential decay rate of error probabilities of the generalized likelihood ratio test for testing two general families of hypotheses. In Chapter 5, we study asymptotic properties of the generalized sequential probability ratio test, the stopping rule of which is the first boundary crossing time of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero. These results have important theoretical implications in hypothesis testing, model selection, and other areas where maximum likelihood is employed.

Call number: LD2668 .T4 1978 H77 / Master of Science

Call number: LD2668 .T4 1978 G92 / Master of Science

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy