Thesis (Ph.D.)--Auburn University, 2005. / Abstract. Vita. Includes bibliographic references (ℓ. )

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2009. / Includes bibliographical references (p. 47-48).

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 112-114) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

D.Sc. (Mathematics) / In this thesis aspects of continuous symmetries of differential equations are studied. In particular the following aspects are studied in detail: Lie algebras, the Lie derivative, the jet bundle formalism for differential equations, Lie point and Lie-Backlund symmetry vector fields, recursion operators, conservation laws, Lax pairs, the Painlcve test, Lie algebra valued differenmtial forms and Dose operators as a representation of differential operators. The purpose of the study is to gain a better understanding of complicated nonlinear dirrerential equations that describe nature and to construct solutions. The differential equations under consideration were derived [rom physics and engineering. They are the following: the Kortcweg-dc Vries equation, Burgers' cquation , the sine-Gordon equation, nonlinear diffusion equations, the Klein Gordon equation, the Schrodinger equation, nonlinear Dirac equations, Yang-Mills equations, the Lorentz model, the Lotka-Volterra model, damped unharrnonic oscillators, and others. The newly found results and insights are discussed in chapters 8 to 17. Details on the COli tents of each chapter and rcfernces to some of my articles arc given in chapter 1.

Lie Groups occur in math and physics as representations of continuous symmetries and are often described in terms of their Lie Algebra. This thesis is concerned with finding a concrete description of a Lie group given its associated Lie algebra. Several calculations toward this end are developed and then implemented in the Maple Differential Geometry package. Examples of the calculations are given.

No description available.

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

No description available.

The purpose of this thesis is the study of the Lie algebras of affine vector fields and curvature collineations of space-time, the aim being, in the first case, to obtain upper bounds on the dimension of the Lie algebra of affine vector fields (under the assumption that the space-time is non-flat) as well as to obtain a characterization of such vector fields in terms of other types of symmetries. In the case of curvature collineations the aim was that of characterizing space-times which may admit an infinite-dimensional Lie algebra of curvature collineations as well as to find local characterizations of such vector fields. Chapters 1 and 2 consist of introductory material, in Differential Geometry (Ch.l) and General Relativity (Ch.2). In Chapter 3 we study homothetic vector fields which admit fixed points. The general results of Alekseevsky (a) and Hall (b) are presented, some being deduced by different methods. Some further details and results are also given. Chapter 4 is concerned with space-times that can admit proper affine vector fields. Using the holonomy classification obtained by Hall (c) it is shown that there are essentially two classes to consider. These classes are analysed in detail and upper bounds on the dimension of the Lie algebra of affine vector fields of such space-times are obtained. In both cases local characterizations of affine vector fields are obtained. Chapter 5 is concerned with space-times which may admit proper curvature collineations. Using the results of Halford and McIntosh (d) , Hall and McIntosh (e) and Hall (f) we were able to divide our study into several classes The last two of these classes are formed by those space-times which admit a (1 or 2-dimensional) non-null distribution spanned by vector fields which contract the Riemann tensor to zero. A complete analysis of each class is made and some general results concerning the infinite-dimensionality problem are proved. The chapter ends with some comments in the cases when the distribution mentioned above is null.