Thesis Supervisor: C. A. Swanson. New comparison and Sturm-type theorems are established which enable us to extend known oscillation and non-oscillation criteria to: (1) non-self-adjoint operators, (2) quasi-linear operators, (3) fourth order operators of a type not previously-considered. Since the classical principle of Courant does not hold for some of the operators considered, the comparison theorems involve, in part, new estimates on the location of the smallest eigenvalue of the operators in question. A description of the behaviour of the eigenvalue as the domain is perturbed is also given for such operators by the use of Schauder's "a priori" estimates. The Sturm-type theorems are proved by topological arguments and extended to quasi-linear as well as to non-self-adjoint operators. The fourth order operators considered are of a type which does not yield forms identical to those arising in second order problems. Some examples illustrating the theory are given. / Science, Faculty of / Mathematics, Department of / Graduate

Criteria will be obtained for a linear self-adjoint elliptic partial differential equation to be oscillatory or nonoscillatory in unbounded domains R of n-dimensional Euclidean space Eⁿ. The criteria are of two main types: (i) those involving integrals of suitable majorants of the coefficients, and (ii) those involving limits of these majorants as the argument tends to infinity. Our theorems constitute generalizations to partial differential equations of well-known criteria of Hille, Leighton, Potter, Moore, and Wintner for ordinary differential equations. In general, our method provides the means for extending in this manner any oscillation criterion for self-adjoint ordinary differential equations. Our results imply Glazman's theorems in the special case of the Schrodinger equation in Eⁿ. In the derivation of the oscillation criteria it is assumed that R is either quasiconical (i.e. contains an infinite cone) or limit-cylindrical (i.e. contains an infinite cylinder). In the derivation of the nonoscillation criteria no special assumptions regarding the shape of the domain are needed. Examples illustrating the theory are given. In particular, it is shown that the limit criteria obtained in the second order case are the best possible of their kind. / Science, Faculty of / Mathematics, Department of / Graduate

Yeung Wai Kong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 51-55). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Some Preliminaries --- p.13 / Chapter 3 --- "Approximate Function we,p" --- p.17 / Chapter 4 --- "The Computation Of Je[we,p]" --- p.21 / Chapter 5 --- The Signs of c1 And c3 --- p.30 / Chapter 6 --- The Asymptotic Behavior of ue and Je[ue] --- p.35 / Chapter 7 --- "The Proofs Of Theorem 1.1, Theorem 1.2 And Corol- lary 11" --- p.40 / Appendix --- p.43 / Bibliography --- p.51

Ng, Chit Yu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 89-90). / Abstracts in English and Chinese.

Thesis (M.S.)--University of Nevada, Reno, 2005. / "May 2005." Includes bibliographical references (leaves 27-28). Online version available on the World Wide Web.

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 40-42).

Inaug.-Diss.--Bonn. / Extra t.p. with thesis statement inserted. Bibliography: p. 54.

Thesis (Ph. D.)--University of Alberta, 2010. / Title from pdf file main screen (viewed on June 18, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.

In Chapter 2 we study the uniqueness problem of sign-changing solutions for a nonlinear scalar equation. It is well-known that positive solution is radially symmetric and unique up to a translation. Recently, there are many works on the existence and multiplicity of sign-changing solutions. However much less is known for uniqueness, even in the radially symmetric class. In Chapter 2, we solve this problem for nearly critical nonlinearity by Lyaponov-Schmidt reduction. Moreover, we can also prove the non-degeneracy. / In Chapter 3 we are concerned with the uniqueness problem for coupled nonlinear Schrodinger equations. The problem is to classify all positive solutions. In Chapter 3, some sufficient conditions are given. In particular, we have a sufficient and necessary condition in one dimension. The proof is elementary because only the implicit function theorem, integration by parts, and the uniqueness for scalar equation are needed. / In Chapter 4 we go back to the nonlinear scalar equation and consider the traveling wave solutions. Using an infinite dimensional Lyaponov-Schmidt reduction, new examples of traveling wave solutions are constructed. Our approach explains the difference between two dimension and higher dimensions, and also explores a connection between moving fronts and the mean curvature flow. This is the first such traveling waves connecting the same states. / This thesis is devoted to the study of nonlinear elliptic equations and systems. It is divided into two parts. In the first part, we study the uniqueness problem, and in the second part, we are concerns with traveling wave solutions. / Yao, Wei. / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 132-142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.