Thesis (doctoral)--University of Bonn, 1988. / Includes bibliographical references (p. 108-109).

Thesis--University of Maryland, College Park. / Typescript. Vita. Includes bibliographical references.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1993. / Includes bibliographical references (p. 49-50).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1989. / Includes bibliographical references.

Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 155-157).

In the first part of this thesis, strong comparison theorems of Sturm's type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to non-self-adjoint quasilinear systems. Application of these results to first order systems leads to criteria for the existence of conjugate points. In the second part, comparison theorems are obtained for elliptic differential operators of arbitrary even order. A description of the behaviour of the smallest eigenvalue for such operators is given under domain perturbations by means of Garding's inequality. New oscillation and nonoscillation criteria are obtained by variational methods. Specialization of our theorems to elliptic equations of fourth order, and to ordinary differential equations yields various generalizations of known results. / Science, Faculty of / Mathematics, Department of / Graduate

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

The aim of this thesis is to provide an introduction to the study of elliptic curves and modular forms over general commutative rings or schemes. We will recall a few aspects of the classical theory of these objects (over the complex numbers) while placing emphasis on the geometric picture. Moreover, we will formulate the theory of elliptic curves in the modern language of algebraic geometry following the work of Katz and Mazur. In addition, we provide an application of p−adic modular forms to the theory of p−adic heights on elliptic curves.

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