A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, March 2015. / The primary purpose of this study is to investigate the asymptotic distribution of the eigenvalues of self-adjoint second order di erential operators. We rst analyse the problem where the functions g and h are equal to zero. To improve on the terms of the eigenvalue problem for g; h = 0, we consider the eigenvalue problem for general functions g and h. Here we calculate explicitly the rst four terms of the eigenvalue asymptotics problem.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1985. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Univ., Diss.--Sendai, 2000.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

We consider a formally self-adjoint elliptic differential operator in IRⁿ, denoted by τ. T₀ and T are operators given by τ with specific domains. We determine conditions under which T₀ is essentially self-adjoint, introducing the topic by means of a brief historical survey of some results in this field. In Part I, we consider an operator of order 4, and in Part II, we generalise the results obtained there to ones for an operator of order 2m. Thus, the two parts run parallel. In Chapter 1, we determine the domain of T₀*, denoted by D(T₀*), where T₀* denotes the adjoint of T₀, and introduce operators <u>T</u>₀ and <u>T</u> which are modifications of T₀ and T. In Chapter 2, we use a theorem of Schechter to give conditions under which <u>T</u>₀ is essentially self-adjoint. Working with the operator <u>T</u>, in Chapter 3 ve show that we can approximate functions u in D(T₀*) by a particular sequence of test-functions, which enables us to derive an identity involving u, Tu and the coefficient functions of the operator concerned. In Chapter 4, we determine an upper bound for the integral of a function involving a derivative of u in D(T₀*) whose order is half the order of the operator concerned, and we use the identity from the previous chapter to reformulate this upper bound. In Chapter 5, we give conditions which are sufficient for the essential self-adjointness of T₀. In the main theorem itself, the major step is the derivation of the integral of the function involving the particular derivative of u in D(T₀*) whose order is half the order of the operator concerned, referred to above, itself as a term of an upper bound of an integral we wish to estimate. Hence, we can employ the upper bound from Chapter 4. This "sandwiching" technique is basic to the approach we have adopted. We conclude with a brief discussion of the operators we considered, and restate the examples of operators which we showed to be essentially self-adjoint.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 148-150). Also available in electronic version. Access restricted to campus users.

No description available.

Thesis (Ph.D.)--York University, 2008. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 102-108). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR51694

First published in the Bulletin of the American Mathematical Society in 1963, published by the American Mathematical Society

This thesis consists of three papers, each touching on a different aspect of the theory of rings of differential operators and D-modules. In particular, an aim is to provide and make explicit good examples of D-module directimages, which are all but absent in the existing literature.The first paper makes explicit the fact that B-splines (a particular class of piecewise polynomial functions) are solutions to D-module theoretic direct images of a class of D-modules constructed from polytopes.These modules, and their direct images, inherit all the relevant combinatorial structure from the defining polytopes, and as such are extremely well-behaved.The second paper studies the ring of differential operator on a reduced monomial ring (aka. Stanley-Reisner ring), in arbitrary characteristic.The two-sided ideal structure of the ring of differential operators is described in terms of the associated abstract simplicial complex, and several quite different proofs are given.The third paper computes the monodromy of the period integrals of Laurent polynomials about the singular point at the origin. The monodromy is describable in terms of the Newton polytope of the Laurent polynomial, in particular the combinatorial-algebraic operation of mutation plays an important role. Special attention is given to the class of maximally mutable Laurent polynomials, as these are one side of the conjectured correspondance that classifies Fano manifolds via mirror symmetry. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Accepted. Paper 2: Manuscript. Paper 3: Manuscript.</p>