http://hdl.handle.net/10125/23255 Thu, 23 May 2013 15:09:55 GMT 2013-05-23T15:09:55Z http://hdl.handle.net/10125/25938 An important and long-standing open problem in universal algebra asks whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. Until this problem is resolved, our understanding of finite algebras is incomplete, since, given an arbitrary finite algebra, we cannot say whether there are any restrictions on the shape of its congruence lattice. If we find a finite lattice that does not occur as the congruence lattice of a finite algebra (as many suspect we will), then we can finally declare that such restrictions do exist. By a well known result of Palfy and Pudlak, the problem would be solved if we could prove the existence of a finite lattice that is not the congruence lattice of a transitive group action or, equivalently, is not an interval in the lattice of subgroups of a finite group. Thus the problem of characterizing congruence lattices of finite algebras is closely related to the problem of characterizing intervals in subgroup lattices. In this work, we review a number of methods for finding a finite algebra with a given congruence lattice, including searching for intervals in subgroup lattices. We also consider methods for proving that algebras with a given congruence lattice exist without actually constructing them. By combining these well known methods with a new method we have developed, and with much help from computer software like the UACalc and GAP, we prove that with one possible exception every lattice with at most seven elements is isomorphic to the congruence lattice of a finite algebra. As such, we have identified the unique smallest lattice for which there is no known representation. We examine this exceptional lattice in detail, and prove results that characterize the class of algebras that could possibly represent this lattice. We conclude with what we feel are the most interesting open questions surrounding this problem and discuss possibilities for future work. Thesis (Ph. D.)--University of Hawaii at Manoa, 2012. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10125/25938 2012-01-01T00:00:00Z DeMeo, William J. http://hdl.handle.net/10125/25937 Ordinary graphs are directed graphs that can be viewed as generalizations of symmetric block designs. They were introduced by Fossorier, Jezek, Nation and Pogel in [2] in an attempt to construct new finite projective planes. In this thesis we investigate some special cases of ordinary graphs, most prominently the case where nonadjacent vertices have no common neighbors. We determine all connected graphs of this type that exist. vii, 65 leaves, bound ; 29 cm.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2005. Sat, 01 Jan 2005 00:00:00 GMT http://hdl.handle.net/10125/25937 2005-01-01T00:00:00Z Kalk, Jonathan W. http://hdl.handle.net/10125/25936 This dissertation introduces triple gluing lattices and proves that a triple gluing lattice is small if the key subcomponents are small. Then attention is turned to triple gluing irreducible small lattices. The triple gluing irreducible [Special characters omitted.] lattices are introduced. The conditions which insure [Special characters omitted.] small are discovered. This dissertation also give some triple gluing irreducible small lattices by gluing [Special characters omitted.] 's. Finally, K-structured lattices are introduced. We prove that a K-structured lattice L is triple gluing irreducible if and only if [Special characters omitted.] . We prove that no 4-element antichain lies in u 1 /v1 of a K-structured small lattice. We also prove that some special lattices with 3-element antichains can not lie in u1 /v1 of a K-structured small lattice. viii, 87 leaves, bound : ill. ; 29 cm.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2000. Sat, 01 Jan 2000 00:00:00 GMT http://hdl.handle.net/10125/25936 2000-01-01T00:00:00Z Heeney, Xiang Xia Huang http://hdl.handle.net/10125/25935 We give a complete characterization of degree two rational maps on P1 with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space M2. The proof includes an algorithm by which to conjugate any degree two rational map corresponding to an integral point in M2 into a map with unit resultant. The local fields result is used to solve the same problem for number fields with class number 1. Some additional results are given for degree 2 rational maps over Q. We also give a full description of post-critically finite maps in M2(Q), including the algorithm used to find them. Thesis (Ph. D.)--University of Hawaii at Manoa, 2012. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10125/25935 2012-01-01T00:00:00Z Yap, Diane http://hdl.handle.net/10125/25934 A mock modular form f+ is the holomorphic part of a harmonic Maass form f. The non-holomorphic part of f is a period integral of a cusp form g, which we call the shadow of f+. The study of mock modular forms and mock theta functions is one of the most active areas in number theory with important works by Bringmann, Ono, Zagier, Zwegers, among many others. The theory has many wide-ranging applications: additive number theory, elliptic curves, mathematical physics, representation theory, and many others. We consider arithmetic properties of mock modular forms in three different settings: zeros of a certain family of modular forms, coupling the Fourier coefficients of mock modular forms and their shadows, and critical values of modular L-functions. For a prime p > 3, we consider j-zeros of a certain family of modular forms called Eisenstein series. When the weight of the Eisenstein series is p - 1, the j-zeros are j-invariants of elliptic curves with supersingular reduction modulo p. We lift these j-zeros to a p-adic field, and show that when the weights of two Eisenstein series are p-adically close, then there are j-zeros of both series that are p-adically close. A direct method for relating the coefficients of shadows and mock modular forms is not known. This is considered to be among the first of Ono's Fundamental Problems for mock modular forms. The fact that a shadow can be cast by infinitely many mock modular forms, and the expected transcendence of generic mock modular forms pose serious obstructions to this problem. We solve these problems when the shadow is an integer weight cusp form. Our solution is p-adic, and it relies on our definition of an algebraic regularized mock modular form. We use mock modular forms to compute generating functions for the critical values of modular L-functions. To obtain this result we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes an "Eichler-Shimura isomorphism", a "multiplicity two" Hecke theory, a correspondence between mock modular periods and classical periods, and a "Haberland-type" formula which expresses Petersson's inner product and a related antisymmetric inner product on M!k in terms of periods. 87 leaves, bound ; 29 cm.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2010. Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25934 2011-01-01T00:00:00Z Kent, Zachary A http://hdl.handle.net/10125/25933 Let T : R [x] → R [x] be a linear operator such that T[ xk] = gammakxk for all k = 0, 1, 2, ..., where gamma k ∈ R . The real sequence gkinfinity k=0 is called a multiplier sequence if for any p ∈ R [x], having only real zeros, T[ p] also has only real zeros. A characterization of all multiplier sequences that can be interpolated by rational functions is given. This partially solves a problem of G. Csordas and T. Craven, who asked for a characterization of all the meromorphic functions, Y(k), such that Yk infinityk=0 is a multiplier sequence. An eight-year-old conjecture of I. Krasikov is proved. Several discrete analogues of classical inequalities for polynomials with only real zeros are obtained, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-Polya class. A study of finite difference operators which preserve reality of zeros is initiated, and new results are proved. Composition theorems and inequalities for polynomials having their zeros in a sector are obtained. These are analogs of classical results by Polya, Schur, and Turan. In addition, a result of Obreschkoff is used to show that the Jensen polynomials related to the Riemann xi-function have only real zeros up to degree 1017. Sufficient conditions are established for a linear transformation to map polynomials having zeros only in a sector to polynomials of the same type, and some multivariate extensions of these results are presented. A complete characterization is given for linear operators which preserve closed ("strict") half-plane stability in the univariate Weyl algebra. These results provide new information about a general stability problem posed formally by G. Csordas and T. Craven. In his 2011 AMS Bulletin article, D. G. Wagner describes recent activity in multivariate stable polynomial theory as "exciting work---elementary but subtle, and with spectacular consequences." Wagner points out that many of the recent advancements in the theory of multivariate stable polynomials are due to the pioneering work of J. Borcea and P. Branden. These results play an important role in the investigation of linear stability preservers in this dissertation. Several different approaches to characterizing linear transformations which map polynomials having zeros only in one region of the complex plane to polynomials of the same type are explored. In addition, an open problem of S. Fisk is solved, and several partial results pertaining to open problems from the 2007 AIM workshop "Polya-Schur-Lax problems: hyperbolicity and stability preservers" are obtained. Thesis (Ph. D.)--University of Hawaii at Manoa, 2011. Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25933 2011-01-01T00:00:00Z Chasse, Matthew Negus http://hdl.handle.net/10125/25932 Motivated by the work of Pólya, Schur, and Turán, a complete characterization of multiplier sequences for the Hermite polynomial basis is given. Laguerre's theorem and a remarkable curve theorem due to Pólya are generalized. Sufficient conditions for the location of zeros in certain strips in the complex plane are determined. Results pertaining to multiplier sequences and complex zero decreasing sequences for other polynomial sets are established. viii, 178 leaves, bound ; 29 cm.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2007. Mon, 01 Jan 2007 00:00:00 GMT http://hdl.handle.net/10125/25932 2007-01-01T00:00:00Z Piotrowski, Andrzej http://hdl.handle.net/10125/11728 Typescript.; Thesis (Ph. D.)--University of Hawaii, 1971.; Bibliography: leaves 73-74.; vi, 74 l Fri, 01 Jan 1971 00:00:00 GMT http://hdl.handle.net/10125/11728 1971-01-01T00:00:00Z Casteren, J. A. van http://hdl.handle.net/10125/11727 Mode of access: World Wide Web.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2005.; Includes bibliographical references (leaves 160-161).; Electronic reproduction.; Also available by subscription via World Wide Web; viii, 161 leaves, bound ill. 29 cm Sat, 01 Jan 2005 00:00:00 GMT http://hdl.handle.net/10125/11727 2005-01-01T00:00:00Z Seffrood, Jiajia Yang Garcia http://hdl.handle.net/10125/11726 Mode of access: World Wide Web.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2004.; Includes bibliographical references (leaves 113-114).; Electronic reproduction.; Also available by subscription via World Wide Web; vii, 114 leaves, bound ill. 29 cm Thu, 01 Jan 2004 00:00:00 GMT http://hdl.handle.net/10125/11726 2004-01-01T00:00:00Z Xiong, Jianfei http://hdl.handle.net/10125/11725 Mode of access: World Wide Web.; Thesis (Ph. D.)--University of Hawaii at Manoa, 2004.; Includes bibliographical references (leaves 159-164).; Electronic reproduction.; Also available by subscription via World Wide Web; x, 164 leaves, bound ill. 29 cm Tue, 01 Jan 2008 00:00:00 GMT http://hdl.handle.net/10125/11725 2008-01-01T00:00:00Z Wills, Luis Alberto http://hdl.handle.net/10125/9963 Thesis (Ph. D.)--University of Hawaii at Manoa, 1996.; Includes bibliographical references (leaves 140-144).; Microfiche.; x, 144 leaves, bound ill. 29 cm Mon, 01 Jan 1996 00:00:00 GMT http://hdl.handle.net/10125/9963 1996-01-01T00:00:00Z Huang, Guoxiang http://hdl.handle.net/10125/9962 Thesis (Ph. D.)--University of Hawaii at Manoa, 1994.; Includes bibliographical references (leaves 29-30); Microfiche.; 30 leaves, bound ill. 29 cm Sat, 01 Jan 1994 00:00:00 GMT http://hdl.handle.net/10125/9962 1994-01-01T00:00:00Z Mackey, John Fletcher http://hdl.handle.net/10125/9961 Thesis (Ph. D.)--University of Hawaii at Manoa, 1993.; Includes bibliographical references (leaf 124); Microfiche.; vii, 124 leaves, bound ill. 29 cm Fri, 01 Jan 1993 00:00:00 GMT http://hdl.handle.net/10125/9961 1993-01-01T00:00:00Z Reiser, Edward J http://hdl.handle.net/10125/9960 Thesis (Ph. D.)--University of Hawaii at Manoa, 1992.; Includes bibliographical references (leaves 48-49); Microfiche.; iv, 49 leaves, bound 29 cm Wed, 01 Jan 1992 00:00:00 GMT http://hdl.handle.net/10125/9960 1992-01-01T00:00:00Z Lewis, Wayne http://hdl.handle.net/10125/9959 Thesis (Ph. D.)--University of Hawaii at Manoa, 1992.; Includes bibliographical references (leaves 102-104); Microfiche.; ix, 104 leaves, bound ill. 29 cm Wed, 01 Jan 1992 00:00:00 GMT http://hdl.handle.net/10125/9959 1992-01-01T00:00:00Z Perry, Patrick Neil http://hdl.handle.net/10125/9958 Typescript.; Thesis (Ph. D.)--University of Hawaii at Manoa, 1990.; Includes bibliographical references.; Microfiche.; vi, 96 leaves, bound ill. 29 cm Mon, 01 Jan 1990 00:00:00 GMT http://hdl.handle.net/10125/9958 1990-01-01T00:00:00Z Zhou, Chiping http://hdl.handle.net/10125/3076 This dissertation introduces the concept of Q-Witt rings and SQ-Witt rings. A Q-Witt ring is defined as a finite quotient of a torsion free abstract Witt ring for an elementary 2-group G. Local Q-Witt rings are characterized using topological and ring theoretic tools. Q-Witt rings of the integral group ring Z[Z2] are classified and several properties are shown. An SQ-Witt ring is formed as a finite quotient of torsion free Witt rings of a formally real field. Recursive construction can be used to locate all SQ-Witt rings. Thesis (Ph. D.)--University of Hawaii at Manoa, 2003.; Includes bibliographical references (leaves 51-52).; Mode of access: World Wide Web.; Also available by subscription via World Wide Web; vi, 52 leaves, bound 29 cm Wed, 01 Jan 2003 00:00:00 GMT http://hdl.handle.net/10125/3076 2003-01-01T00:00:00Z Vo, Monika