http://hdl.handle.net/10125/23249 Thu, 23 May 2013 16:54:53 GMT 2013-05-23T16:54:53Z http://hdl.handle.net/10125/25949 T. Dent, K. Kearnes and A. Szendrei de ne the derivative, 0, of a set of equations and show, for idempotent , that implies congruence modularity if 0 is inconsistent ( 0 j= x y). In this paper we investigate other types of derivatives that give similar results for congruence n-permutable for some n, and for congruence semidistributivity. Fri, 08 Feb 2013 00:00:00 GMT http://hdl.handle.net/10125/25949 2013-02-08T00:00:00Z Freese, Ralph 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/10125/25931 This paper describes an automated algorithm for generating a group code using any unitary group, initial vector, and generating set that satisfy a necessary condition. Examples with exceptional complex reflection groups, as well as an analysis of the decoding complexity, are also included. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2012 Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10125/25931 2012-01-01T00:00:00Z Walker, Catherine http://hdl.handle.net/10125/25930 Invented in the 1960’s, permutation codes have reemerged in recent years as a topic of great interest because of properties making them attractive for certain modern technological applications. In 2011 a decoding method called LP (linear programming) decoding was introduced for a class of permutation codes with a Euclidean distance induced metric. In this paper we comparatively analyze the Euclidean and Kendall tau metrics, ultimately providing conditions and examples for which LP-decoding methods can be extended to permutation codes with the Kendall tau metric. This is significant since contemporary research in permutation codes and their promising applications has incorporated the Kendall tau metric. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2012 Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10125/25930 2012-01-01T00:00:00Z Kong, Justin http://hdl.handle.net/10125/25929 We present a proof of certain congruences modulo powers of an odd prime for the coefficients of a series produced by repeated application of U -operator to a certain weakly holomorphic modular form. This kind of congruences were first observed by Kaneko as a result of numerical experiments, and later proved in a different (but similar) case by Guerzhoy [6]. It is interesting to note that, in our case, the congruences become different, both experimentally and theoretically, depending on whether the prime is congruent to 1 or 3 modulo 4. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2012 Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10125/25929 2012-01-01T00:00:00Z Chi, Mingjing http://hdl.handle.net/10125/25928 In this paper, we study metric structures with a finite number of metrics by extending the model theory developed by Ben Yaacov et al. in themonograph Model theory for metric structures. We first define a metric structure with finitely many metrics, develop the theory of ultraproducts of multimetric structures, and prove some classical model-theoretic theorems about saturation for structures with multiple metrics. Next, we give a characterization of axiomatizability of certain classes of multimetric structures. Finally, we discuss potential avenues of research regarding structures with multiple metrics. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2012 Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10125/25928 2012-01-01T00:00:00Z Caulfield, Erin http://hdl.handle.net/10125/25927 We will consider spaces with nice connectedness properties, and the groups that act on them in such a way that the topology is preserved; we consider looking at the symmetry groups of a surface of genus g. Restricting our view to nite groups, we will develop the concept of covering spaces and illustrate its usefulness to the study of these group actions by generalizing this development to the theory of branched coverings. This theory lets us develop the famous Riemann-Hurwitz Relation, which will in turn allow us to develop Hurwitz's Inequality, an upper bound on the order of a symmetry group of a given surface. We then follow Kulkarni and use the Riemann-Hurwitz Relation to construct a congruence relating the genus g of a surface to the cyclic deciencies of the symmetry groups that can act on it. These developments will then be applied to study a special case of branched coverings, those in which there is only one branch point, yielding a lower bound on the genus of both surfaces involved. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2011 Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25927 2011-01-01T00:00:00Z Turner, Wm. Pitt V http://hdl.handle.net/10125/25926 The Discrete Logarithm Problem (DLP) has been the subject of interest among many mathematicians and cryptographers in recent times because of its computational di¢ culty. For the former, the enormity of the mathematics involved and the intellectual challenge that it entails are certainly motivating factors; for the latter, its usefulness in the ...eld of cryptography. Cryptosystems rest their security on some assumptions that certain mathematical problems are dificult to solve. The ElGamal cryptosystem, for instance, is considered secure because of the computational assumption that it is di¢ cult to solve the Discrete Logarithm Problem. The di¢ culty of the DLP lies in the fact that it has a “one-way” property. Its computational complexity is roughly measured by the computing time of the algorithm used to solve this mathematical problem. This paper is a brief survey of some of the best known algorithms for solving the DLP, examines their computing time, and considers the DLP over two particular ...nite groups. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2011 Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25926 2011-01-01T00:00:00Z Sia, Gretel S. http://hdl.handle.net/10125/25925 This paper presents a detailed study of a-T-menable discrete groups. Starting with several conditions required for a-T-menability, we prove that they are equivalent and hence charaterize a class of a-T-menable discrete groups. We then show that the free groups on two generators is a-T-menable. Using the infnite cyclic group, we succesfully draw a rigid connection –from the perspective of affine isometric actions –between amenable groups and a-T-menable groups. We also prove that the quotient of an a-T-menable group by a finite normal subgroup is a-T-menable. We conclude with a new proof that the free product of two a-T-menable groups is a-T-menable. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2011 Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25925 2011-01-01T00:00:00Z Lu, Ni Yen http://hdl.handle.net/10125/25924 Some Rational elliptic curves whose modular parametrization is given by an Eichler integral were considered. The points, other than cusps, that map to zero under modular parametrization were studied computationally. Surprisingly, these zeros appear to be CM-points. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2011 Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25924 2011-01-01T00:00:00Z Kodgis, Lisa http://hdl.handle.net/10125/25923 This paper considers complex reflection groups for which the generalization of subgroup decoding method does not work at all are considered in this paper. A new method of decoding is introduced to effectively encode and decode the exceptional complex reflection groups. A general decoding algorithm is devised and the results of analysis are presented. Discussion of future research is presented as well. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2011 Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25923 2011-01-01T00:00:00Z Kim, Hye Jung http://hdl.handle.net/10125/25922 Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2011 Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10125/25922 2011-01-01T00:00:00Z Hedley, Mark http://hdl.handle.net/10125/25921 We consider a situation where coalitions are formed to divide a resource. As in real life, the value of a payoff to a given agent is allowed to depend on the payoff to other agents with whom he shares a common interest. There are various notions of equilibrium for this type of game, including the core and no-treat equilibrium. These stabilities may exist or not, depending on the power structure and the rule for allocating the resource. It is shown that under certain conditions, the no-treat equilibrium can exist even though the core is empty. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2010 Fri, 01 Jan 2010 00:00:00 GMT http://hdl.handle.net/10125/25921 2010-01-01T00:00:00Z Mizuno, Ryo http://hdl.handle.net/10125/25920 Polynomial-clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone C, a sequence B ∈ X ω is C-reducible to A ∈ X ω if there is an algorithm that computes B from A using only effectively selected functions from C. We show that if A is a Kurtz random sequence and C1 C2 are distinct polynomial clones, then there is a sequence B that is C1 -reducible to A but not C2 -reducible to A. This implies a generalization of a result first proved by Lachlan for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov-Loveland stochastic. Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2010 Fri, 01 Jan 2010 00:00:00 GMT http://hdl.handle.net/10125/25920 2010-01-01T00:00:00Z Culver, Quinn