Ph.D. Dissertations- Mathematics Department

Permanent URI for this collectionhttps://hdl.handle.net/10125/23255

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Linear and non-linear operators, and the distribution of zeros of entire functions
(2013) Yoshida, Rintaro; Csordas, George
An important chapter in the theory of distribution of zeros of entire functions pertains to the study of linear operators acting on entire functions. This dissertation presents new results involving not only linear, but also some non-linear operators.
On-linear coefficient-wise stability and hyperbolicity preserving transformations
(University of Hawaii at Manoa, 2012) Grabarek, Lukasz; Csordas, George
We study the operation of replacing the coefficients of a real function with a non-linear combination of its coefficients. We are particularly interested in the coefficient-wise transformations that preserve the location of zeros in a prescribed region.
Congruence Lattices of Finite Algebras
(University of Hawaii at Manoa, 2012) DeMeo, William J.; Freese, Ralph; Mathematics
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.
Sparse ordinary graphs
(University of Hawaii at Manoa, 2005) Kalk, Jonathan W.; Mathematics
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.
Small lattices
(University of Hawaii at Manoa, 2000) Heeney, Xiang Xia Huang; Freese, Ralph; Mathematics
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.
Potential Good Reduction of Degree 2 Rational Maps
(University of Hawaii at Manoa, 2012) Yap, Diane; Manes, Michelle; Mathematics
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.
P-adic analysis and mock modular forms
(University of Hawaii at Manoa, 2011) Kent, Zachary A.; Guerzhoy, Pavel; Mathematics
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.
Linear preservers and entire functions with restricted zero loci
(University of Hawaii at Manoa, 2011) Chasse, Matthew Negus; Csordas, George; Mathematics
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.
Linear Operators and the Distribution of Zeros of Entire Functions
(University of Hawaii at Manoa, 2007) Piotrowski, Andrzej; Csordas, George; Mathematics
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.
Generalized Gelfand triples
(University of Hawaii at Manoa, 1971) Casteren, J.A. van
Dual linear spaces generated by a non-Desarguesian configuration
(University of Hawaii at Manoa, 2005) Seffrood, Jiajia Yang Garcia
A dual linear space is a partial projective plane which contains the intersection of every pair of its lines. Every dual linear space can be extended to a projective plane, usually infinite, by a sequence of one line extensions. Moreover, one may describe necessary conditions for the sequence of one line extensions to terminate after finitely many steps with a finite projective plane. A computer program that attempts to construct a finite projective plane from a given dual linear space by a sequence of one line extension has been written by Dr. Nation. In particular, one would like to extend a dual linear space containing a non-Desarguesian configuration to a finite projective plane of nonprime- power order. This dissertation studies the initial dual linear spaces to be used in this algorithm. The main result is that there are 105 non-isomorphic initial dual linear spaces containing the basic non-Desarguesian configuration.
Geometry and singularities of spatial and spherical curves
(University of Hawaii at Manoa, 2004) Xiong, Jianfei
In the first part of this dissertation the spherical evolute, the spherical involute, the spherical orthotomic and the spherical antiorthotomic are investigated and their local diffeomorphic types are determined. The concept of the spherical conic is introduced. It is proven that the incident angle and reflection angle are equal for the spherical conic. The necessary and sufficient conditions for the spherical conic to be a circle are given. In the second part of this dissertation the ruled surfaces of normals and binormals of a regular space curve are locally classified under the left-right action according to the types of the curve. For this purpose some results are obtained on the relationship of the powers of terms in the Taylor series of an invertible function and its inverse.
Finite group graded lie algebraic extensions and trefoil symmetric relativity, standard model, yang mills and gravity theories
(University of Hawaii at Manoa, 2008) Wills, Luis Alberto
We introduce the Quotient Group Graded Lie algebras, which involve graded structure constants. This structure is then used to obtain a graded extension of supersymmetry where diverse features of the Standard Model of elementary particles arise naturally. For the Minimal Vector Clover Extension of the symmetries of special relativity, we develop the extended superspace formalism in differential geometric language. We construct connections, curvature, and prove Bianchi identities both in coordinate and in symmetry covariant bases. We study also a connection making the Lorentz symmetry point dependent, its torsion and curvature. Moreover, we examine a transformation that removes noncommutativity from the Minimal Vector Clover Extension.
Automated reasoning and machine learning
(University of Hawaii at Manoa, 1996) Huang, Guoxiang
This dissertation introduces new theorem-proving strategies and uses these strategies to solve a wide variety of difficult problems requiring logical reasoning. It also shows how to use theorem-proving to solve the problem of learning mathematical concepts. Our first algorithm constructs formulas called Craig interpolants from the refutation proofs generated by contemporary theorem-provers using binary resolution, paramodulation, and factoring. This algorithm can construct the formulas needed to learn concepts expressible in the full first-order logic from examples of the concept. It can also find sentences which distinguish pairs of nonisomorphic finite structures. We then apply case analysis to solve hard problems such as the zebra problem, the pigeonhole problem, and the stable marriage problem. The case analysis technique we use is the first to be fully compatible with resolution and rewriting and powerful enough to solve these problems. Our primary new theorem-proving strategies generate subgoals and efficient sets of rules. We show how to divide problems into smaller parts with intermediate goals by reversing logical implications. We solve these subdivided parts by discovering efficient subsets of rules or by generating efficient new rules. We apply these and other new search strategies to solve difficult problems such as the 15-puzzle, central solitaire, TopSpin, Rubik's cube, and masterball. Our strategies apply universally to all such problems and can solve them quite efficiently: the 15-puzzle, Rubik's cube and masterball can all be done in 300 seconds. Finally we apply our search strategies to solve real-world problems such as sorting, solving equations and inverting nonsingular matrices.
Combinatorial remedies
(University of Hawaii at Manoa, 1994) Mackey, John Fletcher
This dissertation consists of 3 chapters which consider distinct combinatorial problems. In the first chapter we consider objects known as groupies. A non-isolated vertex of a graph G is called a groupie if the average degree of the vertices connected to it is larger than or equal to the average degree of the vertices in G. An isolated vertex is a groupie only if all vertices of G are isolated. It is well known that every graph must contain at least one groupie. The graph Kn - e contains precisely 2 groupie vertices for n ≥2. In this chapter we derive a lower bound for the number of groupies in terms of the number of vertices of each particular degree. This shows, in particular, that any graph with 2 or more vertices must contain at least 2 groupies. In chapter 2 we show that, for a prime number p+n+1 , the number of pth powers in Sn+1 is n+1 times the number of pth powers in Sn. Here Sn denotes the symmetric group on n objects. Our technique also yields a recursion for calculating the number of r": powers in Sn+1. An analogous identity, and corresponding recursion, for pth powers containing a specified number of pi-cycles is also obtained. This generalizes work of Blum in the case p = 2, and Chernoff's work for general p. Finally, in chapter 3, we use techniques introduced by Giraud to obtain lower bounds for certain Ramsey Numbers. The Ramsey Number R(k,j) is defined to be the least positive integer n such that every n-vertex graph contains either a clique of order k or an independent set of order j. In particular we show that R (6, 6) ≤ 165, R (7, 7) ≤ 540, R (8, 8) ≤ 1870, R (9, 9) ≤ 6625, and R (10, 10) ≤ 23854 . These estimates replace bounds of 166, 574, 1982, 7042, and 25082, respectively.
Problems in hyperbolic geometry
(University of Hawaii at Manoa, 1993) Reiser, Edward J.
In this thesis, we discuss the proof that all convex polyhedral metrics can be realized in euclidean and hyperbolic 3-space. This result is accredited to A.D. Alexandrov and is fundamental in modern synthetic differential geometry. Nevertheless, gaps appear in currently acknowledged proofs: (1) It is necessary to prove that strictly convex metrics with 4 real vertices can be realized. (2) It must be shown that, within manifolds of convex polyhedra in E3 or H3, there exist submanifolds of degenerate polyhedra which are "thin" when mapped into manifolds of (abstract) strictly convex metrics. In this thesis we prove these statements. The remainder of the thesis is devoted to general hyperbolic geometry with emphasis on the synthetic point of view. We first construct horocyclic coordinates and use these to derive the Poincare model for the hyperbolic plane. Then we compute useful formulas for the curvature of a surface, and use these formulas to study C2 surfaces in H3, infinitesimal deformations of the horosphere, and curves of constant curvature in H2. Finally, we also prove that certain surfaces of rotation in E3 isometrically imbed in H3. These results, some of which are new, provide a background for synthetic methods underlying the theorem of Alexandrov.
Almost completely decomposable groups with two critical types and their endomorphism rings
(University of Hawaii at Manoa, 1992) Lewis, Wayne
An almost completely decomposable group with two critical types is a direct sum of rank-one groups and indecomposable rank-two groups. A complete set of near isomorphism invariants for an acd group with two critical types is the isomorphism class of the regulator and the isomorphism class of the regulator quotient; with one additional invariant, namely an element of a certain quotient group of (Ζ/m Ζ)^x , a complete set of isomorphism invariants for an acd group with two critical types is obtained. Finally, the endomorphism ring of an acd group with two critical types is computed and the resulting structure is used to give an example of two nearly isomorphic groups with non-isomorphic endomorphism rings.
A covering space approach to (d,k) constrained codes
(University of Hawaii at Manoa, 1992) Perry, Patrick Neil
The capacity of the (d, k) constrained codes and of the (d, k) L level charge constrained codes is considered. The case of rational capacity is examined. and an error in the literature is corrected. This leads to an interesting (0.3)L = 4 level charge constrained code. The error control for this code is done using the finite field GF(3). A table of the ternary convolutional codes of greatest free distance is given for possible applications. The topological properties of the (d, k) constraint graphs are examined. The fundamental group of a constraint graph and covering spaces of a constraint graph are discussed. A constructive process for building a covering space of a constraint graph is given. The construction of (d, k) constrained block codes from covering spaces of the (d, k) constraint graph is examined. Several types of block codes are introduced. The base point codes consist of the (d, k) constrained sequences whose associated edge paths in the covering space are loops at a specified vertex. The parity point codes consist of the (d, k) constrained sequences whose associated edge paths in the covering space conned two specified vertices. It is shown that an [n, k] cyclic code can be constructed as a base point code for a 2^(n-k) fold covering of the (0,∞) constraint graph. Systematic (d, k) constrained block codes are constructed for detecting all single shift errors, drop in errors, and drop out errors. The average probability of an undetected error for the systematic (d, k) constrained block codes is shown to decrease exponentially with the parity length times the capacity of a (d, k) constrained code.
Maximum principles and Liouville theorems for elliptic partial differential equations
(University of Hawaii at Manoa, 1990) Zhou, Chiping
New classes of finite commutative rings
(University of Hawaii at Manoa, 2003-05) Vo, Monika; Craven, Thomas; Mathematics
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.

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