Mathematics Department
http://hdl.handle.net/10125/23249
Thu, 24 Apr 2014 06:51:07 GMT2014-04-24T06:51:07ZGeneralized Analytic Continuation
http://hdl.handle.net/10125/29513
Analytic continuation is the extension of the domain of a given analytic function in the complex plane, to a larger domain of the complex plane. This process has been utilized in many other areas of mathematics, and has given mathematicians new insight into some of the world’s hardest problems. This paper will cover more general forms of analytic continuation, which will be referred to as generalized analytic continuations. The paper will closely follow William Ross’ and Harold Shapiro’s book “Generalized Analytic Continuation” [14], with the proofs worked out with more detail, and a few generalizations are made regarding the Poincare example in Section 3.3.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10125/295132012-01-01T00:00:00ZToyofuku, JustinInformation Processing and Energy Dissipation in Neurons
http://hdl.handle.net/10125/29510
We investigate the relationship between thermodynamic and information theoretic inefficiencies in an individual neuron model, the adaptive exponential integrate-and-fire neuron. Recent work has revealed that minimization of energy dissipation is tightly related to optimal information processing, in the sense that a system has to compute a maximally predictive model. In this thesis we justify the extension of these results to the neuron and quantify the neuron’s thermodynamic and information processing inefficiencies.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10125/295102012-01-01T00:00:00ZMcIntosh, LaneAn Iterated Version of the Generalized Singular Value Decomposition for the Joint Analysis of Two High-Dimensional Data Sets
http://hdl.handle.net/10125/29508
In this work, we developed a new computational algorithm for the integrated analysis of high-dimensional data sets based on the Generalized Singular Value Decomposition(GSVD). We developed an iterative version of the Generalized Singular Value Decomposition (IGSVD) that jointly analyzes two data matrices to identify signals that correlate the rows of two matrices. The IGSVD has been validated on simulated and real genomic data sets and results on simulated show that the algorithm is able to sequentially detect multiple simulated signals that were embedded in high levels of background noise. Results on real DNA microarray data from normal and tumor tissue samples indicate that the IGSVD detects signals that are biologically relevant to the initiation and progression of liver cancer.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10125/295082013-01-01T00:00:00ZZeinalzadeh, AshkanNondeterministic Finite State Complexity
http://hdl.handle.net/10125/29507
We define a new measure of complexity for finite strings using nondeterministic finite automata, called nondeterministic automatic complexity and denoted AN(x). In this paper we prove some basic results for AN(x), give upper and lower bounds, estimate it for some specific strings, begin to classify types of strings with small complexities, and provide AN(x) for |x| ≤ 8.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10125/295072013-01-01T00:00:00ZHyde, KayleighValidating a Food Frequency Questionnaire for Guam
http://hdl.handle.net/10125/29506
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10125/295062012-01-01T00:00:00ZChong, Marie Q.Geometric Path Planning for a Lego AUV
http://hdl.handle.net/10125/29504
For the last thirty years or so, differential geometry and control theory have merged and grown together to produce extraordinary results. When applied to mechanical systems, one sees a system waiting to be exploited for its inherent geometric properties. In this paper, we present the equations of motion for a submerged rigid body from a geometric point of view and use tools from differential geometry to provide solutions to the motion planning problem for an autonomous underwater vehicle. Specifically, the geometry allows us to deduce permissible motions for a vehicle that is underactuated purely from the available degrees of freedom. The geometric equations of motion are then used to path plan for a cost-effective Lego vehicle through simulations and actual implementation as providing a proof of concept.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10125/295042012-01-01T00:00:00ZAndonian, MichaelLinear and non-linear operators, and the distribution of zeros of entire functions
http://hdl.handle.net/10125/29455
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.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10125/294552013-01-01T00:00:00ZYoshida, RintaroOn-linear coefficient-wise stability and hyperbolicity preserving transformations
http://hdl.handle.net/10125/29454
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.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10125/294542012-01-01T00:00:00ZGrabarek, LukaszEquations implying congruence n-permutability and semidistributivity
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 GMThttp://hdl.handle.net/10125/259492013-02-08T00:00:00ZFreese, RalphCongruence Lattices of Finite Algebras
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 GMThttp://hdl.handle.net/10125/259382012-01-01T00:00:00ZDeMeo, William J.Sparse ordinary graphs
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 GMThttp://hdl.handle.net/10125/259372005-01-01T00:00:00ZKalk, Jonathan W.Small lattices
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 GMThttp://hdl.handle.net/10125/259362000-01-01T00:00:00ZHeeney, Xiang Xia HuangPotential Good Reduction of Degree 2 Rational Maps
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 GMThttp://hdl.handle.net/10125/259352012-01-01T00:00:00ZYap, DianeP-adic analysis and mock modular forms
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 GMThttp://hdl.handle.net/10125/259342011-01-01T00:00:00ZKent, Zachary ALinear preservers and entire functions with restricted zero loci
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 GMThttp://hdl.handle.net/10125/259332011-01-01T00:00:00ZChasse, Matthew NegusLinear Operators and the Distribution of Zeros of Entire Functions
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 GMThttp://hdl.handle.net/10125/259322007-01-01T00:00:00ZPiotrowski, AndrzejThe snowflake decoding algorithm
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 GMThttp://hdl.handle.net/10125/259312012-01-01T00:00:00ZWalker, CatherineExtending lp--decoding for permutation codes from euclidean to Kendall tau metric
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 GMThttp://hdl.handle.net/10125/259302012-01-01T00:00:00ZKong, JustinOn Kaneko congruences
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 GMThttp://hdl.handle.net/10125/259292012-01-01T00:00:00ZChi, MingjingMultimetric continuous model theory
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 GMThttp://hdl.handle.net/10125/259282012-01-01T00:00:00ZCaulfield, Erin