M.A. Plan B Theses- Mathematics Department

Permanent URI for this collectionhttps://hdl.handle.net/10125/23256

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Generalized Analytic Continuation
(University of Hawaii at Manoa, 2012) Toyofuku, Justin; Smith, Wayne
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.
Information Processing and Energy Dissipation in Neurons
(University of Hawaii at Manoa, 2012) McIntosh, Lane; Still, Susanne
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.
An Iterated Version of the Generalized Singular Value Decomposition for the Joint Analysis of Two High-Dimensional Data Sets
(University of Hawaii at Manoa, 2013) Zeinalzadeh, Ashkan; Okimoto, Gordon
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.
Nondeterministic Finite State Complexity
(University of Hawaii at Manoa, 2013) Hyde, Kayleigh; Kjos-Hanssen, Bjørn
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.
Validating a Food Frequency Questionnaire for Guam
(University of Hawaii at Manoa, 2012) Chong, Marie Q.; Wilkens, Lynne
Geometric Path Planning for a Lego AUV
(University of Hawaii at Manoa, 2012) Andonian, Michael; Chyba, Monique
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.
The snowflake decoding algorithm
(University of Hawaii at Manoa, 2012) Walker, Catherine; Nation, J.B.
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.
Extending lp--decoding for permutation codes from euclidean to Kendall tau metric
(University of Hawaii at Manoa, 2012) Kong, Justin; Craven, Thomas
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.
On Kaneko congruences
(University of Hawaii at Manoa, 2012) Chi, Mingjing; Guerzhoy, Pavel
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.
Multimetric continuous model theory
(University of Hawaii at Manoa, 2012) Caulfield, Erin; Ross, David
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.
Riemann, Hurwitz, and branched covering spaces: an exposition in mathematics
(University of Hawaii at Manoa, 2011) Turner, Wm. Pitt V.; Little, Robert
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.
A brief survey of the discrete logarithm problem
(University of Hawaii at Manoa, 2011) Sia, Gretel S.; Manes, Michelle
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.
A-T-menability of groups
(University of Hawaii at Manoa, 2011) Lu, Ni Yen; Guentner, Erik
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.
Zeros of the modular parametrization of rational elliptic curves
(University of Hawaii at Manoa, 2011) Kodgis, Lisa; Guerzhoy, Pavel
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.
Complex reflection group coding
(University of Hawaii at Manoa, 2011) Kim, Hye Jung; Nation, J.B.
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.
A generalization of Euler's constant
(University of Hawaii at Manoa, 2011) Hedley, Mark
Core and no-treat equilibrium in tournament games with externalities
(University of Hawaii at Manoa, 2010) Mizuno, Ryo; Nation, J.B.
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.
Polynomial-clone reducibility
(University of Hawaii at Manoa, 2010) Culver, Quinn; Kjos-Hanssen, Bjørn
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.
Almost global feedack control of autonomous underwater vehicles
(University of Hawaii at Manoa, 2009) Singh, Shashi Bhushan; Chyba, Monique
An Autonomous Underwater Vehicle (AUV) is expected to operate in an aquatic environment and compensate for poorly known disturbance forces and moments. Due to uncertain environment, it is difficult to apply an open-loop control scheme for tracking the desired trajectory. The objective of this thesis is to develop a robust feedback trajectory tracking control scheme for an AUV that can track a prescribed trajectory amidst such disturbances.
Omega function: a theoretical introduction
(University of Hawaii at Manoa, 2009) Nguyen, Vu Ngoc; Ramsey, Thomas
This paper investigates the theory behind a new universal performance measure (the so called Omega function), which was frst introduced by Con Keating and William F. Shadwick in 2002 (see [1]). In the frst section, we review some rudimentary probability. We then defne the Omega function, introduce some of its properties, and prove these properties without continuity assumptions. We also defne the standard dispersion, a new statistic derived from the Omega function. We prove one new theorem about the range of the standard dispersion for a fnite sample. The structure of the second section on the Omega function follows closely that of a recent talk given by Ana Cascon and William Shadwick in [4]. In the last section, we demonstrate these properties with real-life data.

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