M.A. Plan B Theses- Mathematics Department
http://hdl.handle.net/10125/23256
Sat, 05 Sep 2015 05:48:46 GMT2015-09-05T05:48:46ZGeneralized 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, MichaelThe 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, ErinRiemann, Hurwitz, and branched covering spaces: an exposition in mathematics
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 GMThttp://hdl.handle.net/10125/259272011-01-01T00:00:00ZTurner, Wm. Pitt V.A brief survey of the discrete logarithm problem
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 GMThttp://hdl.handle.net/10125/259262011-01-01T00:00:00ZSia, Gretel S.A-T-menability of groups
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 GMThttp://hdl.handle.net/10125/259252011-01-01T00:00:00ZLu, Ni YenZeros of the modular parametrization of rational elliptic curves
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 GMThttp://hdl.handle.net/10125/259242011-01-01T00:00:00ZKodgis, LisaComplex reflection group coding
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 GMThttp://hdl.handle.net/10125/259232011-01-01T00:00:00ZKim, Hye JungA generalization of Euler's constant
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 GMThttp://hdl.handle.net/10125/259222011-01-01T00:00:00ZHedley, MarkCore and no-treat equilibrium in tournament games with externalities
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 GMThttp://hdl.handle.net/10125/259212010-01-01T00:00:00ZMizuno, RyoPolynomial-clone reducibility
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 GMThttp://hdl.handle.net/10125/259202010-01-01T00:00:00ZCulver, QuinnAlmost global feedack control of autonomous underwater vehicles
http://hdl.handle.net/10125/25919
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.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2009
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10125/259192009-01-01T00:00:00ZSingh, Shashi BhushanOmega function: a theoretical introduction
http://hdl.handle.net/10125/25918
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.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2009
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10125/259182009-01-01T00:00:00ZNguyen, Vu NgocSatellite orbital control
http://hdl.handle.net/10125/25917
In this project we will be looking at the change in control required to transfer a satellite between two elliptic Keplerian orbits. We will first derive the equations of motion for our satellite and then study the controllability properties of our system. We will introduce a simple feedback controller and prove local asymptotic stability of the target orbit. The goal of this paper is to prove stability using both geometric control theory as well as stabilization methods and thus link prior work done on orbital control in both fields.Our primary tool to accomplish this will be LaSalle’s invariance principle.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2009
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10125/259172009-01-01T00:00:00ZEvans, Regina M.Time optimal control of a right invariant system on a compact Lie group
http://hdl.handle.net/10125/25916
In this paper we will study the pulse sequences in NMR spectroscopy and quantum computing as a time control problem. Radio frequency pulses are used to execute a unitary transfer of state. Sequences of pulses should be as short as possible to minimize decoherence. We model the problem as a controllable right invariant system on a compact Lie group. We investigate the minimum time required to steer the system from an initial point to a specified final point.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2008
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10125/259162008-01-01T00:00:00ZStorm, Jody LynnLinear differential operators and the distribution of zeros of polynomials
http://hdl.handle.net/10125/25915
The purpose of this paper is fourfold: (1) to survey some classical and recent results in the theory of distribution of zeros of entire functions, (2) to demonstrate a novel proof answering a question of Raitchinov, (3) to present some new results in the theory of complex zero decreasing operators, and (4) to initiate the study of the location of zeros of complex polynomials under the action of certain linear operators. In addition, several open problems are given.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2008
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10125/259152008-01-01T00:00:00ZSo, EugeneStabilization of a cart and double pendulum system
http://hdl.handle.net/10125/25914
This Master’s Plan B report for the University of Hawai‘i at M ̄noa is the result of Daniel Langdon’s examination of methods of modeling and stabilizing a cart and double-pendulum system against small disturbances. The Euler-Lagrange Equations, a fundamental result of the calculus of variations, combined with the Principle of Least Action and the Lagrange D’Alembert Principle are used to describe the equations of motion for a cart and double-pendulum system in terms of the kinetic and potential energy of the system, which is in turn described in terms of the positions and velocities of the cart and two pendulum bobs. Theorems from differential equations are combined with a linear approximation of the equations of motion and the notion of feedback to compute an algorithm for stabilization, whose action of stabilization against a sample small disturbance is demonstrated.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2007
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10125/259142007-01-01T00:00:00ZLangdon, Daniel AllenSolving the Dirichlet problem via Brownian motion
http://hdl.handle.net/10125/25913
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2007
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10125/259132007-01-01T00:00:00ZKrot, TatianaSymmetry group solutions to differential equations
http://hdl.handle.net/10125/25912
In this project we will be looking at Sophus Lie’s desire (his so called idee fixe) to apply Contact Transformations (what would eventually develop into the modern idea of a Lie Algebra) in order to arrive at symmetries of differential equations, and thus certain solutions. Our goal—as well as Lie’s—is to develop a more universal method for solving differential equations than the familiar cook-book methods we learn in an introductory ordinary or partial differential equations class. We answer three questions. What was the historical underpinning of Sophus Lie’s theory? How do we find the symmetry Lie algebras? How do we use the symmetry Lie algebras to find solutions to the differential equation? (In order to answer these questions we will need to fill in some background material and our answers will also result in a novel derivation of the “Fundamental Source Solution.”) Our second objective will be to establish a connection between solvability in Galois Theory and in Differential Equations. We will assume a familiarity with certain concepts from Abstract Algebra.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2007
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10125/259122007-01-01T00:00:00ZBurkman, Jacob HarryNormality of stock prices
http://hdl.handle.net/10125/25911
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2006
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10125/259112006-01-01T00:00:00ZShi, BoThe Laguerre inequality and the distribution of zeros of entire functions
http://hdl.handle.net/10125/25910
The purpose of this paper is fourfold: (1) to survey some classical and recent results in the theory of distribution of zeros of entire functions, (2) to generalize the Laguerre inequality, (3) to establish several special cases of the Hawai‘i Conjecture, and (4) to present some new results dealing with the polar derivatives of polynomials. In addition, applications of the extended Laguerre inequalities are given. The paper concludes with several open problems.
Plan B paper, M.A., Mathematics, University of Hawaii at Manoa, 2003
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10125/259102003-01-01T00:00:00ZMuranaka, BrandonRiemann, Hurwitz, and Branched Covering Spaces: An Exposition in Mathematics
http://hdl.handle.net/10125/23260
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 de ciencies 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.
Mon, 23 Jul 2012 00:00:00 GMThttp://hdl.handle.net/10125/232602012-07-23T00:00:00ZTurner, Wm. Pitt V.