http://hdl.handle.net/10125/23256 Sat, 25 May 2013 07:37:37 GMT 2013-05-25T07:37:37Z 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 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 GMT http://hdl.handle.net/10125/25919 2009-01-01T00:00:00Z Singh, Shashi Bhushan 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 GMT http://hdl.handle.net/10125/25918 2009-01-01T00:00:00Z Nguyen, Vu Ngoc 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 GMT http://hdl.handle.net/10125/25917 2009-01-01T00:00:00Z Evans, Regina M. 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 GMT http://hdl.handle.net/10125/25916 2008-01-01T00:00:00Z Storm, Jody Lynn 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 GMT http://hdl.handle.net/10125/25915 2008-01-01T00:00:00Z So, Eugene 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 GMT http://hdl.handle.net/10125/25914 2007-01-01T00:00:00Z Langdon, Daniel Allen 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 GMT http://hdl.handle.net/10125/25913 2007-01-01T00:00:00Z Krot, Tatiana 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 GMT http://hdl.handle.net/10125/25912 2007-01-01T00:00:00Z Burkman, Jacob Harry