M.A. Plan B Theses Mathematics Department
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ItemGeneralized Analytic Continuation(University of Hawaii at Manoa, 2012)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.

ItemInformation Processing and Energy Dissipation in Neurons(University of Hawaii at Manoa, 2012)We investigate the relationship between thermodynamic and information theoretic inefficiencies in an individual neuron model, the adaptive exponential integrateandfire 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.

ItemAn Iterated Version of the Generalized Singular Value Decomposition for the Joint Analysis of Two HighDimensional Data Sets(University of Hawaii at Manoa, 2013)In this work, we developed a new computational algorithm for the integrated analysis of highdimensional 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.

ItemNondeterministic Finite State Complexity(University of Hawaii at Manoa, 2013)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.

ItemValidating a Food Frequency Questionnaire for Guam(University of Hawaii at Manoa, 2012)

ItemGeometric Path Planning for a Lego AUV(University of Hawaii at Manoa, 2012)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 costeffective Lego vehicle through simulations and actual implementation as providing a proof of concept.

ItemThe snowflake decoding algorithm(University of Hawaii at Manoa, 2012)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.

ItemExtending lpdecoding for permutation codes from euclidean to Kendall tau metric(University of Hawaii at Manoa, 2012)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 LPdecoding 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.

ItemOn Kaneko congruences(University of Hawaii at Manoa, 2012)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.

ItemMultimetric continuous model theory(University of Hawaii at Manoa, 2012)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 modeltheoretic 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.