Applied Mathematics Commons^™

Applied Statistics In Environmental Monitoring: Case Studies And Analysis For The Michigan Bald Eagle Biosentinel Program, Katherine Leith

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The bald eagle (Haliaeetus leucocephalus) is an extensively researched tertiary predator. Its life history and the impact of various stressors on its reproductive outcomes have been documented in many studies, and over many years. Furthermore, the bald eagle population recovery in Michigan has been closely monitored since the 1960s, as it has continued to recover from a contaminant-induced bottleneck. Because of its position at the top of the aquatic food web and the large body of ethological knowledge, the bald eagle has become a sentinel species for the Michigan aquatic ecosystem. In April 1999, the Michigan Department of Environmental Qualtity, …

Grobner Bases: Degree Bounds And Generic Ideals, Juliane Golubinski Capaverde

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In this thesis, we study two problems related to Gröbner basis theory: degree bounds for general ideals and Gröbner bases structure for generic ideals. We start by giving an introduction to Gröbner bases and their basic properties and presenting a recent algorithm by Gao, Volny and Wang. Next, we survey degree bounds for the ideal membership problem, the effective Nullstellensatz, and polynomials in minimal Gröbner bases. We present general upper bounds, and bounds for several classes of special ideals. We provide classical examples showing some of these bounds cannot be improved in general. We present a comprehensive study of a …

On Numerical Algorithms For Fluid Flow Regularization Models, Abigail Bowers

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This thesis studies regularization models as a way to approximate a flow simulation at a lower computational cost. The Leray model is more easily computed than the Navier-Stokes equations (NSE), and it is more computationally attractive than the NS-α regularization because it admits a natural linearization which decouples the mass/momentum system and the filter system, allowing for efficient and stable computations. A major disadvantage of the Leray model lies in its inaccuracy. Thus, we study herein several methods to improve the accuracy of the model, while still retaining many of its attractive properties. This thesis is arranged as follows. Chapter …

Level Stripping Of Genus 2 Siegel Modular Forms, Rodney Keaton

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In this Dissertation we consider stripping primes from the level of genus 2 cuspidal Siegel eigenforms. Specifically, given an eigenform of level Nlr which satisfies certain mild conditions, where l is a prime not dividing N, we construct an eigenform of level N which is congruent to our original form. To obtain our results, we use explicit constructions of Eisenstein series and theta functions to adapt ideas from a level stripping result on elliptic modular forms. Furthermore, we give applications of this result to Galois representations and provide evidence for an analog of Serre's conjecture in the genus 2 case.

Convergence Of A Reinforcement Learning Algorithm In Continuous Domains, Stephen Carden

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In the field of Reinforcement Learning, Markov Decision Processes with a finite number of states and actions have been well studied, and there exist algorithms capable of producing a sequence of policies which converge to an optimal policy with probability one. Convergence guarantees for problems with continuous states also exist. Until recently, no online algorithm for continuous states and continuous actions has been proven to produce optimal policies. This Dissertation contains the results of research into reinforcement learning algorithms for problems in which both the state and action spaces are continuous. The problems to be solved are introduced formally as …

Approximation In Multiobjective Optimization With Applications, Lakmali Weerasena

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Over the last couple of decades, the field of multiobjective optimization has received much attention in solving real-life optimization problems in science, engineering, economics and other fields where optimal decisions need to be made in the presence of trade-offs between two or more conflicting objective functions. The conflicting nature of objective functions implies a solution set for a multiobjective optimization problem. Obtaining this set is difficult for many reasons, and a variety of approaches for approximating it either partially or entirely have been proposed.

In response to the growing interest in approximation, this research investigates developing a theory and methodology …

Mathematical Optimization For Engineering Design Problems, Brian Dandurand

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Applications in engineering design and the material sciences motivate the development of optimization theory in a manner that additionally draws from other branches of mathematics including the functional, complex, and numerical analyses.
The first contribution, motivated by an automotive design application, extends multiobjective optimization theory under the assumption that the problem information is not available in its entirety to a single decision maker as traditionally assumed in the multiobjective optimization literature. Rather, the problem information and the design control are distributed among different decision makers. This requirement appears in the design of an automotive system whose subsystem components themselves correspond …

On The Cuspidality Of Maass-Gritsenko And Mixed Level Lifts, Dania Zantout

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your words

Advancements In Finite Element Methods For Newtonian And Non-Newtonian Flows, Keith Galvin

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This dissertation studies two important problems in the mathematics of computational fluid dynamics. The first problem concerns the accurate and efficient simulation of incompressible, viscous Newtonian flows, described by the Navier-Stokes equations. A direct numerical simulation of these types of flows is, in most cases, not computationally feasible. Hence, the first half of this work studies two separate types of models designed to more accurately and efficient simulate these flows. The second half focuses on the defective boundary problem for non-Newtonian flows. Non-Newtonian flows are generally governed by more complex modeling equations, and the lack of standard Dirichlet or Neumann …

Polyhedral Approximations Of Quadratic Semi-Assignment Problems, Disjunctive Programs, And Base-2 Expansions Of Integer Variables, Frank Muldoon

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This research is concerned with developing improved representations for special families of mixed-discrete programming problems. Such problems can typically be modeled using different mathematical forms, and the representation employed can greatly influence the problem's ability to be solved. Generally speaking, it is desired to obtain mixed 0-1 linear forms whose continuous relaxations provide tight polyhedral outer-approximations to the convex hulls of feasible solutions. This dissertation makes contributions to three distinct problems, providing new forms that improve upon published works.
The first emphasis is on devising solution procedures for the classical quadratic semi-assignment problem(QSAP), which is an NP-hard 0-1 quadratic program. …

Sensitivity Analysis In Magnetic Resonance Elastography And A Local Wavelength Reconstruction Based On Wave Direction, Christopher Gillam

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or the detection of early stage cancer. MRE utilizes interior data for its inverse problems, which greatly reduces the ill-posedness from which most traditional inverse problems suffer.
In this thesis, we first establish a sensitivity analysis for viscoelastic scalar medium with complex wave number and compare it with the purely elastic case. Also we estimate the smallest detectable inclusion for breast and liver, which is about twice larger than using the purely elastic model. We also found the existence of optimal frequency (50 Hz) that maximizes the detectability when the Voigt model is used.
Second, we propose a local wavelength …

Physicic-Based Algorithms And Divergence Free Finite Elements For Coupled Flow Problems, Nicholas Wilson

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This thesis studies novel physics-based methods for
simulating incompressible fluid flow described by the Navier-Stokes equations (NSE) and
magnetohydrodynamics equations (MHD).
It is widely accepted in computational fluid dynamics (CFD) that numerical schemes which are more
physically accurate lead to more precise flow simulations especially over long time intervals.
A prevalent theme throughout will be the inclusion of as much
physical fidelity in numerical solutions as efficiently possible. In algorithm design, model
selection/development, and element choice, subtle changes can provide better physical accuracy,
which in turn provides better overall accuracy (in any measure). To this end we develop and study …

Sensitivity Anaylsis And Detectability For Magnetic Resonance Elastography, Catherine White

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This thesis is for a sensitivity analysis of magnetic resonance elastography, a hybrid imaging technique used in early-stage cancer screening. To quantitatively analyze the sensitivity, we introduce a notion of detectability, which is dened as a relative amplitude
drop in a small sti tumor region. This analysis is accomplished in both the full elastic and viscoelastic models and compared with that of the simpler scalar model which is frequently used in the actual application.
Some of the highlights are 1) a useful formula for detectability in terms of physical parameters, which will help the design of experiments; 2) the discrepancy …

Sparsity Regularization In Diffuse Optical Tomography, John Cooper

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The purpose of this dissertation is to improve image reconstruction in Diffuse Optical Tomography (DOT), a high contrast imaging modality that uses a near infrared light source. Because the scattering and absorption of a tumor varies significantly from healthy tissue, a reconstructed spatial representation of these parameters serves as tomographic image of a medium. However, the high scatter and absorption of the optical source also causes the inverse problem to be severely ill posed, and currently only low resolution reconstructions are possible, particularly when using an unmodulated direct current (DC) source.
In this work, the well posedness of the forward …

Modular Forms, Elliptic Curves And Drinfeld Modules, Catherine Trentacoste

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In this thesis we explore three different subfields in the area of number theory. The first topic we investigate involves modular forms, specifically nearly holomorphic eigenforms. In Chapter 3, we show the product of two nearly holomorphic eigenforms is an eigenform for only a finite list of examples. The second type of problem we analyze is related to the rank of elliptic curves. Specifically in Chapter 5 we give a graph theoretical approach to calculating the size of 3-Selmer groups for a given family of elliptic curves. By calculating the size of the 3-Selmer groups, we give an upper bound …

On Factoring Hecke Eigenforms, Nearly Holomorphic Modular Forms, And Applications To L-Values, Jeff Beyerl

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This thesis is a presentation of some of my research activities while at Clemson University. In particular this includes joint work on the factorization of eigenforms and their relationship to Rankin- Selberg L-values, and nearly holomorphic eigenforms. The main tools used on the factorization of eigenforms are linear algebra, the j function, and the Rankin-Selberg Method. The main tool used on nearly holomorphic modular forms is the Rankin-Cohen bracket operator.

A Collection Of Problems In Combinatorics, Janine Janoski

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We present several problems in combinatorics including the partition function, Graph Nim, and the evolution of strings.
Let p(n) be the number of partitions of n. We say a sequence a_n is log-concave if for every n, a_n2 &ge a_n+1 a_n-1. We will show that p(n) is log-concave for n &ge 26. We will also show that for n<26, p(n) alternatively satisfies and does not satisfy the log-concave property. We include results for the Sperner property of the partition function.
The second problem we present is the game of Graph Nim. We use the Sprague-Grundy theorem to analyze modified versions of Nim played on various graphs. We include progress made towards proving that all G-paths …

An Optimization Approach To A Geometric Packing Problem, Bradley Paynter

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We investigate several geometric packing problems (derived from an industrial setting) that involve fitting patterns of regularly spaced disks without overlap. We first derive conditions for achieving the feasible placement of a given set of patterns and construct a network formulation that, under certain conditions, allows the calculation of such a placement. We then discuss certain related optimization problems (e.g., fitting together the maximum number of patterns) and broaden the field of application by showing a connection to the well-known Periodic Scheduling Problem. In addition, a variety of heuristics are developed for solving large-scale instances of these provably difficult packing …

Bases And Applications Of Riemann-Roch Spaces Of Function Fields With Many Rational Places, Justin Peachey

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Algebraic geometry codes are generalizations of Reed-Solomon codes, which are implemented in nearly all digital communication devices. In ground-breaking work, Tsfasman, Vladut, and Zink showed the existence of a sequence of algebraic geometry codes that exceed the Gilbert-Varshamov bound, which was previously thought unbeatable. More recently, it has been shown that multipoint algebraic geometry codes can outperform comparable one-point algebraic geometry codes. In both cases, it is desirable that these function fields have many rational places. The prototypical example of such a function field is the Hermitian function field which is maximal. In 2003, Geil produced a new family of …

Fractal Jackson Networks, Mahmoud Rezaei

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In this dissertation, Gaussian random measures that arise as limits of Jackson networks. The support of the random measure is a fractal having Hausdorff dimension delta . The variance measure is the Hausdorff measure also of dimension delta.

Inference In Reversible Markov Chains, Tara Steuber

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This dissertation describes the research that we have done concerning
reversible Markov chains. We first present definitions for what it means
for a Markov chain to be reversible. We then give applications of where
reversible Markov chains are used and give a brief history of Markov chain
inference. Finally, two journal articles are found in the paper, one that
is already published and another which is currently being submitted.
The first article examines estimation of the one-step-ahead
transition probabilities in a reversible Markov chain on a countable state
space. A symmetrized moment estimator is proposed that exploits the
reversible structure. …

Pseudocodewords Of Parity-Check Codes, Wittawat Kositwattanarerk

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The success of modern algorithms for the decoding problem such as message-passing iterative decoding and linear programming decoding lies in their local nature. This feature allows the algorithms to be extremely fast and capable of correcting more errors than guaranteed by the classical minimum distance of the code. Nonetheless, the performance of these decoders depends crucially on the Tanner graph representation of the code. In order to understand this choice of representation, we need to analyze the pseudocodewords of the Tanner graph of a code. These pseudocodewords are outputs of local decoding algorithms which may not be legitimate codewords. In …

New Algorithms For Computing Groebner Bases, Frank Volny

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In this thesis, we present new algorithms for computing Groebner bases. The first algorithm, G2V, is incremental in the same fashion as F5 and F5C. At a typical step, one is given a Groebner basis G for an ideal I and any polynomial g, and it is desired to compute a Groebner basis for the new ideal , obtained from I by joining g. Let (I : g) denote the colon ideal of I divided by g. Our algorithm computes Groebner bases for I, g and (I : g) simultaneously. In previous algorithms, S-polynomials that reduce to zero are useless, …

Biologically Relevant Classes Of Boolean Functions, Lori Layne

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A large influx of experimental data has prompted the development of
innovative computational techniques for modeling and reverse
engineering biological networks. While finite dynamical systems,
in particular Boolean networks, have gained attention as relevant
models of network dynamics, not all Boolean functions reflect the
behaviors of real biological systems. In this work, we focus on two
classes of Boolean functions and study their applicability as
biologically relevant network models: the nested and partially nested
canalyzing functions.
We begin by analyzing the nested canalyzing functions} (NCFs),
which have been proposed as gene regulatory network models due to
their stability properties. We …

Source Optimization In Abstract Function Spaces For Maximizing Distinguishability: Applications To The Optical Tomography Inverse Problem, Bonnie Jacob

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The focus of this thesis is to formulate an optimal source problem for the medical imaging technique of optical tomography by maximizing certain distinguishability criteria. We extend the concept of distinguishability in electrical impedance tomography to the frequency-domain diffusion approximation model used in optical tomography.
We consider the dependence of the optimal source on the choice of appropriate function spaces, which can be chosen from certain Sobolev or Lp spaces. All of the spaces we consider are Hilbert spaces; we therefore exploit the inner product in several ways. First, we define and use throughout an inner product on the Sobolev …

Improved Accuracy For Fluid Flow Problems Via Enhanced Physics, Michael Case

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This thesis is an investigation of numerical methods for approximating solutions to fluid flow problems, specifically the Navier-Stokes equations (NSE) and magnetohydrodynamic equations (MHD), with an overriding theme of enforcing more physical behavior in discrete solutions. It is well documented that numerical methods with more physical accuracy exhibit better long-time behavior than comparable methods that enforce less physics in their solutions. This work develops, analyzes and tests finite element methods that better enforce mass conservation in discrete velocity solutions to the NSE and MHD, helicity conservation for NSE, cross-helicity conservation in MHD, and magnetic field incompressibility in MHD.

Some New Problems In Changepoint Analysis, Jonathan Woody

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Climatological studies have often neglected changepoint effects when modeling
various physical phenomena. Here, changepoints are plausible whenever a station location moves or its instruments are changed. There is frequently meta-data to
perform sound statistical inferences that account for changepoint
information. This dissertation focuses on two such problems in changepoint analysis.
The first problem we investigate involves assessing trends
in daily snow depth series. Here, we introduce a stochastic storage model. The model allows for seasonal features, which permits the
analysis of daily data. Changepoint times are shown to greatly influence estimated trends in one snow depth series and are accounted …

Decoding Of Multipoint Algebraic Geometry Codes Via Lists, Nathan Drake

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Algebraic geometry codes have been studied greatly since their introduction by Goppa . Early study had focused on algebraic geometry codes CL(D;G) where G was taken to be a multiple of a single point. However, it has been shown that if we allow G to be supported by more points, then the associated code may have better parameters. We call such a code a multipoint code and if G is supported by m points, then we call it an m-point code. In this dissertation, we wish to develop a decoding algorithm for multipoint codes. We show how we can embed …

Change-Point Analysis: Asymptotic Theory And Applications, Michael Robbins

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The problem of undocumented change-points in data sets appears in many areas of science. Mathematical fundamentals of asymptotic methods used in change-point analysis are discussed, and several important maximally selected change-point statistics are introduced. First, the likelihood ratio method is applied to abstract data models within the setting of precipitation series. Basic inference as to the legitimacy and effectiveness of asymptotic methods at detecting undocumented change-points is provided. Next, maximally selected chi-square statistics are discussed in detail and applied to data on tropical cyclone behavior, where a widely available and widely analyzed data set on Atlantic basin cyclones is studied. …

Multiobjective Optimization For Complex Systems, Melissa Gardenghi

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Complex systems are becoming more and more apparent in a variety of disciplines, making solution methods for these systems valuable tools. The solution of complex systems requires two significant skills. The first challenge of developing mathematical models for these systems is followed by the difficulty of solving these models to produce preferred solutions for the overall systems. Both issues are addressed by this research.
This study of complex systems focuses on two distinct aspects. First, models of complex systems with multiobjective formulations and a variety of structures are proposed. Using multiobjective optimization theory, relationships between the efficient solutions of the …