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Articles 31 - 60 of 66
Full-Text Articles in Physical Sciences and Mathematics
On Factoring Hecke Eigenforms, Nearly Holomorphic Modular Forms, And Applications To L-Values, Jeff Beyerl
On Factoring Hecke Eigenforms, Nearly Holomorphic Modular Forms, And Applications To L-Values, Jeff Beyerl
All Dissertations
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
A Collection Of Problems In Combinatorics, Janine Janoski
All Dissertations
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 an is log-concave if for every n, an2 &ge an+1 an-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
An Optimization Approach To A Geometric Packing Problem, Bradley Paynter
All Dissertations
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
Bases And Applications Of Riemann-Roch Spaces Of Function Fields With Many Rational Places, Justin Peachey
All Dissertations
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
Fractal Jackson Networks, Mahmoud Rezaei
All Dissertations
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
Inference In Reversible Markov Chains, Tara Steuber
All Dissertations
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
Pseudocodewords Of Parity-Check Codes, Wittawat Kositwattanarerk
All Dissertations
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
New Algorithms For Computing Groebner Bases, Frank Volny
All Dissertations
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
Biologically Relevant Classes Of Boolean Functions, Lori Layne
All Dissertations
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
All Dissertations
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
Improved Accuracy For Fluid Flow Problems Via Enhanced Physics, Michael Case
All Dissertations
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
Some New Problems In Changepoint Analysis, Jonathan Woody
All Dissertations
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
Decoding Of Multipoint Algebraic Geometry Codes Via Lists, Nathan Drake
All Dissertations
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 …
Variations On Graph Products And Vertex Partitions, Jobby Jacob
Variations On Graph Products And Vertex Partitions, Jobby Jacob
All Dissertations
In this thesis we investigate two graph products called double vertex graphs and complete double vertex graphs, and two vertex partitions called dominator partitions and rankings.
We introduce a new graph product called the complete double vertex graph and study its properties. The complete double vertex graph is a natural extension of the Cartesian product and a generalization of the double vertex graph.
We establish many properties of complete double vertex graphs, including results involving the chromatic number of a complete double vertex graph and the characterization of planar complete double vertex graphs. We also investigate the important problem of …
Multiobjective Optimization For Complex Systems, Melissa Gardenghi
Multiobjective Optimization For Complex Systems, Melissa Gardenghi
All Dissertations
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 …
Factoring Polynomials And Groebner Bases, Genhua (Yinhua) Guan
Factoring Polynomials And Groebner Bases, Genhua (Yinhua) Guan
All Dissertations
Factoring polynomials is a central problem in computational algebra and number theory and is a basic routine in most
computer algebra systems (e.g. Maple, Mathematica, Magma, etc). It has been extensively studied
in the last few decades by many mathematicians and computer scientists. The main approaches include Berlekamp's method
(1967) based on the kernel of Frobenius map, Niederreiter's method (1993) via an ordinary differential equation,
Zassenhaus's modular approach (1969), Lenstra, Lenstra and Lovasz's lattice reduction (1982), and Gao's method via a partial differential equation (2003). These methods and their recent improvements due to van Hoeij (2002) and
Lecerf et al …
Asymptotics Of Families Of Polynomials And Sums Of Hurwitz Class Numbers, Timothy Flowers
Asymptotics Of Families Of Polynomials And Sums Of Hurwitz Class Numbers, Timothy Flowers
All Dissertations
In a note in the American Mathematical Monthly in 1960, Strodt mentions a way to prove both the Euler-Maclaurin summation formula and the Boole summation formula using operators. In a 2009 article in the Monthly, Borwein, Calkin, and Manna expand on this idea. Therein, they define Strodt operators and Strodt polynomials and show that the classical Bernoulli polynomials and Euler polynomials are examples of Strodt polynomials.
It is well known that both Bernoulli polynomials and Euler polynomials on a fixed interval are asymptotically sinusoidal. Borwein, Calkin, and Manna show that a similar result holds for the uniform Strodt polynomials. We …
Discrete Dynamics Over Finite Fields, Jang-Woo Park
Discrete Dynamics Over Finite Fields, Jang-Woo Park
All Dissertations
A dynamical system consists of a set V and a map f : V → V . The primary goal is to characterize points in V according to their limiting behaviors under iteration of the map f . Especially understanding dynamics of nonlinear maps is an important but difficult problem, and there are not many methods available. This work concentrates on dynamics of certain nonlinear maps over finite fields. First we study monomial dynamics over finite fields. We show that determining the number of fixed points of a boolean monomial dynamics is #P–complete problem and consider various cases in which …
Quality Representation In Multiobjective Programming, Stacey Faulkenberg
Quality Representation In Multiobjective Programming, Stacey Faulkenberg
All Dissertations
In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective programming problems. This manuscript presents two methods for generating discrete representations with equidistant points for multiobjective programs with solution sets determined by convex cones. The Bilevel Controlled Spacing (BCS) method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing. The Constraint Controlled Spacing (CCS) method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. Both methods (under certain assumptions) are proven to produce (weakly) nondominated points. Along …
Change-Point Analysis: Asymptotic Theory And Applications, Michael Robbins
Change-Point Analysis: Asymptotic Theory And Applications, Michael Robbins
All Dissertations
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. …
On Elliptic Curves, Modular Forms, And The Distribution Of Primes, Ethan Smith
On Elliptic Curves, Modular Forms, And The Distribution Of Primes, Ethan Smith
All Dissertations
In this thesis, we present four problems related to elliptic curves, modular forms, the distribution of primes, or some combination of the three. The first chapter surveys the relevant background material necessary for understanding the remainder of the thesis. The four following chapters present our problems of interest and their solutions. In the final chapter, we present our conclusions as well as a few possible directions for future research.
Hurwitz class numbers are known to have connections to many areas of number theory. In particular, they are intimately connected to the theory of binary quadratic forms, the structure of imaginary …
New Directions In Multivariate Public Key Cryptography, Raymond Heindl
New Directions In Multivariate Public Key Cryptography, Raymond Heindl
All Dissertations
Most public key cryptosystems used in practice are based on integer factorization or discrete logarithms (in finite fields or elliptic curves). However, these systems suffer from two potential drawbacks. First, they must use large keys to maintain security, resulting in decreased efficiency. Second, if large enough quantum computers can be built, Shor's algorithm will render them completely insecure.
Multivariate public key cryptosystems (MPKC) are one possible alternative. MPKC makes use of the fact that solving multivariate polynomial systems over a finite field is an NP-complete problem, for which it is not known whether there is a polynomial algorithm on quantum …
Intersections And Representations Of Graphs, John Light
Intersections And Representations Of Graphs, John Light
All Dissertations
Given two graphs G and H sharing the same vertex set, the edge-intersection spectrum of G and H is the set of possible
sizes of the intersection of the edge sets of both graphs. For example,
the spectrum of two copies of the cycle C5 is {0, 2, 3, 5}, and the spectrum of two copies of the star K1,r is {1, r}. The intersection spectrum was initially studied for designs by Lindner and Fu and others and was originally extended to graphs by Eric Mendelsohn. Several examples are studied, both when G and H are isomorphic and …
Modeling Hiv Drug Resistance, Mingfu Zhu
Modeling Hiv Drug Resistance, Mingfu Zhu
All Dissertations
Despite the development of antiviral drugs and the optimization of therapies, the emergence of drug resistance remains one of the most challenging issues for successful treatments of HIV-infected patients. The availability of massive HIV drug resistance data provides us not only exciting opportunities for HIV research, but also the curse of high dimensionality.
We provide several statistical learning methods in this thesis to analyze sequence data from different perspectives. We propose a hierarchical random graph approach to identify possible covariation among residue-specific mutations. Viral progression pathways were inferred using an EM-like algorithm in literature, and we present a normalization method …
Fast Fourier Transform Algorithms With Applications, Todd Mateer
Fast Fourier Transform Algorithms With Applications, Todd Mateer
All Dissertations
This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the 'Fast Fourier Transform' as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the 'Fast Fourier Transform' as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the 'Fast Fourier Transform' that can also be understood by someone who has an …
Portfolio Selection Under Various Risk Measures, Hariharan Kandasamy
Portfolio Selection Under Various Risk Measures, Hariharan Kandasamy
All Dissertations
Portfolio selection has been a major area of study after Markowitz's ground-breaking paper. Risk quantification for portfolio selection is studied in the literature extensively and many risk measures have been proposed.
In this dissertation we study portfolio selection under various risk measures. After exploring important risk measures currently available we propose a new risk measure, Unequal Prioritized Downside Risk (UPDR). We illustrate the formulation of UPDR for portfolio selection as a mixed-integer program. We establish conditions under which UPDR can be formulated as a linear program.
We study single-period portfolio selection using two risk measures simultaneously. We propose four alternate …
Numerical Analysis Of A Fractional Step Theta-Method For Fluid Flow Problems, John Chrispell
Numerical Analysis Of A Fractional Step Theta-Method For Fluid Flow Problems, John Chrispell
All Dissertations
The accurate numerical approximation of viscoelastic fluid flow poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. Specifically, the viscoelastic modeling equations have a hyperbolic constitutive equation coupled to a parabolic conservation of momentum equation. An appealing approximation approach is to use a fractional step $\theta$-method. The $\theta$-method is an operator splitting technique that may be used to decouple mathematical equations of different types as well as separate the updates of distinct modeling equation variables when modeling mixed systems of partial …
Homomorphisms Of Graphs, Samuel Lyle
Homomorphisms Of Graphs, Samuel Lyle
All Dissertations
Understanding the structure of graphs is fundamental to advances in many areas of graph theory, as well as in many applications. In many cases, an analysis of the structure of graphs follows one of two approaches; either many structural properties are considered over a restricted class of graphs, or a particular structural property is considered over many classes of graphs. Both approaches will be considered in this dissertation.
Graphs which do not contain a clique of size r, i.e., Kr-free graphs, are of fundamental importance in the area of extremal graph theory. Many results have been obtained …
Issues In Model Selection, Minimax Estimation, And Censored Data Analysis, Meng Zhao
Issues In Model Selection, Minimax Estimation, And Censored Data Analysis, Meng Zhao
All Dissertations
In this dissertation, we address several research problems in statistical inference. We obtain results in the following four directions: linear model selection, minimax estimation of linear functionals, Bayes type estimators for the survival functions based on right censored data, and estimation of survival functions based on doubly censored data.
Automorphic Decompositions Of Graphs, Robert Beeler
Automorphic Decompositions Of Graphs, Robert Beeler
All Dissertations
Let G and H be graphs. A G-decomposition D of a graph H is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. It is well known that a graceful labelling (or more generally a rho-valuation) of a graph G induces a cyclic G-decomposition of a complete graph. We will extend these notions to that of a general valuation in a cyclic group. Such valuations yield decompositions of circulant graphs. We will show that every graph has a valuation and hence is a …