Mathematics Doctoral Work

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Almost Dedekind Domains and Atomicity
(North Dakota State University, 2012) Hasenauer, Richard Erwin
The objective of this dissertation was to determine the class of domains that are both almost Dedekind and atomic. To investigate this question we constructed a global object called the norm, and used it to determine properties that a domain must have to be both atomic and almost Dedekind. Additionally we use topological notions on the spectrum of a domain to determine atomicity. We state some theorems with regard to ACCP and class groups. The lemmas and theorems in this dissertation answer in part the objective. We conclude with a chapter of future study that aims to approach a complete answer to the objective.
Analysis of Variance Based Financial Instruments and Transition Probability Densities Swaps, Price Indices, and Asymptotic Expansions
(North Dakota State University, 2018) Issaka, Aziz
This dissertation studies a couple of variance-dependent instruments in the financial market. Firstly, a number of aspects of the variance swap in connection to the Barndorff-Nielsen and Shephard model are studied. A partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated. Under appropriate assumptions for the first four cumulants of the driving subordinator, a Ve\v{c}e\v{r}-type theorem is proved. The bounds of the arbitrage-free variance swap price are also found. Finally, a price-weighted index modulated by market variance is introduced. The large-basket limit dynamics of the price index and the ``error term" are derived. Empirical data driven numerical examples are provided in support of the proposed price index. We implemented Feynman path integral method for the analysis of option pricing for certain L\'evy process-driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain L\'evy process-driven markets where the interest rate is stochastic.
Applications of Groups of Divisibility and a Generalization of Krull Dimension
(North Dakota State University, 2011) Trentham, William Travis
Groups of divisibility have played an important role in commutative algebra for many years. In 1932 Wolfgang Krull showed in [12] that every linearly ordered Abelian group can be realized as the group of divisibility of a valuation domain. Since then it has also been proven that every lattice-ordered Abelian group can be recognized as the group of divisibility of a Bezont domain. Knowing these two facts allows us to use groups of divisibility to find examples of rings with highly exotic properties. For instance, we use them here to find examples of rings which admit elements that factor uniquely as the product of uncountably many primes. In addition to allowing us to create examples, groups of divisibility can he used to characterize some of the most important rings most commonly encountered in factorization theory, including valuation domains, UFD's, GCD domains, and antimatter domains. We present some of these characterizations here in addition to using them to create many examples of our own, including examples of rings which admit chains of prime ideals in which there are uncountably many primes in the chain. Moreover, we use groups of divisibility to prove that every fragmented domain must have infinite Krull dimension.
Atomicity in Rings with Zero Divisors
(North Dakota State University, 2011) Trentham, Stacy Michelle
In this dissertation, we examine atomicity in rings with zero divisions. We begin by examining the relationship between a ring’s level of atomicity and the highest level of irreducibility shared by the ring’s irreducible elements. Later, we chose one of the higher forms of atomicity and identify ways of building large classes of examples of rings that rise to this level of atomicity but no higher. Characteristics of the various types of irreducible elements will also be examined. Next, we extend our view to include polynomial extensions of rings with zero divisors. In particular, we focus on properties of the three forms of maximal common divisors and how a ring’s classification as an MCD, SMCD, or VSMCD ring affects its atomicity. To conclude, we identify some unsolved problems relating to the topics discussed in this dissertation.
Codualizing Modules and Complexes
(North Dakota State University, 2013) Wicklein, Richard Kenneth
See dissertation for full abstract and properly formatted mathematical formulas.
DG Homological Algebra, Properties of Ring Homomorphisms, and the Generalized Auslander-Reiten Conjecture
(North Dakota State University, 2013) Nasseh, Saeed
This dissertation contains three aspects of my research that are listed as three joint papers with my advisor and a solo paper in the bibliography [56]--[59]. Each paper will be discussed in a different chapter. Chapter 1 contains the introduction to this dissertation. In this chapter we give the statements of the most important results discussed in Chapters 3--6. Chapter 2 contains notation and background material for use in the subsequent chapters. In Chapter 3, we prove lifting results for DG modules that are akin to Auslander, Ding, and Solberg's famous lifting results for modules. Chapter 4 contains the complete answer to a question of Vasconcelos from 1974. We show that a local ring has only finitely many shift-isomorphism classes of semidualizing complexes. Our proof relies on certain aspects of deformation theory for DG modules over a finite dimensional DG algebra, which we develop. In Chapter 5, we investigate Cohen factorizations of local ring homomorphisms from three perspectives. First, we prove a “weak functoriality” result for Cohen factorizations: certain morphisms of local ring homomorphisms induce morphisms of Cohen factorizations. Second, we use Cohen factorizations to study the properties of local ring homomorphisms in certain commutative diagrams. Third, we use Cohen factorizations to investigate the structure of quasi-deformations of local rings, with an eye on the question of the behavior of CI-dimension in short exact sequences. In Chapter 6 we show under some conditions that a Gorenstein ring R satisfies the Generalized Auslander-Reiten Conjecture if and only if so does R[x]. When R is a local ring we prove the same result for some localizations of R[x].
The First Exit-Time Analysis of an Approximate Barndorff-Nielsen and Shephard Model, with Data Science-Based Applications in the Commodity Market
(North Dakota State University, 2021) Awasthi, Shantanu
In this dissertation, an approximate version of the Barndorff-Nielsen and Shephard model, driven by a Brownian motion and a Lévy subordinator, is formulated. The first-exit time of the log-return process for this model is analyzed. It is shown that with a certain probability, the first-exit time process of the log-return is decomposable into the sum of the first exit time of the Brownian motion with drift, and the first exit time of a Lévy subordinator with drift. Subsequently, the probability density functions of the first exit time of some specific Lévy subordinators, connected to stationary, self-decomposable variance processes, are studied. Analytical expressions of the probability density function of the first-exit time of three such Lévy subordinators are obtained in terms of various special functions. The results are implemented to empirical S&P 500 dataset. After this exit time analysis, in this dissertation, we propose a model for the soybean export market share dynamics and analyze the empirical data using machine and deep learning algorithms. We justify the proposed general model and provide several theoretical analyses related to a special case of the general model. The empirical data set is a time series with weekly observations over the period January 6, 2012, through January 3, 2020. This is a period of growing intense competition, and during which a trade war had influenced the results. The target variable is the share of soybean exports made from the US Gulf to China. We implement machine and deep learning-based techniques to analyze the empirical data. Various numerical results are obtained. The results indicate that export market shares, which are otherwise highly volatile, can be effectively explained (predicted) using machine/deep learning methodologies and a set of logical feature variables. We conclude this dissertation with an analysis of option pricing and implied volatility in the case when the market is driven by a jump-stochastic volatility model. We find the price of the European call option in this case. In addition, we implement Malliavin calculus to analyze the implied volatility.
The Game of Nim on Graphs
(North Dakota State University, 2011) Erickson, Lindsay Anne
The ordinary game of Nim has a long history and is well-known in the area of combinatorial game theory. The solution to the ordinary game of Nim has been known for many years and lends itself to numerous other solutions to combinatorial games. Nim was extended to graphs by taking a fixed graph with a playing piece on a given vertex and assigning positive integer weight to the edges that correspond to a pile of stones in the ordinary game of Nim. Players move alternately from the playing piece across incident edges, removing weight from edges as they move. Few results in this area have been found, leading to its appeal. This dissertation examines broad classes of graphs in relation to the game of Nim to find winning strategies and to solve the problem of finding the winner of a game with both unit weighting assignments and with arbitrary weighting assignments. Such classes of graphs include the complete graph, the Petersen graph, hypercubes, and bipartite graphs. We also include the winning strategy for even cycles.
Gorenstein Dimension over Some Rings of the Form R [0 plus]C
(North Dakota State University, 2015) Aung, Pye Phyo
Commutative algebra is the study of commutative rings and other abstract structures based on commutative rings. Modules can be viewed as a common generalization of several of those structures, and some invariants, e.g. homological dimensions, of modules are used to characterize certain properties of the base ring. Some generalizations of such invariants include C-Gorenstein dimensions, where C is a semidualizing module over a commutative noetherian ring. Holm and Jørgensen [16] investigate some connections between C-Gorenstein dimensions of an R-complex and Gorenstein dimensions of the same complex viewed as a complex over the "trivial extension" R × C. I generalize some of their results to a certain type of retract diagram. I also investigate some examples of such retract diagrams, namely D'Anna and Fontana's amalgamated duplication [8] and Enescu's pseudocanonical cover [9].
The Half-Factorial Property in Polynomial Rings
(North Dakota State University, 2014) Batell, Mark Thomas
This dissertation investigates the following question: If R is a half-factorial domain (HFD) and x is an indeterminate, under what conditions is the polynomial ring R[x] an HFD? The question has been answered in a few special cases. A classical result of Gauss states that if R is a UFD, then R[x] is a UFD. Also, Zaks showed that if R is a Krull domain with class group Cl(R), then R[x] is an HFD if and only if jCl(R)j 6 2. In the proof of his result, Zaks did not use Gauss's methods. We give a new proof that does. We also study the question in domains other than Krull domains.
Homological Dimensions with Respect to a Semidualizing Complex
(North Dakota State University, 2015) Totushek, Jonathan
See Dissertation Document for Full Abstract (Mathematical Symbols Included)
Hypothesis Testing on Time Series Driven by Underlying Lévy Processes, with Machine Learning Applications
(North Dakota State University, 2021) Roberts, Michael
In this dissertation, we study the testing of hypotheses on streams of observations that are driven by Lévy processes. This is applicable for sequential decision making on the state of two-sensor systems. In one case, each sensor receives or does not receive a signal obstructed by noise. In another, each sensor receives data driven by Lévy processes with large or small jumps. In either case, these give rise to four possible outcomes for the hypotheses. Infinitesimal generators are presented and analyzed. Bounds for likelihood functions in terms of super-solutions} and sub-solutions are computed. As an application, we study a change point detection hypothesis test for the detection of the distribution of jump size in one-dimensional Lévy processes. This is shown to be implementable in relation to various classification problems for a crude oil price data set. Machine and deep learning algorithms are implemented to extract a specific deterministic component from the data set, and the deterministic component is implemented to improve the Barndorff-Nielsen & Shephard (BN-S) model, a commonly used stochastic model for derivative and commodity market analysis.
Ideal Graphs
(North Dakota State University, 2014) Al-Kaseasbeh, Saba Zakariya
In this dissertation, we explore various types of graphs that can be associated to a commutative ring with identity. In particular, if R is a commutative ring with identity, we consider a number of graphs with the vertex set being the set of proper ideals; various edge sets defined via different ideal theoretic conditions give visual insights and structure theorems pertaining to the multiplicative ideal theory of R. We characterize the interplay between the ideal theory and various properties of these graphs including diameter and connectivity.
Integral Closure and the Generalized Multiplicity Sequence
(North Dakota State University, 2015) Dunn, Thomas Boyd
See Dissertation Document for Full Abstract (Mathematical Symbols Included)
Knot Groups and Bi-Orderable HNN Extensions of Free Groups
(North Dakota State University, 2020) Martin, Cody Michael
Suppose K is a fibered knot with bi-orderable knot group. We perform a topological winding operation to half-twist bands in a free incompressible Seifert surface Σ of K. This results in a Seifert surface Σ' with boundary that is a non-fibered knot K'. We call K a fibered base of K'. A fibered base exists for a large class of non-fibered knots. We prove K' has a bi-orderable knot group if Σ' is obtained from applying the winding operation to only one half-twist band of Σ. Utilizing a Seifert surface gluing technique, we obtain HNN extension group presentations for both knot groups that differ by only one relation. To show the knot group of K' is bi-orderable, we apply the following: Let G be a bi-ordered free group with order preserving automorphism ɑ. It is well known that the semidirect product ℤ ×ɑG is a bi-orderable group. If X is a basis of G, a presentation of ℤ ×ɑG is ⟨ t,X | R ⟩, where the relations are R = {txt-1}ɑ(x)-1 : x ∈ X}. If R' is any subset of R, we prove that the group H =⟨ t,X | R' ⟩ is bi-orderable. H is a special case of an HNN extension of G. Finally, we add new relations to the group presentation of H such that bi-orderability is preserved.
L1 Approximation in De Branges Spaces
(North Dakota State University, 2015) Spanier, Mark Andrew
In this thesis we study bandlimited approximations to various functions. Bandlimited functions have compactly supported Fourier transforms, which is a desirable feature in many applications. In particular, we address the problem of determining best approximations that minimize a weighted integral error. By utilizing the theory of Hilbert spaces of entire functions developed by L. de Branges, we are able to obtain optimal solutions for several weighted approximation problems. As an application, we determine extremal majorants and minorants that vanish at a prescribed point for a class of functions, which may be used to remove contributions from undesirable points.
Mathematical Modeling of Epidemics: Parametric Heterogeneity and Pathogen Coexistence
(North Dakota State University, 2020) Sarfo Amponsah, Eric
No two species can indefinitely occupy the same ecological niche according to the competitive exclusion principle. When competing strains of the same pathogen invade a homogeneous population, the strain with the largest basic reproductive ratio R0 will force the other strains to extinction. However, over 51 pathogens are documented to have multiple strains [3] coexisting, contrary to the results from homogeneous models. In reality, the world is heterogeneous with the population varying in susceptibility. As such, the study of epidemiology, and hence the problem of pathogen coexistence should entail heterogeneity. Heterogeneous models tend to capture dynamics such as resistance to infection, giving more accurate results of the epidemics. This study will focus on the behavior of multi-pathogen heterogeneous models and will try to answer the question: what are the conditions on the model parameters that lead to pathogen coexistence? The goal is to understand the mechanisms in heterogeneous populations that mediate pathogen coexistence. Using the moment closure method, Fleming et. al. [22] used a two pathogen heterogeneous model (1.9) to show that pathogen coexistence was possible between strains of the baculovirus under certain conditions. In the first part of our study, we consider the same model using the hidden keystone variable (HKV) method. We show that under some conditions, the moment closure method and the HKV method give the same results. We also show that pathogen coexistence is possible for a much wider range of parameters, and give a complete analysis of the model (1.9), and give an explanation for the observed coexistence. The host population (gypsy moth) considered in the model (1.9) has a year life span, and hence, demography was introduced to the model using a discrete time model (1.12). In the second part of our study, we will consider a multi-pathogen compartmental heterogeneous model (3.1) with continuous time demography. We show using a Lyapunov function that pathogen coexistence is possible between multiple strains of the same pathogen. We provide analytical and numerical evidence that multiple strains of the same pathogen can coexist in a heterogeneous population.
Maximally Edge-Colored Directed Graph Algebras
(North Dakota State University, 2017) Brownlee, Erin Ann
Graph C*-algebras are constructed using projections corresponding to the vertices of the graph, and partial isometries corresponding to the edges of the graph. Here, we use the gauge-invariant uniqueness theorem to first establish that the C*-algebra of a graph composed of a directed cycle with finitely many edges emitting away from that cycle is Mn+k(C(T)), where n is the length of the cycle and k is the number of edges emitting away. We use this result to establish the main results of the thesis, which pertain to maximally edge-colored directed graphs. We show that the C*-algebra of any finite maximally edge-colored directed graph is *Mn(C){ Mn(C(T))}k, where n is the number of vertices of the graph and k depends on the structure of the graph. Finally, we show that this algebra is in fact isomorphic to Mn(*C{ C(T)}k).
Modeling Financial Swaps and Geophysical data Using the Barndorff-Nielsen and Shephard Model
(North Dakota State University, 2015) Habtemicael, Semere Kidane
This dissertation uses Barndoff-Nielsen and Shephard (BN-S) models to model swap, a type of financial derivative, and analyze geophysical data for estimation of major earthquakes. From empirical observation of the stock market activity and earthquake occurrence, we observe that the distributions have high kurtosis and right skewness. Consequently, such data cannot be captured by stochastic models driven by a Wiener process. Non-Gaussian processes of Ornstein-Uhlenbeck type are one of the most significant candidates for being the building blocks of models of financial economics. These models offer the possibility of capturing important distributional deviations from Gaussianity and thus these are more practical models of dependence structures. Introduced by Barndorff-Nielsen and Shephard these processes are not only convenient to model volatility in financial market, but have an independent interest for modeling stationary time series of various kinds. For the financial data we use BN-S models to price the arbitrage-free value of volatility, variance, covariance, and correlation swap. Such swaps are used in financial markets for volatility hedging and speculation. We use the S&P500 and NASDAQ index for parameter estimation and numerical analysis. We show that with this model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. For any given time interval, the earthquake magnitude data have three main properties: (1) magnitude is a non-negative stationary stochastic process; (2) for any finite interval of time there are only finite number of jumps; (3) the sample path of the magnitude of an earthquake consists of upward jumps (significant earthquake) and a gradual decrease (aftershocks). For such geophysical data we specifically use Gamma Ornstein Uhlenbeck processes driven by a Levy process to estimate a major earthquake in a certain region in California. Rigorous regression analysis is provided, and based on that, first-passage times are computed for different sets of data. Both applications demonstrate the significance of BN-S models to phenomena that follow non-Gaussian distributions.
Modeling Swap and Geophysical Analysis using Barndorff-Nielson and Shephard Model
(North Dakota State University, 2015) Habtemicael, Semere Kidane
Video summarizing Ph.D. dissertation for a non-specialist audience.

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