Mathematics Doctoral Work
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The First ExitTime Analysis of an Approximate BarndorffNielsen and Shephard Model, with Data ScienceBased Applications in the Commodity Market
(North Dakota State University, 2021)In this dissertation, an approximate version of the BarndorffNielsen and Shephard model, driven by a Brownian motion and a Lévy subordinator, is formulated. The firstexit time of the logreturn process for this model is ... 
The Game of Nim on Graphs
(North Dakota State University, 2011)The ordinary game of Nim has a long history and is wellknown 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 ... 
On Partial Permutations and Alternating Sign Matrices: Bijections and Polytopes
(North Dakota State University, 2021)Motivated by the study of chained permutations and alternating sign matrices, we investigate partial permutations and alternating sign matrices. We give a length generating function for partial permutations and show ... 
Some Results on Semicrossed Products and Related Operator Algebras
(North Dakota State University, 2021)We investigate various properties of two classes of operator algebras: directed graph operator algebras and semicrossed products. First we consider analytic structure in the form of derivations and point derivations on ... 
Hypothesis Testing on Time Series Driven by Underlying Lévy Processes, with Machine Learning Applications
(North Dakota State University, 2021)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 twosensor systems. In one case, each ... 
Mathematical Modeling of Epidemics: Parametric Heterogeneity and Pathogen Coexistence
(North Dakota State University, 2020)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 ... 
Knot Groups and BiOrderable HNN Extensions of Free Groups
(North Dakota State University, 2020)Suppose K is a fibered knot with biorderable knot group. We perform a topological winding operation to halftwist bands in a free incompressible Seifert surface Σ of K. This results in a Seifert surface Σ' with boundary ... 
Analysis of Variance Based Financial Instruments and Transition Probability Densities Swaps, Price Indices, and Asymptotic Expansions
(North Dakota State University, 2018)This dissertation studies a couple of variancedependent instruments in the financial market. Firstly, a number of aspects of the variance swap in connection to the BarndorffNielsen and Shephard model are studied. A partial ... 
Atomicity in Rings with Zero Divisors
(North Dakota State University, 2011)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 ... 
Applications of Groups of Divisibility and a Generalization of Krull Dimension
(North Dakota State University, 2011)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 ... 
Sign Matrix Polytopes
(North Dakota State University, 2018)Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, several new families of polytopes are defined as convex hulls of sign matrices, which are certain ... 
Multidimensional Toggle Dynamics
(North Dakota State University, 2018)J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. One set they investigated was order ideals of partially ordered ... 
Maximally EdgeColored Directed Graph Algebras
(North Dakota State University, 2017)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 gaugeinvariant uniqueness theorem to first ... 
Integral Closure and the Generalized Multiplicity Sequence
(North Dakota State University, 2015)See Dissertation Document for Full Abstract (Mathematical Symbols Included) 
Modeling Financial Swaps and Geophysical data Using the BarndorffNielsen and Shephard Model
(North Dakota State University, 2015)This dissertation uses BarndoffNielsen and Shephard (BNS) models to model swap, a type of financial derivative, and analyze geophysical data for estimation of major earthquakes. From empirical observation of the stock ... 
L1 Approximation in De Branges Spaces
(North Dakota State University, 2015)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 ... 
Gorenstein Dimension over Some Rings of the Form R [0 plus]C
(North Dakota State University, 2015)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. ... 
Semidualizing DG Modules over Tensor Products
(North Dakota State University, 2015)In this dissertation, we study rings: sets with addition, subtraction, and multiplication. One way to study a ring is by studying its modules: the algebraic objects the ring acts on. Since it is impractical to study all ... 
The HalfFactorial Property in Polynomial Rings
(North Dakota State University, 2014)This dissertation investigates the following question: If R is a halffactorial 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 ... 
Ideal Graphs
(North Dakota State University, 2014)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 ...