Mathematics Doctoral Work

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Experimental investigation of novel designs for aerodynamic flow control over airfoils
(North Dakota State University, 2024) Refling, William
In this study, an experimental investigation was performed to characterize and validate three novel flow control strategies for unsteady aerodynamics. These three strategies are based on active and passive flow control designs. The two active control strategies make use of smart material alloy’s (SMA) located on the leading and trailing edge of a Boeing-Vertol VR-12 airfoil used in rotorcraft wings. The SMA used were macro fiber composites (MFC) a piezo-electric actuator. These actuators were located at the 25% and 85% of the chord length. Two different implementation strategies are used: one as an active morphing of the leading and trailing edge, second as acoustic resonators on the leading edge. The third strategy was of a passive flow control structure located on the pressure side of the leading edge of a NACA 0012 airfoil. This strategy makes use a microcavity to mitigate the transient separation and dynamic stall. In addition to the three novel strategies, traditional approaches such as drooping of the leading and trailing edge were studies. All systems were tested in both a static condition where the airfoil is held stationary as a freestream velocity is applied to the airfoil. As well as testing with a dynamic motion of the airfoil simulating a sinusoidal pitching motion. All of these tests were performed in the open loop wind tunnel located in the advanced flow diagnostics lab. The details of the wings design, manufacturing, actuation, programing, control, and test implementation is reported herein. The flow fields were measured through use of 2-D particle image velocimetry (PIV) an optical diagnostic flow methodology used for characterization and validation of the designs. To capture more detail of the unsteady and unpredictable nature of the flow, time resolved 2-D PIV, is implemented to provide full details of the flow while each strategy undergoes multiple pitching cycles. All three flow control strategies showed positive improvements to the airfoil performance. The active morphing provided the largest performance boost as the flow remained attached throughout the pitching experimentation. Showcasing the improvement that active morphing has over traditional methods of droop. The acoustic resonance was a close follow-up showing improvement in both pitching and static conditions, however for the case of pitching the system was inconsistent. This has been attributed to the need to adjust the frequency generated while the angle of attack changes. Lastly the passive cavity structure showed limited improvement during light dynamic stall, improving the flow when compared to the baseline. However, the flow conditions needed to concisely prove the control strategy were not possible with the current equipment.
Ramification and Infinite Extensions of Dedekind Domains
(North Dakota State University, 2010) Hashbarger, Carl Stanley
This dissertation presents methods for determining the behavior of prime ideals m an integral extension of a Dedekind domain. One tool used to determine this behavior is an algorithm that computes which prime ideals ramify in a finite separable extension. Other results about factorization of prime ideals are improved and applied to finite extensions. By considering a set of finite extensions whose union is an infinite extension, it is possible to predict ideal factorization in the infinite extension as well. Among other things, this ideal factorization determines whether a given infinite extension is almost Dedekind. These methods and results yield some interesting facts when they are demonstrated on a pair of classical rings of algebraic number theory.
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.
On Partial Permutations and Alternating Sign Matrices: Bijections and Polytopes
(North Dakota State University, 2021) Heuer, Dylan
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 bijections relating certain subsets to decorated permutations and set partitions. We prove bijections among partial alternating sign matrices and several other combinatorial objects as well as results related to their dynamics, analogous to those in the usual alternating sign matrix setting. We also study families of polytopes which are the convex hulls of these matrices. We determine inequality descriptions, facet enumerations, and face lattice descriptions. Finally, we study partial permutohedra which arise naturally as projections of these polytopes, revealing connections to graph associahedra.
Some Results on Semicrossed Products and Related Operator Algebras
(North Dakota State University, 2021) Duchsherer, Melissa
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 these algebras. Our two main results describe the structure of derivations on graph operator algebras and point derivations on semicrossed product operator algebras. We then investigate multivariate semicrossed products and the maps on the associated, underlying compact Hausdorff space. We consider potential generalizations of classical 1-dimensional variants and look for which of our multivariate analogs have nice structure with a proposed invariant for multivariate dynamical systems. We close by developing a component-wise look at the maximal C∗-algebra of the n×n matrices, the simplest of the direct graph operator algebras. This is the first concrete example of a maximal C∗-algebra since the one example that accompanied the definition in the original paper about maximal C∗-envelopes.
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.
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.
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.
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.
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.
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.
Sign Matrix Polytopes
(North Dakota State University, 2018) Solhjem, Sara Louise
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 {0,1,-1}--matrices in bijection with semistandard Young tableaux. This bijection is refined to include standard Young tableau of certain shapes. One such shape is counted by the Catalan numbers, and the convex hull of these standard Young tableaux form a Catalan polytope. This Catalan polytope is shown to be integrally equivalent to the order polytope of the triangular poset: therefore the Ehrhart polynomial and volume can be combinatorial interpreted. Various properties of all of these polytope families are investigated, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes, and transportation polytopes.
Multidimensional Toggle Dynamics
(North Dakota State University, 2018) Vorland, Corey
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 sets (posets). They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion or promotion exhibits homomesy. We prove an analogous result in the case of the product of three chains where one chain has two elements. In order to prove this result, we generalize from two to n dimensions the recombination technique that D. Einstein and Propp developed to study homomesy. We see that our main homomesy result does not fully generalize to an arbitrary product of three chains, nor to larger products of chains; however, we have a partial generalization to an arbitrary product of three chains. Additional corollaries include refined homomesy results in the product of three chains and a new result on increasing tableaux. We also generalize recombination to any ranked poset and from this, obtain a homomesy result for a type B minuscule poset cross a two-element chain. We conclude by extending the definition of promotion to infinite posets, exploring homomesy, recombination, and a connection to monomial ideals.
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).
Integral Closure and the Generalized Multiplicity Sequence
(North Dakota State University, 2015) Dunn, Thomas Boyd
See Dissertation Document for Full Abstract (Mathematical Symbols Included)
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.
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.
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].
Semidualizing DG Modules over Tensor Products
(North Dakota State University, 2015) Altmann, Hannah Lee
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 of its modules, I study its semidualizing modules. These modules have proven useful in the study of the composition of local ring homomorphisms of finite G-dimension and Bass numbers of local rings. Let R be a commutative, noetherian ring with identity. A finitely generated R-module C is semidualizing if the homothety map χ(R/C) : R → HomR(C,C) is an isomorphism and Ext(i/R)(C,C) = 0 for all i > 0. For example, the ring R is semidualizing over itself, as is a dualizing module, if R has one. In some sense the number of semidualizing modules a ring has gives a measure of the "complexity" of the ring. I am interested in that number. More generally in this dissertation we use the definition of semidualizing differential graded (DG) module, pioneered by Christensen and Sather-Wagstaff. In particular, I construct semidualizing DG modules over the tensor product of two DG k-algebras, say A' and A''. This gives us a lower bound on the number of semidualizing DG modules over the tensor product A' ⊗ k A''. Therefore, as far as semidualizing DG modules can detect, the singularity of A' ⊗ k A'' is at least as bad as the singularities of both A' and A'' combined.

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