Mathematics

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Research from the Department of Mathematics. The department website may be found at https://www.ndsu.edu/math/

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Absolute Stability of a Class of Second Order Feedback Non-Linear, Time-Varying Systems
(North Dakota State University, 2010) Omotoyinbo, Tayo
In this thesis, we consider the problem of absolute stability of continuous time feedback systems with a single, time-varying nonlinearity. Necessary and sufficient conditions for absolute stability of second-order systems in terms of system parameters are developed, which are characterized by eigenvalue locations on the complex plane. More specifically, our results are presented in terms of the associated matrix-pencil {A+ bvc*, v E [11,1, /L2]}, where /Li, /J,2 E ffi., A is n x n-matrix, b and c are n-vectors. The stability conditions require that the eigenvalues of all matrices A +bvc*, p1 ~ v ~ μ2, lie in the interior of a specific region of the complex plane ( a cone to be specific). Thus, we have the following reformulation of the problem. Find the maximal cone satisfying the following condition: If all eigenvalues of corresponding linear systems belong to this cone, then system is absolutely stable. Known results show that this cone is not smaller than { z E
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.
Bifurcation Portrait of a Hybrid Spiking Neuron Model
(North Dakota State University, 2012) Goodell, Brandon Grae
A bifurcation portrait classifies the behavior of a dynamical system and how it transitions between different behaviors. A hybrid dynamical system displays both continuous and discrete dynamics and may display nonsmooth bifurcations. Herein, we analyze a novel hybrid model of a spiking neuron proposed by E.M. Izhikevich [9] that is based on a previous hybrid model with a convex spike-activation function f(x), but modified with a conductance reversal potential term. We analyze the model proposed by Izhikevich and obtain a bifurcation portrait for the continuous dynamics for an arbitrary convex spike activation function f(x). Both subcritical and supercritical Andronov-Hopf bifurcations are possible, and we numerically confirm the presence of a Bautin bifurcation for a particular choice of spike activation function. The model is capable of simulating common cortical neuron types and presents several possibilities for generalizations that may be capable of more complicated behavior.
Codualizing Modules and Complexes
(North Dakota State University, 2013) Wicklein, Richard Kenneth
See dissertation for full abstract and properly formatted mathematical formulas.
Colorings of Zero-Divisor Graphs of Commutative Rings
(North Dakota State University, 2015) Ramos, Rebecca Elizabeth
We will focus on Beck’s conjecture that the chromatic number of a zero-divisor graph of a ring R is equal to the clique number of the ring R. We begin by calculating the chromatic number of the zero-divisor graphs for some finite rings and characterizing rings whose zero-divisor graphs have finite chromatic number, known as colorings. We will discuss some properties of colorings and elements called separating elements, which will allow us to determine that Beck’s conjecture holds for rings that are principal ideal rings and rings that are reduced. Then we will characterize the finite rings whose zero-divisor graphs have chromatic number less than or equal to four. In the general case, we will discuss a local ring that serves as a counterexample to Beck’s conjecture.
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)
Hook Formula For Skew Shapes
(North Dakota State University, 2019) Jensen, Megan Lisa
The number of standard Young tableaux is given by the hook-length formula of Frame, Robinson, and Thrall. Recently, Naruse found a hook-length formula for the number of skew shaped standard Young tableaux. In a series of papers, Morales, Pak, and Panova prove the Naruse hook-length formula as well as q-analogues of Naruse's formula. In this paper, we will discuss their work, including connections between excited diagrams and Dyck paths.
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)
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.
Local Risk Minimization Under Time-Varying Transaction Costs
(North Dakota State University, 2010) Nitschke, Matthew Cody
Closely following the results of Lamberton, Pham, and Schweizer [5] we construct a locally risk-minimizing strategy in a general incomplete market including transactiou costs. This is done in dbcrete time under the assurnptious of a bounded meanvariance tracleoff and substantial risk. Once we establbh all the required integrability conditions, a backward induction argument is implemented to obtain the desired strategy for every square-integrable contingent claim. \Ve model the trnusactiou costs as an adapted stochastic process aud provide all necessary proofs in detail.

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