Mathematics Doctoral Work

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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.
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.
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.
NAK for Ext, Ascent of Module Structures, and the Blindness of Extended Modules
(North Dakota State University, 2012) Anderson, Benjamin John
This dissertation investigates the interplay between properties of Ext modules and ascent of module structures along ring homomorphisms. First, we consider a flat local ring homomorphism ϕ: (R, [special characters omitted], k) → (S, [special characters omitted]S, k). We show that if M is a finitely generated R-module such that [special characters omitted](S, M) satisfies NAK (e.g. if [special characters omitted](S, M) is finitely generated over S) for i = 1,…, dimR( M), then [special characters omitted](S, M) = 0 for all i ≠ 0 and M has an S-module structure via ϕ. We also provide explicit computations of [special characters omitted](S, M) to indicate how large it can be when M does not have a compatible S-module structure. Next, we consider the properties of an R-module M that has a compatible S-module structure via the flat local ring homomorphism ϕ. Our results in this direction show that M cannot see the difference between the rings R and S. Specifically, many homological invariants of M are the same when computed over R and over S. Finally, we investigate these ideas in the non-local setting. We consider a faithfully flat ring homomorphism ϕ: R → S such that for all [special characters omitted] ∈ m-Spec R, the map R/[special characters omitted] → S/[special characters omitted]S is an isomorphism and the induced map ϕ*: Spec( S) → Spec(R) is such that ϕ*(m-Spec( S)) ⊆ m-Spec(R), and show that if M is a finitely generated R-module such that [special characters omitted](S, M) satisfies NAK for i = 1,…,dim R(M), then M has an S-module structure via ϕ, and obtain the same Ext vanishing as in the local case.
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.
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].
Codualizing Modules and Complexes
(North Dakota State University, 2013) Wicklein, Richard Kenneth
See dissertation for full abstract and properly formatted mathematical formulas.
Variational Methods for Polycrystal Plasticity and Related Topics in Partial Differential Equations
(North Dakota State University, 2013) Abdullayev, Farhod
In the first part of the thesis the effective yield set of ionic polycrystals is characterized by means of variational principles in L∞ that are associated to supremal functionals acting on matrix-valued divergence-free fields. The second part of the thesis is concerned with the study of the asymptotic behavior, as p → ∞, of the first and second eigenvalues and the corresponding eigenfunctions for the p(x)-Laplacian with Robin and Neumann boundary conditions, in an open, bounded domain with smooth boundary. We obtain uniform bounds for the sequences of eigenvalues (suitably rescaled), and we prove that the positive eigenfunctions converge uniformly to viscosity solutions of problems involving the ∞-Laplacian subject to appropriate boundary conditions.
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.
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.
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.
Homological Dimensions with Respect to a Semidualizing Complex
(North Dakota State University, 2015) Totushek, Jonathan
See Dissertation Document for Full Abstract (Mathematical Symbols Included)
Integral Closure and the Generalized Multiplicity Sequence
(North Dakota State University, 2015) Dunn, Thomas Boyd
See Dissertation Document for Full Abstract (Mathematical Symbols Included)
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.
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.
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.
Subfractals Induced by Subshifts
(North Dakota State University, 2016) Sattler, Elizabeth
In this thesis, a subfractal is the subset of points in the attractor of an iterated function system in which every point in the subfractal is associated with an allowable word from a subshift on the underlying symbolic space. In the case in which (1) the subshift is a subshift of nite type with an irreducible adjacency matrix, (2) the iterated function system satis es the open set condition, and (3) contractive bounds exist for each map in the iterated function system, we nd bounds for both the Hausdor and box dimensions of the subfractal, where the bounds depend both on the adjacency matrix and the contractive bounds on the maps. We extend this result to so c subshifts, a more general subshift than a subshift of nite type, and to allow the adjacency matrix to be reducible. The structure of a subfractal naturally de nes a measure on Rn. For an iterated function system which satis es the open set condition and in which the maps are similitudes, we construct an invariant measure supported on a subfractal induced by a subshift of nite type. For this speci c measure, we calculate the local dimension for almost every point, and hence calculate the Hausdor dimension for the measure.
Optimization Problems Arising in Stability Analysis of Discrete Time Recurrent Neural Networks
(North Dakota State University, 2016) Singh, Jayant
We consider the method of Reduction of Dissipativity Domain to prove global Lyapunov stability of Discrete Time Recurrent Neural Networks. The standard and advanced criteria for Absolute Stability of these essentially nonlinear systems produce rather weak results. The method mentioned above is proved to be more powerful. It involves a multi-step procedure with maximization of special nonconvex functions over polytopes on every step. We derive conditions which guarantee an existence of at most one point of local maximum for such functions over every hyperplane. This nontrivial result is valid for wide range of neuron transfer functions.

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