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Item DG Homological Algebra, Properties of Ring Homomorphisms, and the Generalized Auslander-Reiten Conjecture(North Dakota State University, 2013) Nasseh, SaeedThis 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].Item Modeling Financial Swaps and Geophysical data Using the Barndorff-Nielsen and Shephard Model(North Dakota State University, 2015) Habtemicael, Semere KidaneThis 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.Item Ideal Graphs(North Dakota State University, 2014) Al-Kaseasbeh, Saba ZakariyaIn 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.Item NAK for Ext, Ascent of Module Structures, and the Blindness of Extended Modules(North Dakota State University, 2012) Anderson, Benjamin JohnThis 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.Item Homological Dimensions with Respect to a Semidualizing Complex(North Dakota State University, 2015) Totushek, JonathanSee Dissertation Document for Full Abstract (Mathematical Symbols Included)Item Almost Dedekind Domains and Atomicity(North Dakota State University, 2012) Hasenauer, Richard ErwinThe 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.Item Applications of Groups of Divisibility and a Generalization of Krull Dimension(North Dakota State University, 2011) Trentham, William TravisGroups 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.Item The Half-Factorial Property in Polynomial Rings(North Dakota State University, 2014) Batell, Mark ThomasThis 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.Item Integral Closure and the Generalized Multiplicity Sequence(North Dakota State University, 2015) Dunn, Thomas BoydSee Dissertation Document for Full Abstract (Mathematical Symbols Included)Item Subfractals Induced by Subshifts(North Dakota State University, 2016) Sattler, ElizabethIn 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.