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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 ...
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 ...
Optimization Problems Arising in Stability Analysis of Discrete Time Recurrent Neural Networks
(North Dakota State University, 2016)
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 ...
NAK for Ext, Ascent of Module Structures, and the Blindness of Extended Modules
(North Dakota State University, 2012)
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], ...
Almost Dedekind Domains and Atomicity
(North Dakota State University, 2012)
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 ...
Integral Closure and the Generalized Multiplicity Sequence
(North Dakota State University, 2015)
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)
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. ...
The Half-Factorial Property in Polynomial Rings
(North Dakota State University, 2014)
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 ...
DG Homological Algebra, Properties of Ring Homomorphisms, and the Generalized Auslander-Reiten Conjecture
(North Dakota State University, 2013)
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 ...