Science and Mathematics Education Commons^™

Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis concerns three topics in commutative algebra:

1) The projective line over the integers (Chapter 2),

2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),

3) Fiber products and connected sums of local rings (Chapter 4),

In the first chapter we introduce basic terminology used in this thesis for all three topics.

In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate identification. …

An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich

Department of Mathematics: Dissertations, Theses, and Student Research

In this work we provide an analysis of both fractional- and integer-order boundary value problems, certain of which contain explicit nonlocal terms. In the discrete fractional case we consider several different types of boundary value problems including the well-known right-focal problem. Attendant to our analysis of discrete fractional boundary value problems, we also provide an analysis of the continuity properties of solutions to discrete fractional initial value problems. Finally, we conclude by providing new techniques for analyzing integer-order nonlocal boundary value problems.

Adviser: Lynn Erbe and Allan Peterson

Commutative Rings Graded By Abelian Groups, Brian P. Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might …

Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

Department of Mathematics: Dissertations, Theses, and Student Research

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …

Combinatorics Using Computational Methods, Derrick Stolee

Department of Mathematics: Dissertations, Theses, and Student Research

Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. Some results in space-bounded computational complexity are also presented.