Science and Mathematics Education Commons^™

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

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 …

Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov

Department of Mathematics: Dissertations, Theses, and Student Research

Given a C*-dynamical system (A, G, σ) the crossed product C*-algebra A x _σG encodes the action of G on A. By the universal property of A x _σG there exists a one to one correspondence between the set all covariant representations of the system (A, G, σ) and the set of all *-representations of A x _σG. Therefore, the study of representations of A x _σG is equivalent to that of covariant representations of (A, G, σ).

We study induced covariant representations of systems involving compact groups. We prove that every irreducible (resp. factor) covariant …

On Morrey Spaces In The Calculus Of Variations, Kyle Fey

Department of Mathematics: Dissertations, Theses, and Student Research

We prove some global Morrey regularity results for almost minimizers of functionals of the form u → ∫_Ω f(x, u, ∇u)dx. This regularity is valid up to the boundary, provided the boundary data are sufficiently regular. The main assumption on f is that for each x and u, the function f(x, u, ·) behaves asymptotically like the function h(|·|)^α(x), where h is an N-function.

Following this, we provide a characterization of the class of Young measures that can be generated by a sequence …

On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate the structure and properties of a variety of generalized Wiener spaces. Our main focus is on Wiener-type measures on spaces of continuous functions; our generalizations include an extension to multiple parameters, and a method of adjusting the distribution and covariance structure of the measure on the underlying function space.

In the second chapter, we consider single-parameter function spaces and extend a fundamental integration formula of Paley, Wiener, and Zygmund for an important class of functionals on this space. In the third chapter, we discuss measures on very general function spaces and introduce the specific example of a generalized …

The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm

Department of Mathematics: Dissertations, Theses, and Student Research

The author's purpose in this dissertation is to introduce, develop and apply the tools of discrete fractional calculus to the arena of fractional difference equations. To this end, we develop the Fractional Composition Rules and the Fractional Laplace Transform Method to solve a linear, fractional initial value problem in Chapters 2 and 3. We then apply fixed point strategies of Krasnosel'skii and Banach to study a nonlinear, fractional boundary value problem in Chapter 4.

Adviser: Lynn Erbe and Allan Peterson

Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette

Department of Mathematics: Dissertations, Theses, and Student Research

In this work, I offer an alternative presentation theory for C*-algebras with applicability to various other normed structures. Specifically, the set of generators is equipped with a nonnegative-valued function which ensures existence of a C*-algebra for the presentation. This modification allows clear definitions of a "relation" for generators of a C*-algebra and utilization of classical algebraic tools, such as Tietze transformations.

Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for …

Combinatorial And Commutative Manipulations In Feynman's Operational Calculi For Noncommuting Operators, Duane Einfeld

Department of Mathematics: Dissertations, Theses, and Student Research

In Feynman's Operational Calculi, a function of indeterminates in a commutative space is mapped to an operator expression in a space of (generally) noncommuting operators; the image of the map is determined by a choice of measures associated with the operators, by which the operators are 'disentangled.' Results in this area of research include formulas for disentangling in particular cases of operators and measures. We consider two ways in which this process might be facilitated. First, we develop a set of notations and operations for handling the combinatorial arguments that tend to arise. Second, we develop an intermediate space for …