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Full-Text Articles in Education

Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov Jul 2011

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

Dissertations, Theses, and Student Research Papers in Mathematics

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 ...


On Morrey Spaces In The Calculus Of Variations, Kyle Fey May 2011

On Morrey Spaces In The Calculus Of Variations, Kyle Fey

Dissertations, Theses, and Student Research Papers in Mathematics

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 of functions {fj} uniformly bounded in the Morrey space Lp, λ ...


Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell May 2011

Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell

Dissertations, Theses, and Student Research Papers in Mathematics

The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and ...


Global Well-Posedness For A Nonlinear Wave Equation With P-Laplacian Damping, Zahava Wilstein May 2011

Global Well-Posedness For A Nonlinear Wave Equation With P-Laplacian Damping, Zahava Wilstein

Dissertations, Theses, and Student Research Papers in Mathematics

This dissertation deals with the global well-posedness of the nonlinear wave equation
utt − Δu − Δput = f (u) in Ω × (0,T),
{u(0), ut(0)} = {u0,u1} ∈ H10 (Ω) × L 2 (Ω),
u = 0 on Γ × (0, T ),
in a bounded domain Ω ⊂ ℜ n with Dirichlét boundary conditions. The nonlinearities f (u) acts as a strong source, which is allowed to have, in some cases, a super-supercritical exponent. Under suitable restrictions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and ...


Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer May 2011

Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

Dissertations, Theses, and Student Research Papers in Mathematics

This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti ...


Groups And Semigroups Generated By Automata, David Mccune May 2011

Groups And Semigroups Generated By Automata, David Mccune

Dissertations, Theses, and Student Research Papers in Mathematics

In this dissertation we classify the metabelian groups arising from a restricted class of invertible synchronous automata over a binary alphabet. We give faithful, self-similar actions of Heisenberg groups and upper triangular matrix groups. We introduce a new class of semigroups given by a restricted class of asynchronous automata. We call these semigroups ``expanding automaton semigroups''. We show that this class strictly contains the class of automaton semigroups, and we show that the class of asynchronous automaton semigroups strictly contains the class of expanding automaton semigroups. We demonstrate that undecidability arises in the actions of expanding automaton semigroups and semigroups ...


On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce May 2011

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

Dissertations, Theses, and Student Research Papers in Mathematics

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 ...


Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest Apr 2011

Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest

Dissertations, Theses, and Student Research Papers in Mathematics

This dissertation focuses on the intersection of two classical and fundamental areas in graph theory: graph packing and degree sequences. The question of packing degree sequences lies naturally in this intersection, asking when degree sequences have edge-disjoint realizations on the same vertex set. The most significant result in this area is Kundu's k-Factor Theorem, which characterizes when a degree sequence packs with a constant sequence. We prove a series of results in this spirit, and we particularly search for realizations of degree sequences with edge-disjoint 1-factors.

Perhaps the most fundamental result in degree sequence theory is the Erdos-Gallai Theorem ...


Extremal Trees And Reconstruction, Andrew Ray Apr 2011

Extremal Trees And Reconstruction, Andrew Ray

Dissertations, Theses, and Student Research Papers in Mathematics

Problems in two areas of graph theory will be considered.

First, I will consider extremal problems for trees. In these questions we examine the trees that maximize or minimize various invariants. For instance the number of independent sets, the number of matchings, the number of subtrees, the sum of pairwise distances, the spectral radius, and the number of homomorphisms to a fixed graph. I have two general approaches to these problems. To find the extremal trees in the collection of trees on n vertices with a fixed degree bound I use the certificate method. The certificate is a branch invariant ...


Annihilators Of Local Cohomology Modules, Laura Lynch Apr 2011

Annihilators Of Local Cohomology Modules, Laura Lynch

Dissertations, Theses, and Student Research Papers in Mathematics

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.

Adviser: Thomas Marley


The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm Apr 2011

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

Dissertations, Theses, and Student Research Papers in Mathematics

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 Mar 2011

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

Dissertations, Theses, and Student Research Papers in Mathematics

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.


On The Betti Number Of Differential Modules, Justin Devries Jan 2011

On The Betti Number Of Differential Modules, Justin Devries

Dissertations, Theses, and Student Research Papers in Mathematics

Let R = k[x1, ..., xn] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d2 = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rankR D ≥ 2n. This relates to a problem of Buchsbaum ...