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University of Nebraska - Lincoln
Department of Mathematics: Dissertations, Theses, and Student Research
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- Operator Algebras (2)
- Operator Theory (2)
- Time scales (2)
- Adjoint functor (1)
- Almost Minimizer (1)
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- Analysis (1)
- Asymptotically Convex (1)
- Blowup (1)
- C*-algebra (1)
- Calculus of Variations (1)
- Cauchy function (1)
- Compact group (1)
- Cone (1)
- Continuous fractional calculus (1)
- Convergence (1)
- Convolution (1)
- Cowen-Douglas Operator (1)
- Crossed product (1)
- Derivations (1)
- Difference equations (1)
- Discrete Fractional Calculus (1)
- Discrete fractional calculus (1)
- Disentangling (1)
- Disturbance Specialists (1)
- Dynamical system (1)
- Feynman's operational calculi (1)
- Fixed Point Theory (1)
- Fourth order (1)
- Fractional Boundary Value Problem (1)
- Fractional Boundary Value Problems (1)
Articles 1 - 18 of 18
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Admissibility Of C*-Covers And Crossed Products Of Operator Algebras, Mitchell A. Hamidi
Admissibility Of C*-Covers And Crossed Products Of Operator Algebras, Mitchell A. Hamidi
Department of Mathematics: Dissertations, Theses, and Student Research
In 2015, E. Katsoulis and C. Ramsey introduced the construction of a non-self-adjoint crossed product that encodes the action of a group of automorphisms on an operator algebra. They did so by realizing a non-self-adjoint crossed product as the subalgebra of a C*-crossed product when dynamics of a group acting on an operator algebra by completely isometric automorphisms can be extended to self-adjoint dynamics of the group acting on a C*-algebra by ∗-automorphisms. We show that this extension of dynamics is highly dependent on the representation of the given algebra and we define a lattice structure for an operator algebra's …
Unbounded Derivations Of C*-Algebras And The Heisenberg Commutation Relation, Lara M. Ismert
Unbounded Derivations Of C*-Algebras And The Heisenberg Commutation Relation, Lara M. Ismert
Department of Mathematics: Dissertations, Theses, and Student Research
This dissertation investigates the properties of unbounded derivations on C*-algebras, namely the density of their analytic vectors and a property we refer to as "kernel stabilization." We focus on a weakly-defined derivation δD which formalizes commutators involving unbounded self-adjoint operators on a Hilbert space. These commutators naturally arise in quantum mechanics, as we briefly describe in the introduction.
A first application of kernel stabilization for δD shows that a large class of abstract derivations on unbounded C*-algebras, defined by O. Bratteli and D. Robinson, also have kernel stabilization. A second application of kernel stabilization provides a sufficient condition …
Operator Algebras Generated By Left Invertibles, Derek Desantis
Operator Algebras Generated By Left Invertibles, Derek Desantis
Department of Mathematics: Dissertations, Theses, and Student Research
Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space.We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames.
The primary object of this thesis is the norm-closed operator algebra generated by a left invertible $T$ together with its Moore-Penrose inverse $T^\dagger$. We denote this algebra by $\mathfrac{A}_T$. In the isometric case, $T^\dagger = T^*$ and $\mathfrac{A}_T$ is a representation of the Toeplitz algebra. Of particular interest …
Fractional Difference Operators And Related Boundary Value Problems, Scott C. Gensler
Fractional Difference Operators And Related Boundary Value Problems, Scott C. Gensler
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation we develop a fractional difference calculus for functions on a discrete domain. We start by showing that the Taylor monomials, which play a role analagous to that of the power functions in ordinary differential calculus, can be expressed in terms of a family of polynomials which I will refer to as the Pochhammer polynomials. These important functions, the Taylor monomials, were previously described by other scholars primarily in terms of the gamma function. With only this description it is challenging to understand their properties. Describing the Taylor monomials in terms of the Pochhammer polynomials has made it …
Properties And Convergence Of State-Based Laplacians, Kelsey Wells
Properties And Convergence Of State-Based Laplacians, Kelsey Wells
Department of Mathematics: Dissertations, Theses, and Student Research
The classical Laplace operator is a vital tool in modeling many physical behaviors, such as elasticity, diffusion and fluid flow. Incorporated in the Laplace operator is the requirement of twice differentiability, which implies continuity that many physical processes lack. In this thesis we introduce a new nonlocal Laplace-type operator, that is capable of dealing with strong discontinuities. Motivated by the state-based peridynamic framework, this new nonlocal Laplacian exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow better representation of physical phenomena at different scales and in materials with different …
Local And Nonlocal Models In Thin-Plate And Bridge Dynamics, Jeremy Trageser
Local And Nonlocal Models In Thin-Plate And Bridge Dynamics, Jeremy Trageser
Department of Mathematics: Dissertations, Theses, and Student Research
This thesis explores several models in continuum mechanics from both local and nonlocal perspectives. The first portion settles a conjecture proposed by Filippo Gazzola and his collaborators on the finite-time blow-up for a class of fourth-order differential equations modeling suspension bridges. Under suitable assumptions on the nonlinearity and the initial data, a finite-time blowup is demonstrated as a result of rapid oscillations with geometrically growing amplitudes. The second section introduces a nonlocal peridynamic (integral) generalization of the biharmonic operator. Its action converges to that of the classical biharmonic as the radius of nonlocal interactions---the ``horizon"---tends to zero. For the corresponding …
Invariant Basis Number And Basis Types For C*-Algebras, Philip M. Gipson
Invariant Basis Number And Basis Types For C*-Algebras, Philip M. Gipson
Department of Mathematics: Dissertations, Theses, and Student Research
We develop the property of Invariant Basis Number (IBN) in the context of C*-algebras and their Hilbert modules. A complete K-theoretic characterization of C*- algebras with IBN is given. A scheme for classifying C*-algebras which do not have IBN is given and we prove that all such classes are realized. We investigate the invariance of IBN, or lack thereof, under common C*-algebraic construction and perturbation techniques. Finally, applications of Invariant Basis Number to the study of C*-dynamical systems and the classification program are investigated.
Adviser: David Pitts
Boundary Value Problems For Discrete Fractional Equations, Pushp R. Awasthi
Boundary Value Problems For Discrete Fractional Equations, Pushp R. Awasthi
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation we develop certain aspects of the theory of discrete fractional calculus. The author begins with an introduction to the discrete delta calculus together with the fractional delta calculus which is used throughout this dissertation. The Cauchy function, the Green's function and some of their important properties for a fractional boundary value problem for are developed. This dissertation is comprised of four chapters. In the first chapter we introduce the delta fractional calculus. In the second chapter we give some preliminary definitions, properties and theorems for the fractional delta calculus and derive the appropriate Green's function and give …
Regularity For Solutions To Parabolic Systems And Nonlocal Minimization Problems, Joe Geisbauer
Regularity For Solutions To Parabolic Systems And Nonlocal Minimization Problems, Joe Geisbauer
Department of Mathematics: Dissertations, Theses, and Student Research
The goal of this dissertation is to contribute to both the nonlocal and local settings of regularity within the calculus of variations. We provide analogues of higher differentiability results in the context of Besov spaces for minimizers of nonlocal functionals. We also establish the Holder continuity of solutions to a system of parabolic partial differential equations.
Advisor: Mikil Foss
An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich
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
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
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
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
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
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
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
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
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 …