### Recent Submissions

• #### Forcing ℵ1-free groups to be free. ﻿

(2021-04-12)
ℵ1-free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and set-theoretic interest. Illustrating this, we note that ℵ1-free groups, and in particular the question of when ℵ1-free ...
• #### Orbits, pseudo orbits, and the characteristic polynomial of q-nary quantum graphs. ﻿

(2020-07-17)
Quantum graphs provide a simple model of quantum mechanics in systems with complex geometry and can be used to study quantum chaos. We evaluate the variance of the coefficients of a quantum binary graph’s associated ...
• #### An explicit description of Pieri inclusions. ﻿

(2020-06-09)
By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called ...
• #### Oscillating satellites about the straight line equilibrium points. ﻿

(1952)
The problem as studied in this thesis is the behavior of a satellite at the three equilibrium points. When a satellite is displaced from rest at one of these points it will either oscillate or rapidly leave the system. The ...
• #### Polynomial preconditioning with the minimum residual polynomial. ﻿

(2019-12-17)
Krylov subspace methods are often used to solve large, sparse systems of linear equations Ax=b. Preconditioning can help accelerate the Krylov iteration and reduce costs for solving the problem. We study a polynomial ...
• #### Preconditioning mixed finite elements for tide models. ﻿

(2019-07-24)
We describe finite element methods for the linearized rotating shallow water equations which govern tides. Symplectic Euler and Crank-Nicolson time-stepping strategies have good energy preservation properties, which is ...
• #### On Birman--Hardy--Rellich-type inequalities. ﻿

(2019-07-24)
In 1961, Birman proved a sequence of inequalities on the space of m-times continuously di↵erentiable functions of compact support Cm 0 ((0, 1)) ⇢ L2((0, 1)), containing the classical (integral) Hardy inequality and the ...
• #### Spectral properties of quantum graphs with symmetry. ﻿

(2019-07-15)
Quantum graphs were first introduced as a simple model for studying quantum mechanics in geometrically complex systems. For example, Pauling used quantum graphs to study organic molecules by viewing the atomic nuclei as ...
• #### Stable up-downwind finite difference methods for solving Heston stochastic volatility equations. ﻿

(2019-07-15)
This dissertation explores effective and efficient computational methodologies for solving two-dimensional Heston stochastic volatility option pricing models with multiple financial engineering applications. Dynamically ...
• #### The sixth-order Krall differential expression and self-adjoint operators. ﻿

(2019-03-22)
We first provide an overview of classical GNK Theory for symmetric, or symmatrizable, differential expressions in L2((a, b);w). Then we review how this theory was applied to find a self-adjoint operator in L2 \mu(-1, 1) ...
• #### Moment representations of exceptional orthogonal polynomials. ﻿

(2018-07-12)
Exceptional orthogonal polynomials (XOPs) can be viewed as an extension of their classical orthogonal polynomial counterparts. They exclude polynomials of a certain order(s) from being eigenfunctions for their corresponding ...
• #### Boundary data smoothness for solutions of nonlocal boundary value problems for nth order differential equations. ﻿

(2018-04-18)
The method involves application of Peano's Theorem for initial value problems.
• #### Boundary conditions associated with left-definite theory and the spectral analysis of iterated rank-one perturbations. ﻿

(2018-05-24)
This dissertation details the development of several analytic tools that are used to apply the techniques and concepts of perturbation theory to other areas of analysis. The main application is an efficient characterization ...
• #### Similarity of blocks in parabolic category O and a wonderful correspondence for modules of covariants. ﻿

(2018-07-13)
Let (g,k) be the pair of complexified Lie algebras corresponding to an irreducible Hermitian symmetric space of noncompact type. Then we have an associated triangular decomposition g = p− ⊕ k ⊕ p+, and q = k ⊕ p+ is a ...
• #### Conventional and asymptotic stabilities of decomposed compact methods for solving highly oscillatory wave problems. ﻿

(2018-07-12)
This dissertation explores the numerical stabilities of decomposed compact finite difference methods for solving Helmholtz partial differential equation problems with large wave numbers. It is known that many practically ...
• #### Local automorphisms of finitary incidence algebras. ﻿

(2017-07-07)
Let $R$ be a commutative, indecomposable ring with identity and let $(P,\le)$ be a locally finite partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le)$ over $R$. In this case, the finitary ...
• #### Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods. ﻿

(2017-07-08)
In this dissertation, we explore and analyze highly effective and efficient computational procedures for solving a class of nonlinear and stochastic partial differential equations. We are particularly interested in ...
• #### Glazman-Krein-Naimark theory, left-definite theory, and the square of the Legendre polynomials differential operator. ﻿

(2016-02-27)
As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential ...
• #### Eigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem. ﻿

(2016-03-23)
Comparison of smallest eigenvalues for certain two point boundary value problems for a fifth order linear differential equation are first obtained. The results are extended to (2n+1)-order and (3n+2)-order boundary value ...
• #### Chaotic properties of set-valued dynamical systems. ﻿

(2016-04-04)
In this thesis, many classical results of topological dynamics are adapted to the set-valued case. In particular, focus is given to the notions of topological entropy and the specification property. These properties are ...