Theses/Dissertations - Mathematics
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Generalized inverse limits and the intermediate value property.
(2021-07-26)We introduce and discuss various notions of the intermediate value property applicable to upper-semicontinuous set-valued functions f : [0, 1] → 2^[0,1]. In the first part, we present sufficient conditions such that an ... -
On functions related to the spectral theory of Sturm--Liouville operators.
(2022-04-01)Functions related to the spectral theory of differential operators have been extensively studied due to their many applications in mathematics and physics. In this dissertation, we will consider spectral ζ-functions, ... -
Approximation and interpolation with Bernstein polynomials.
(2021-11-03)Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. In this dissertation, we investigate fundamental problems in ... -
On various notions of the shadowing property in non-compact spaces.
(2021-07-25)We discuss various notions of the shadowing property on non-compact spaces. In the first part, we discuss the shadowing property acting on general sequence spaces. We develop criteria upon the weights of the space in which ... -
Group automorphisms of incidence algebras.
(2021-07-01)Let (P,≤) be an arbitrary partially ordered set and I(P) its incidence space. Then FI(P) denotes the associated finitary incidence algebra, where FI(P) = I(P) for locally finite posets (P,≤). We investigate the group of ... -
Poncelet polygons through the lenses of orthogonal polynomials on the unit circle, finite Blaschke products, and numerical ranges.
(2021-08-06)The topics in this thesis fall in the intersection of projective geometry, complex analysis, and linear algebra. Each of these three fields gives a canonical construc-tion of families of polygons inscribed in the unit ... -
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 ...
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