Department of Mathematics
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Approximation and interpolation with Bernstein polynomials.
(20211103)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 noncompact spaces.
(20210725)We discuss various notions of the shadowing property on noncompact 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.
(20210701)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.
(20210806)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 construction of families of polygons inscribed in the unit ... 
Forcing ℵ1free groups to be free.
(20210412)ℵ1free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and settheoretic interest. Illustrating this, we note that ℵ1free groups, and in particular the question of when ℵ1free ... 
Orbits, pseudo orbits, and the characteristic polynomial of qnary quantum graphs.
(20200717)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.
(20200609)By the Pieri rule, the tensor product of an exterior power and a finitedimensional irreducible representation of a general linear group has a multiplicityfree 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.
(20191217)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.
(20190724)We describe finite element methods for the linearized rotating shallow water equations which govern tides. Symplectic Euler and CrankNicolson timestepping strategies have good energy preservation properties, which is ... 
On BirmanHardyRellichtype inequalities.
(20190724)In 1961, Birman proved a sequence of inequalities on the space of mtimes 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.
(20190715)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 updownwind finite difference methods for solving Heston stochastic volatility equations.
(20190715)This dissertation explores effective and efficient computational methodologies for solving twodimensional Heston stochastic volatility option pricing models with multiple financial engineering applications. Dynamically ... 
The sixthorder Krall differential expression and selfadjoint operators.
(20190322)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 selfadjoint operator in L2 \mu(1, 1) ... 
Moment representations of exceptional orthogonal polynomials.
(20180712)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.
(20180418)The method involves application of Peano's Theorem for initial value problems. 
Boundary conditions associated with leftdefinite theory and the spectral analysis of iterated rankone perturbations.
(20180524)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.
(20180713)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.
(20180712)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.
(20170707)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 ...