Department of Mathematics
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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) ... 
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 ... 
Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.
(20170708)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 ... 
GlazmanKreinNaimark theory, leftdefinite theory, and the square of the Legendre polynomials differential operator.
(20160227)As an application of a general leftdefinite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the leftdefinite theory associated with the classical Legendre selfadjoint secondorder differential ... 
Eigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem.
(20160323)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 setvalued dynamical systems.
(20160404)In this thesis, many classical results of topological dynamics are adapted to the setvalued case. In particular, focus is given to the notions of topological entropy and the specification property. These properties are ... 
Applications of full rank factorization to solving matrix equations
(199212)In the study of matrices, we are always searching for tools which allow us to simplify our investigations. Because full rank factorizations exist for all matrices and their properties often help to simplify arguments, their ... 
A multigrid Krylov method for eigenvalue problems.
(20150731)We are interested in computing eigenvalues and eigenvectors of matrices derived from differential equations. They are often large sparse matrices, including both symmetric and non symmetric cases. Restarted Arnoldi methods ... 
Krylov methods for solving a sequence of large systems of linear equations.
(20150722)Consider solving a sequence of linear systems A_{(i)}x^{(i)}=b^{(i)}, i=1, 2, ... where A₍ᵢ₎ ϵℂⁿᵡⁿ and b⁽ⁱ⁾ϵℂⁿ using some variations of Krylov subspace methods, like GMRES. For a single system Ax=b, it is wellknown that ... 
Boundary condition dependence of spectral zeta functions.
(20150714)In this work, we provide the analytic continuation of the spectral zeta function associated with the onedimensional regular SturmLiouville problem and the twodimensional Laplacian on the annulus. In the onedimensional ... 
Quadratic Lyapunov theory for dynamic linear switched systems.
(, 20140128)In this work, a special class of timevarying linear systems in the arbitrary time scale setting is examined by considering the qualitative properties of their solutions. Building on the work of J. DaCunha and A. Ramos, ... 
Existence and uniqueness of solutions of boundary value problems by matching solutions.
(, 20130924)In this dissertation, we investigate the existence and uniqueness of boundary value problems for the third and nth order differential equations by matching solutions. Essentially, we consider the interval [a, c] of a BVP ... 
A combinatorial property of BernsteinGelfandGelfand resolutions of unitary highest weight modules.
(, 20130924)It follows from a formula by Kostant that the difference between the highest weights of consecutive parabolic Verma modules in the BernsteinGelfandGelfandLepowsky resolution of the trivial representation is a single ... 
On a ring associated to F[x].
(, 20130924)For a ﬁeld F and the polynomial ring F [x] in a single indeterminate, we deﬁne Ḟ [x] = {α ∈ End_F(F [x]) : α(ƒ) ∈ ƒF [x] for all ƒ ∈ F [x]}. Then Ḟ [x] is naturally isomorphic to F [x] if and only if F is inﬁnite. If F ... 
Existence of positive solutions to right focal three point singular boundary value problems.
(, 20130916)In this dissertation, we prove the existence of positive solutions to two classes of three point right focal singular boundary value problems of at least fourth order. 
Orbit structures of homeomorphisms.
(, 20121129)In this dissertation we answer the following question: If X is a Cantor set and T: X → to X is a homeomorphism, what possible orbit structures can T have? The answer is given in terms of the orbit spectrum of T. If X is a ... 
Spectral functions for generalized piston configurations.
(, 20121129)In this work we explore various piston configurations with different types of potentials. We analyze Laplacetype operators P=g^ij ∇^E_i ∇ ^E_j+V where V is the potential. First we study delta potentials and rectangular ...