Department of Mathematicshttp://hdl.handle.net/2104/47732016-10-01T10:24:20Z2016-10-01T10:24:20ZGlazman-Krein-Naimark theory, left-definite theory, and the square of the Legendre polynomials differential operator.http://hdl.handle.net/2104/96542016-09-07T17:46:23Z2016-02-27T00:00:00ZGlazman-Krein-Naimark theory, left-definite theory, and the square of the Legendre polynomials differential operator.
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 operator A in L² (−1, 1) which has the Legendre polynomials {Pn} ∞ n=0 as eigenfunctions. As a consequence, they explicitly determined the domain D(A2) of the self-adjoint operator A2 . However, this domain, in their characterization, does not contain boundary conditions in its formulation. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each endpoint x = ±1 in L2 (−1, 1), so D(A2) should exhibit four boundary conditions. In this thesis, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman-Krein-Naimark) theory. In addition, we determine a new characterization of D(A2) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple and natural, and are equivalent to the boundary conditions obtained from the GKN theory.
2016-02-27T00:00:00ZEigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem.http://hdl.handle.net/2104/96312016-09-09T15:28:59Z2016-03-23T00:00:00ZEigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem.
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 problems. Methods used for these results involve the theory of $u_0$-positive operators with respect to a cone in conjunction with sign properties of Green's functions. Finally, initial results are established for the existence of positive solutions for singular two point boundary value problems for a fifth order nonlinear differential equation. The methods involve application of a fixed point theorem for decreasing operators.
2016-03-23T00:00:00ZChaotic properties of set-valued dynamical systems.http://hdl.handle.net/2104/96052016-09-07T18:16:37Z2016-04-04T00:00:00ZChaotic properties of set-valued dynamical systems.
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 defined in the context of set-valued functions, and examples are given both of classical theorems that extend naturally to this setting and of theorems which have no clear analogue. Particular attention is paid to a result which states that if a dynamical system has the specification property, there exists invariant non-atomic measures with full support.
2016-04-04T00:00:00ZApplications of full rank factorization to solving matrix equationshttp://hdl.handle.net/2104/95272016-01-21T20:16:22Z1992-12-01T00:00:00ZApplications of full rank factorization to solving matrix equations
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 uses are abundant. There exist many matrix equations for which solutions are otherwise quite difficult to find. Full rank factorizations and generalized inverses allow us to easily find solutions to many such equations. Their properties can also be used to study the diagonalization of non-square matrices and to develop conditions under which matrices are simultaneously diagonalizable. Finally, the full rank factorization can be used to derive canonical forms and other factorizations such as the singular value decomposition.
1992-12-01T00:00:00Z