Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 81-84). Also available on the Internet.

Thesis (M.Eng.(Electronic Engineering))--University of Pretoria, 2009. / Summaries in Afrikaans and English. Includes bibliographical references.

Thesis (M.A.)--Boston University

Let μ be an eigenvalue of the graph G with multiplicity k. A star set corresponding to μ in G is a subset of V(G) such that [x] = k and μ is not an eigenvalue of G - X. It is always the case that the vertex set of G can be partitioned into star sets corresponding to the distinct eigenvalues of G. Such a partition is called a star partition. We give some examples of star partitions and investigate the dominating properties of the set V (G) \ X when μ ε {-I, a}. The induced subgraph H = G - X is called a star complement for μ in G. The Reconstruction Theorem states that for a given eigenvalue μ of G, knowledge of a star complement corresponding to μ, together with knowledge of the edge set between X and its complement X, is sufficient to reconstruct G. Pursuant to this we explore the idea that the adjacencies of pairs of vertices in X is determined by the relationship between the H-neighbourhoods of these vertices. We give some new examples of cubic graphs in this context. For a given star complement H the range of possible values for the corresponding eigenvalue μ is constrained by the condition that μ must be a simple eigenvalue of some one-vertex extension of H, and a double eigenvalue of some two-vertex extension of H. We apply the Reconstruction Theorem to the generic form of a two-vertex extension of H, thereby obtaining sufficient information to construct a graph containing H as a star complement for one of the possible eigenvalues. We give examples of graph characterizations arising in the case where the star complement is (to within isolated vertices) a complete bipartite graph.

This work concerns the description of eigenvalue independent: partitioning theory, and its application to quantum mechanical calculations of interest in chemistry. The basic theory for an m-fold partitioning of a hermitian matrix H, (2 < m < n, the dimension of the matrix), is developed in detail, with particular emphasis on the 2x2 partitioning, which is the most' useful. It consists of the partitioning of the basis space into two subspaces — an n[sub A]-dimensional subspace (n[sub A] > 1), and the complementary n-n[sub A] = n[sub B]-dimensional subspace. Various n[sub A]-(or n[sub B]-) dimensional effective operators, and projections onto n[sub A]- (or n[sub B] dimensional eigenspaces of H, are defined in terms of a mapping, f, relating the parts of eigenvectors lying im each of the partitioned subspaces. This mapping is shown to be determined by a simple nonlinear operator equation, which can be solved by iterative methods exactly, or by using a pertur-bation expansion. Properties of approximate solutions, and various alternative formulas for effective operators, are examined. The theory is developed for use with both orthonormal and non-orthonormal bases. Being a generalization of well known one-dimensional partitioning formalisms, this eigenvalue independent partitioning theory has a number of important areas of application. New and efficient methods are developed for the simultaneous determination of several eigenvalues and eigenvectors of a large hermitian matrix, which are based on the construction and diagonalization of an appropriate effective operator. Perturbation formulas are developed both for effective operators defined in terms of f, and for projections onto whole eigen-spaces of H. The usefulness of these formulas, especially when the zero order states of interest are degenerate, is illustrated by a number of examples, including a formal uncoupling of the four component Dirac hamiltonian to obtain a two component hamiltonian for electrons only, the construction of an effective nuclear spim hamiltonian in esr theory, and the derivation of perturbation series for the one-particle density matrix in molecular orbital theory (in both Huckel-type and closed shell self-consistent field contexts). A procedure is developed for the direct minimization of the total electronic energy in closed shell self-consistent field theory in terms of the elements of f, which are unconstrained and contain no redundancies. This formalism is extended straightforwardly to the general multi-shell single determinant case. The resulting formulas, along with refinements of the basic conjugate gradient minimization algorithm* which involve the use of scaled variables and frequent basis modification, lead to efficient, rapidly convergent methods for the determination of stationary values of the electronic energy* This is illustrated by some numerical calculations in the closed shell and unrestricted Hartree-Fock cases. / Science, Faculty of / Chemistry, Department of / Graduate

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Rotor angle stability is the ability of the interconnected synchronous machines of a power system to remain in synchronism. This stability problem is concerned with the behavior of one or more synchronous machine after they have been perturbed. These perturbations can be small or large depending upon the type of disturbances considered. The work presented in this thesis is focused on the power system behavior when subjected to small disturbances. The ?small signal? disturbances are considered sufficiently small for the linearization of system equations to be permissible for the purpose of the analysis. The first step in the small signal stability studies is to obtain initial steady state conditions using load flow solutions. After establishing initial conditions, an unregulated mathematical model of the power system is formed. The mathematical model obtained is a set of nonlinear coupled first order differential equations. The method of small changes, called the perturbation method, is used to linearize these nonlinear differential equations. The equations are then written in a linear state space model form. The eigenvalues and the participation factors are obtained from the state matrix and the contribution of a particular machine in a particular mode or oscillations (or eigenvalue) can be examined for the small signal stability studies.

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