A random variable that is defined as the absorption time of an evanescent finite-state continuous-time Markov chain is said to have a phase-type distribution. A phase-type distribution is said to have a representation (α,T ) where α is the initial state probability distribution and T is the infinitesimal generator of the Markov chain. The distribution function of a phase-type distribution can be expressed in terms of this representation. The wider class of matrix-exponential distributions have distribution functions of the same form as phase-type distributions, but their representations do not need to have a simple probabilistic interpretation. This class can be equivalently defined as the class of all distributions that have rational Laplace-Stieltjes transform. There exists a one-to-one correspondence between the Laplace-Stieltjes transform of a matrix- exponential distribution and a representation (β,S) for it where S is a companion matrix. In order to use matrix-exponential distributions to fit data or approximate probability distributions the following question needs to be answered: “Given a rational Laplace-Stieltjes transform, or a pair (β,S) where S is a companion matrix, when do they correspond to a matrix-exponential distribution?” In this thesis we address this problem and demonstrate how its solution can be applied to the abovementioned fitting or approximation problem. / Thesis (Ph.D.)--School of Applied Mathematics, 2003.

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Thesis (Ph. D.)--University of Wisconsin--Madison, 1971. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 54-55).

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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 100-102. / by Robert Alan Proctor. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: p. 135-138. / by James William Walker. / Ph.D.

In this thesis, we present a number of results in set theory, particularly in the areas of forcing, large cardinals, and combinatorial set theory. Chapter 2 concerns covering matrices, combinatorial structures introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of this proof and subsequent work with Sharon, Viale isolated two reflection principles, CP and S, which can hold of covering matrices. We investigate covering matrices for which CP and S fail and prove some results about the connections between such covering matrices and various square principles. In Chapter 3, motivated by the results of Chapter 2, we introduce a number of square principles intermediate between the classical and (+). We provide a detailed picture of the implications and independence results which exist between these principles when is regular. In Chapter 4, we address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik-Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik-Sharon model and various other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales. In Chapter 5, we prove that, assuming large cardinals, it is consistent that there are many singular cardinals such that every stationary subset of + reflects but there are stationary subsets of + that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth and is joint work with James Cummings. In Chapter 6, we extend a result of Gitik, Kanovei, and Koepke regarding intermediate models of Prikry-generic forcing extensions to Radin generic forcing extensions. Specifically, we characterize intermediate models of forcing extensions by Radin forcing at a large cardinal using measure sequences of length less than. In the final brief chapter, we prove some results about iterations of w1-Cohen forcing with w1-support, answering a question of Justin Moore.

Thesis(M.S.)--Auburn University, 2005. / Abstract. Vita. Includes bibliographic references.

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Robert Coulter, Dept. of Mathematical Sciences. Includes bibliographical references.