A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive(respectively, negative, zero) entry of B by + (respectively, -, 0). For a sign pattern matrix A, the sign pattern class of A, denoted Q(A), is defined as {B: sgn(B) = A}. An n by n sign pattern matrix A requires all distinct eigenvalues if every real matrix whose sign pattern is represented by A has n distinct eigenvalues. In this thesis, a number of sufficient and/or necessary conditions for a sign pattern to reuiqre all distinct eigenvalues are reviewed. In addition, for n=2 and 3, the n by n sign patterns that require all distinct eigenvalues are surveyed. We determine most of the 4 by 4 irreducible sign patterns that require four distinct eigenvalues.

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<p> An investigation is carried out to see whether a nonsingular two-nucleon velocity-dependent potential can give similar results as those of a model hard-core potential in the a-state in nuclear-matter. Reasons for failure of some earlier velocity-dependent model potentials to give saturation in nuclear-matter at reasonable densities are examined and a new velocity-dependent potential is proposed which gives results similar to the model hard-core potential. </p> / Thesis / Doctor of Philosophy (PhD)