M.S.

Ph. D.

Let G = (a,b,c,...) be a groupoid and T a topology for G with U_a denoting an open set in T that contains the element a. The topology T is admissible for G if for every a·b=c and U_c there exist U_a and U_b such that U_a·U_b c U_c. G is said to be topologically trivial if the only admissible topologies for G are the discrete and indiscrete. It is shown that finite groups are topologically trivial if and only if they are simple. It is shown that finite topologically trivial semigroups are necessarily groups. Various classes of topologically trivial groupoids are examine, and it is shown that there exist topologically trivial groupoids of every order. G is said to be right (analogously left) topologically trivial if one can find elements a·b = c in G and U_c in T such that a·U_b ⊈ U_c for all U_b in T whenever T is not trivial. G is said to be totally topologically trivial if one can find a·b = c in G and U_c in T such that a·U_b ⊈ U_c and U_a·b ⊈ U_c for all U_a and U_b in T whenever T is not trivial. Right, left, and total topologically triviality are studies for various algebraic systems. A continuity condition that always holds is exhibited as are new proofs for several old theorems. Consequences of imposing the tower topology on various algebraic systems are examined. If the proper subset I contained in the groupoid G is such that the null set, the set G, and each singleton set of the elements in G-I form the basis for an admissible topology for G, then I is called a generalized ideal in G. Properties of generalized ideals are studied at length. A function t from a groupoid G to another groupoid is called a local homomorphism if for each a and b in G there exist r and s in G such that a·b = r·s and such that t(r·s) = t(r)·t(s). Several properties of local homomorphisms are examined. / Ph. D.

A software package has been written to compute the Hopf bifurcation structure in functional differential equations. The package is modular, and consists of several routines which perform one or more tasks. In conjunction with the routines available in this package, the user is required to provide a few routines which describe the specific system under analysis. Three example systems (from epidemiology, biochemistry and aerospace engineering) have been analyzed to illustrate the use of this package. / M.S.

M.S.

If (X,Ƭ) is a topological space, then a quasi-uniformity U on X is compatible with Ƭ if the quasi-uniform topology, Ƭ_u = Ƭ. This paper is concerned with local properties of quasi-uniformities on a set X that are compatible with a given topology on X. Chapter II is devoted to the construction of Hausdorff completions of transitive quasi-uniform spaces that are members of the Pervin quasi-proximity class. Chapter III discusses locally complete, locally precompact, locally symmetric and locally transitive quasi-uniform spaces. Chapter IV is devoted to function spaces of quasi-uniform spaces. Chapter V and the Appendix are concerned with the topological homeomorphism groups of quasi-uniform spaces. / Ph. D.

Control systems governed by nonlinear neutral functional differential equations are formulated as abstract evolution equations in product spaces. At this point existence and uniqueness of solutions are studied. This formulation is used to develop a general approximation scheme for those systems. Convergence of this scheme is analyzed. It is also shown how spline based approximating methods fall within this general framework. An illustrative example is presented. / Ph. D.

The effects of polymer particle size, temperature, and time on the continued condensation of two polyamides: poly(hexamethylene adipamide) and poly(hexamethylene sebacamide), and a polyester, poly(ethylene terephthalate), were studied experimentally. The polyamides were held at elevated temperatures from 120 to 180 °C for periods of 5 to 20 hours in a nitrogen atmosphere. A similar procedure was followed with the polyester except that the range of temperature was 160 to 200 °C. The number-average molecular weights of the polymers before and after treatment were calculated from the polymer intrinsic viscosities. The thermal behavior of selected polymers was also examined by differential thermal analysis. Poly(hexamethylene adipamide) showed an increase in the number-average molecular weight from approximately 10,000 to 22,000 when treated at 180 °C for 20 hours. Under the same conditions, poly(hexamethylene sebacamide) showed a change from about 10,000 to 14,000. Poly- (ethylene terephthalate) treated at 200 °C for 20 hours exhibited an increase from approximately 18,000 to 34,000. Two kinetic equations were derived and were successfully applied to the experimental data. From these equations the specific reaction rates were obtained. The temperature dependency of the reaction rates was expressed in the form of the Arrhenius equation. The effect of particle size on the reaction was noted. Also noted were changes in polymer thermal behavior as the reaction temperature was raised. At low temperatures the transport of reaction by-products from the interior to the surface of solids controlled the reaction. At high temperatures chemical kinetics determined the reaction. / Doctor of Philosophy

We begin this paper with a survey of characterizations of pseudocompact spaces and relate pseudocompactness to other forms of compactness such as light compactness, countable compactness, weak compactness, etc. Some theorems on properties of subspaces of pseudocompact spaces are presented. In particular, conditions are given for the intersection of two pseudocompact spaces to be pseudocompact. First countable pseudocompact spaces are investigated and turn out to be maximally pseudocompact and minimally first countable in the class of completely regular spaces. We define a pseudocompactification of a space X to be a pseudocompact space in which Xis embedded as a dense subspace. In particular, for a completely regular space X, we consider the pseudocompactification αX = (βX - ζX) U X. We investigate this space and in general all pseudocompact subspaces of βX which contain X. There are many pseudocompact spaces between X and βX, but we may characterize αx as follows: 1) αx is the smallest subspace of βX containing X such that every free hyperreal z-ultrafilter on X is fixed in αx. 2) αx is the largest subspace of βX containing X such that every point in αX - X is contained in a zero set which doesn't intersect X. The space αx also has the nice property that any subset of X which is closed and relatively pseudocompact in X is closed in αx. The relatively pseudocompact subspaces of a space are important and are investigated in Chapter 4. We further relate relative pseudocompactness to the hyperreal z-ultrafilter on X and obtain the following characterizations of a relatively pseudocompact zero set: 1) A zero set Z is relatively pseudocompact if and only if Z is contained in no hyperreal z-ultrafilter. 2) A zero set Z is relatively pseudocompact if and only if every countable cover of Z by cozero sets of X has a finite subcover. In the next chapter we consider locally pseudocompact spaces and obtain results analogous to those for locally compact spaces. Then we relate pseudocompactness and the property of being C* - or C-embedded in a space X. Included in this is a study of certain weak normality properties and their relationship to pseudocompact spaces. We develop two types of one-point pseudocompactifications and investigate the properties of each. It turns out that a space X is never C* -embedded in its one-point pseudocompactification. Also one space has the property that closed pseudocompact subsets are closed in the one-point pseudocompactification while the other may not have this property but will be completely regular. We present survey material on products of pseudocompact spaces and unify these results. As an outgrowth of this study we investigate certain functions which are related to pseudocompactness. / Doctor of Philosophy

Let G be a finite connected graph. The circularity of G has been previously defined as σ(G) = max{r ε N| G has a circular covering of r elements, each element being a closed, connected subset of G containing at least one vertex of G}. This definition is known to be equivalent to the combinatorial description, σ(G) = max{r ε N| there is an admissible map f:V(G)→A(r)}. In this thesis, co-admissible maps are introduced and the co-circularity of a graph, G, is defined as η(G) = max{n ε N| there is a co-admissible map g:V(G)→Z_n}. It is shown that σ(G) = 2η(G) or 2η(G) + 1. It is also shown that if G is a graph and g:V(G)→Z_n is a co-admissible map, then G contains a cycle, J, called a co-admissible cycle, for which g:V(J)→Z_n is also co-admissible. Necessary and sufficient conditions are given for extending a co-admissible map on a cycle of a graph to the entire graph. If G is a graph with σ(G) = r, it is shown that any suspended (v,w)-path P in G induces, under any admissible map f:V(G)→A(r), either at most four elements of Z_r or every vertex of P with valency two induces exactly two elements of Z_r not induced by any other vertex of G. Finally it is shown that if G is a planar graph and if g:V(G)→Z_n is a co-admissible map, then any planar representation of G has exactly two faces bounded by co-admissible cycles. / Doctor of Philosophy