Let R be a ring with an identity 1. Let A, B, and C be R-modules. The sequence A → [f above arrow] B→[g above arrow] C is exact providing f and g are R-homomorphisms and Im f =Ker g. Let 0 represent the R-module with precisely one element. An R-module J is injective if and only if for every exact sequence 0→A→ [f above arrow] B of R-modules and R-homomorphisms and every R-homomorphism g: A→J there exists an R-homomorphism h: B→J such that hf = g. This is a dual concept to that of a projective R-module. In the second chapter the idea of an injective R-module is studied quite intensively, and several different characterizations of injective · modules are proved. One of the principal results obtained is that every R-module is a submodule of an injective R-module. Further properties of injective R-modules are given in Chapter 3, including the concepts of injective dimension and an injective resolution of an R-module. Using these concepts the Shifting Theorem for injectives is proved. The basic definitions and results necessary for the development of the concept of injective for abstract categories are included in Chapter 4. An injective object is then defined in this general setting. Then the concept of an injective envelope is defined. The problems that arise, in the effort to restrict the category of topological groups to the appropriate subcategory so that the concept of an injective topological group is of interest, are investigated in Chapter 5. The development of the concept for one such restriction concludes this thesis. / Master of Science

Three different definitions for the determinant of a matrix over arbitrary lattices have been developed to determine which properties and relations were reminiscent of the determinant or permanent of elementary algebra. In each determinant there are properties concerning: the elements of the matrix in the expansion of its determinant; the determinant of a matrix and its transpose; a principle of duality for rows and columns; the interchange of rows and columns; the determinant of a matrix formed from another by a row or column meet of certain elements; and evaluations of certain special matrices. An expansion by row or column is given for one determinant and a lemma on inverses is proven in light of another. A preliminary section on Lattice Theory is also included. / Master of Science

This thesis is intended to be an introductory mathematical presentation of analog computation. An attempt was made to explain in concise mathematical language, how an electronic analog computer works, why it works, and the simplicity of its use. The components of the computer are considered as operational blocks, each block performing an indicated operation. Hence, the electrical knowledge presented is meager. The methods of solution and the corresponding computer solutions obtained for several types of mathematical problems are presented; such as, the determination of the characteristic vectors and characteristic values of a given matrix. In each case, a 15-amplifier Heath Kit analog computer model number ES-400 was used. Since this type of computer contains no devices for multiplying variable quantities, the only types of problems that could be considered were those that can be represented by a system of linear, ordinary differential equations with constant coefficients. However, similar techniques are applicable to the analogous non-linear systems and systems with variable coefficients, on a fully-equipped analog computer. / Master of Science

This thesis is concerned with the study of quadratic forms over fields of characteristic 2. First, we consider the extension of quadratic forms to fields of characteristic ≠ 2. Then we make the adjustments necessary to make the characteristic 2 case non-trivial, and investigate the structure of quadratic spaces. We define equivalence of quadratic forms and investigate a set of invariants. We develop a set of necessary and sufficient conditions for equivalence and give a characterization of non-equivalent forms over certain finite fields. / Master of Science

Master of Science

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

R. J. Warne has defined an wⁿ-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (I^o)ⁿ, where I^o is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωⁿ-right cancellative semigroup into an ωⁿ-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ωⁿ-right cancellative semigroup into an ω^m-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(I^o)ⁿ , γ₁,...,γ_n, w₁,…,w_Ø(n)) and P^* = (G ,(I^o)ⁿ , α₁,...,α_n, t₁,…,t_Ø(n)) be ωⁿ-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,z_n be elements of G^* and let f be a homomorphism of G into G^* such that (1) (Af)^(U_kg)C_{z_k} = (Aγ_kf) for A ∈ G where 1 ≤ k ≤ n and (2) ((z_k+s)^(U_kg)(U_kg)^(U_k+sg)C_{z_k} = w_Ø(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements U_k (1 ≤ k ≤ n) are generators of (I^o)ⁿ, xC_{z_k} = z_kxz_k⁻¹ for x ∈ G^*, and x^a,a^b in G^* (x ∈ G^*; a,b ∈ (I^o)ⁿ are specified. Define, for (A,a₁,...,a_n) ∈ P, (A,a₁,...,a_n)M = [(Af)(a₁,...,a_n)h,(a₁,...,a_n)g] where h is a specified function from (I^o)ⁿ into G* and g is a determined endomorphism of (I^o)ⁿ. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ω^m-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let E_S denote the set of idempotents of S. S is called ωⁿ-bisimple if and only if E_S, under its natural order, is order isomorphic to I x (I^o)ⁿ under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if E_S, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ωⁿ-bisimple, ωⁿI-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ωⁿI-bisimple Semigroups," to appear], and ["I-bisimple Semigroups," Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωⁿ-right cancellative semigroups, for the cases (i) S ωⁿ-bisimple and S* ω^m-bisimple and (ii) S I-bisimple or ωⁿI-bisimple and S* I-bisimple or ω^mI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωⁿ-bisimple, ωⁿI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date. / Ph. D.