This paper will be devoted to an exposition of some of the basic properties of paracompact spaces. In particular, it will be shown that every pseudo-metrizable space is paracompact and countably paracompact.

This report presents both the most essential known results and new results in the theory of partially ordered groups and rings. This report deals with partially ordered groups and rings in an algebraic aspect because it is more important than partially ordered, fully ordered and lattice-ordered semigroup theory.

The development of the concept of a filter leads to a theory of convergence in topological spaces. There is a close relationship between the concept of a net and that of a filter.

In this paper, elementary properties of relative extreme points are investigated. The properties are defined in linear and topological terms. Proofs of many of these properties require the use of topological concepts.

This thesis examines linear programming problems, the theoretical foundations of the simplex method, and how a liner programming problem can be solved with the simplex method.

This thesis presents a series of theorems and proofs describing a twenty-five point geometry in which exists exactly thirty lines.

This thesis describes various forms of metric spaces and establishes some of the properties of functions defined on metric spaces. No attempt is made in this paper to examine a particular type of function in detail. Instead, some of properties of several kinds of functions will be observed as the functions are defined on various forms of metric spaces such as connected spaces, compact spaces, complete spaces, etc.

The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.

This thesis presents a development of some useful concepts concerning topological spaces. Most of the theorems given apply to the most general form of topological space.

The purpose of this paper is to define an integral for real-valued functions which are defined on a field of sets and to demonstrate several properties of such an integral.

The primary purpose of this paper is to develop a rigorous study of the Schwarz derivative. This study will be based primarily on the comparison of the ordinary derivative to the Schwarz derivative.

A special case of the linear programming problem, the transportation problem, is the subject of this thesis. The development of a solution to the transportation problem is based on fundamental concepts from the theory of linear algebra and matrices.

This paper establishes a basis for abelian categories, then gives the statement and proof of two equivalent definitions of an abelian category, the development of the basic theory of such categories, and the proof of some theorems involving this basic theory.

The purpose of this thesis is to investigate certain properties of rings, ideals, and a special type of ring called a Boolean ring.

In mathematical physics, Laplace's equation plays an especially significant role. It is fundamental to the solution of problems in electrostatics, thermodynamics, potential theory and other branches of mathematical physics. It is for this reason that this investigation concerns the development of some general properties of continuous solutions of this equation.

The purpose of this thesis is to develop some properties of a special class of functions called valuations. The study begins with and examination of the properties of valuations defined on an arbitrary field, F, and later, consideration is given to valuations defined on the field of rational numbers. The concept of a pseud-valuation is introduced and an investigation is made of the properties of pseudo-valuations.

In this thesis we consider integrals of a certain class of interval functions. Specifically we consider a nondegenerate number interval [a,b], a real valued function m, defined and nondecreasing on [a,b], and the set Hm, of real valued functions f, defined on [a,b] such that: 1) f(a)=0; 2) for each subinterval [p,q] of [a,b], if m(q)-m(p)=0, then f(q)-f(p)=0; and 3) the set of all sums of the form Σ(Δf)2/Δm for subdivisions D of [a,b] is bounded above.

The purpose of this thesis is to explore some of the characteristics of 0-simple semigroups and completely 0-simple semigroups.

This thesis is an investigation of some of the properties of polynomial rings, unique factorization domains, Euclidean domains, and principal ideal domains. The nature of some of the relationships between each of the above systems is also developed in this paper.

This paper was written in response to the following question: what conditions are sufficient to guarantee that if a compact subset A of a topological linear space L^3 is not convex, then for every point x belonging to the complement of A relative to the convex hull of A there exists a line segment yz such that x belongs to yz and y belongs to A and z belongs to A? Restated in the terminology of this paper the question bay be given as follow: what conditions may be imposed upon a compact subset A of L^3 to insure that A is braced?

The purpose of this paper is to study nth order integral equations. The integrals studied in this paper are of the Riemann type.

This thesis is a study of primary Abelian groups and height.

The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers.

In the numerous problems of matrix algebra, one finds the problem of determining the eigenvalues of eigenvectors of a matrix quite frequently. The theory and methods leading to the solution of the eigenvalue and eigenvector problem are of considerable interest. The relation between vector spaces, matrices, eigenvalues, and eigenvectors is to be considered in this chapter, with particular concentration directed toward the eigenvalues and eigenvectors shall be developed in the following chapters with detailed examples of the methods.