This paper will be devoted to an examination of several general and specific equations and systems of equations of the diophantine type. Only algebraic equations with integral coefficients, not all zero, will considered. The elementary properties of the integers will be assumed.

The notion of a topological group follows naturally from a combination of the properties of a group and a topological space. Since a group consists of a set G of elements which may be either finite or infinite and since this is also common to a topological space, a question is opened as to whether or not it is possible to assign a topology to a set of elements which form a group under a certain operation. Now it is possible to assign a topology to any set of elements if no restriction is placed on the topology assigned and hence this study would be of little value from the standpoint of the group itself. If however it is required that the group operation be continuous in the topological space then a very interesting theory is developed.

The main purpose of this paper is to make a detailed study of a certain class T of complex functions. The functions of the class T have a special mapping property and are meromorphic in every region. As an application of this study, certain elementary functions are defined and studied in terms of a special T-function.

This thesis is a study of fundamental properties of Hilbert space, properties of linear manifold, and realizations of Hilbert space.

The object of this thesis is to examine properties of an abstract vector space of finite dimension n. The properties of the set of complex numbers are assumed, and the definition of a field and of an abelian group are not stated, although reference to these systems is made.

This thesis explores the fundamental properties of the contingent.

The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

Axioms for a topological space are generally based on neighborhoods where "neighborhood" is an undefined term. Then, limit points are defined in terms of neighborhoods. However, limit points seem to be the basic concept of a topological space, rather than neighborhoods. For this reason, it will be attempted to state a set of axioms for a topological space, using limit point as the undefined concept, and to delete the idea of neighborhoods from the theory.

This thesis investigates some of the properties of certain transformations. Some properties are considered in general; others, only in the xy-plane.

It is intended that this paper present an acceptable proof of the existence of a solution for the wave equation.

The purpose of this paper is to present a study of partly ordered sets. It includes a rigorous development of relations based on the notion of a relation as a set, lattices, and theorems concerning the lattice of subgroups of a group.

This paper investigates the Runge-Kutta method of numerically integrating ordinary differential equations. An existence theorem is given insuring a solution to the differential equation, then the theorem is modified to yield an analytic solution. The derivation of the method itself is followed by an analysis of the inherent error.

This paper will investigate the geometry that results when Euclid's fifth postulate is replaced by the Lobachevskian assumption. The investigations will be limited to the plane.

This thesis presents a series of theorems along with proofs to establish a theory of congruences.

The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.

This paper is the result of a study of triangular matrices with particular emphasis on those which are convergence preserving transformations.

The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.

This paper is primarily concerned with the fundamental properties of a linear algebra of finite order over a field. A discussion of linear sets of finite order over a field is used as an introduction to these properties.

The purpose of this paper is to examine the properties of the envelope and the extraneous loci associated with the solution curves of ordinary differential equations of the first order and degree greater than one.

The purpose of this paper is to develop some of the more basic Fourier transforms which are the outgrowth of the Fourier theorem. Although often approached from the stand-point of the series, this paper will approach the theorem from the standpoint of the integral.

This study of the Euclidean N-space looks at some definitions and their characteristics, some comparisons, boundedness and compactness, and transformations and mappings.

This thesis is a study of linear spaces and linear transformations in normed linear spaces. The notion of a field, in particular the complex number field, is assumed in this paper.

The purpose of this paper is the investigation of various properties of ordered triples whose components are elements of certain sets.

The purpose of this thesis is to investigate the relation Z for point sets in a topological space. There were two original goals which caused the study.