This paper is devoted to finding integral solutions of algebraic equations. Only algebraic equations with integral coefficients are considered. The elementary properties of integers are assumed.

The main purpose of this paper is the study of transformations in topological space and relationships between special types of transformations.

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.

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 thesis is to present a definition and some properties of a curved arc in a plane and to present a definition and some properties of the Jordan curve.

The purpose of this paper is the analysis of functions of real numbers which have a special additive property, namely, f(x+y) = f(x)+f(y).

The purpose of this paper is to develop the Peano postulates from a weaker axiom system than the system used by John L. Kelley in General Topology. The axiom of regularity which states "If X is a non-empty set, then there is a member Y of X such that the intersection of X and Y is empty." is not assumed in this thesis. The axiom of amalgamation which states "If X is a set, then the union of the elements of X is a set." is also not assumed. All other axioms used by Kelley relevant to the Peano postulates are assumed. The word class is never used in the thesis, though the variables can be interpreted as classes.

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

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

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

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

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.

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.

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

This thesis explores the fundamental properties of the contingent.

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 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 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.

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.

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.