The purpose of this thesis is to define equivalence classes of Cauchy sequences of rational numbers and the operations of taking a sum and a product and then to show that this system is an uncountable, ordered, complete field. In so doing, a mathematical system is obtained which is isomorphic to the real number system.

The purpose of this thesis is to examine general properties, convergence, and limit points of sequences of real numbers.

This thesis investigates ideals in semigroups.

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.

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

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.

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 thesis analyzes tensor associations with applications to Reimann spaces.

The purpose of this paper is to study convergence of certain continued fractions.

The purpose of this paper is to define and investigate some of the properties of chains. Particular attention is given to a natural topology for chains, called the interval topology, and how the chain properties and topological properties of chains affect each other.

This thesis provides a development of integration from two different points of view. In Chapter I, a measure and a measurable function are defined. A theory of integration is then developed in Chapter II based on the measure. In Chapter III, the integral is introduced directly without first going through the process of defining a measure, and a measure is developed from the integral. The concluding chapter shows the equivalence of the two integrals under rather general conditions.

This paper is an introductory, algebraic study of semigroups.

The objective here is to give a logical development of some mathematical systems beginning with those having relatively little algebraic structure, and progressing to systems rich in structure. The discussion will commence with a mathematical system called the groupoid, a nonempty set with a binary operation.

The purpose of this paper is to present a discussion on the basic fundamentals of the theory of sets. Primarily, the discussion will be confined to the study of cardinal and ordinal numbers. The concepts of sets, classes of sets, and families of sets will be undefined quantities, and the concept of the class of all sets will be avoided.

Because lattice theory is so vast, the primary purpose of this paper will be to present some of the general properties of lattices, exhibit examples of lattices, and discuss the properties of distributive and modular lattices.

Some of the properties of the specific summation methods will be investigated, such as what type of divergent series a method can or cannot sum, if the insertion of zeros into a series does change the sum, and when different methods give the same sum for a series.

This paper is concerned with equations in which all derivatives are ordinary rather than partial derivatives. The customary meanings of differential order and difference order of an equation are observed.

This paper will be devoted to an exposition of some of the relationships existing between a field and certain of its extension fields. In particular, it will be shown that many fields may be characterized rather simply in terms of their subfields which, in turn, may be directly correlated with the subgroups of a finite group of automorphisms of the given field.

One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

This thesis discusses groups, modules, the module of homomorphisms, and extension of modules.

The purpose of this paper is to investigate some of the more basic properties of peripherally continuous functions, graph maps and connectivity maps.

This thesis is presented as an introduction to the Fibonacci sequence of integers. It is hoped that this thesis will create in the reader more interest in this type of sequence and especially the Fibonacci sequence. It seems that this particular area of mathematics is often ignored in the classroom or touched upon far too briefly to stimulate curiosity and develop further interest in this field.

The usual practice in any discussion of an inner-product space is to restrict the field over which the inner-product space is defined to the field of complex numbers. In defining the inner-product function, (x,y), a second function is needed; namely the conjugate function (x,y)* so that (x,y) ± (y,x)*. We will attempt to generalize this concept by investigating the existence of a conjugate function defined on fields other than the field of complex numbers and relate this function to an inner-product function defined on a linear space L over these fields.