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A Classification of Regular Planar Graphs
The purpose of this paper is the investigation and classification of regular planar graphs. The motive behind this investigation was a desire to better understand those properties which allow a graph to be represented in the plane in such a manner that no two edges cross except perhaps at vertices.
Completing the Space of Step Functions
In this thesis a study is made of the space X of all step functions on [0,1]. This investigation includes determining a completion space, X*, for the incomplete space X, defining integration for X*, and proving some theorems about integration in X*.
Concerning Measure Theory
The purpose of this thesis is to study the concept of measure and associated concepts. The study is general in nature; that is, no particular examples of a measure are given.
Continuous Multifunctions
This paper is a discussion of multifunctions, various types of continuity defined on multifunctions, and implications of continuity for the range and domain sets of the multifunctions.
Differentiation in Banach Spaces
This thesis investigates the properties and applications of derivatives of functions whose domain and range are Banach spaces.
G-domains, G-ideals, and Hilbert Rings
The problem with which this investigation is concerned is that of determining the properties of the following: a particular type of integral domain, the G-domain; a type of prime ideal, the G-ideal; and a special type of ring, the Hilbert ring.
A Generalization of the Weierstrass Approximation Theorem
A presentation of the Weierstrass approximation theorem and the Stone-Weierstrass theorem and a comparison of these two theorems are the objects of this thesis.
Integration of Vector Valued Functions
This paper develops an integral for Lebesgue measurable functions mapping from the interval [0, 1] into a Banach space.
The Lp Spaces of Equivalence Classes of Lebesgue Integrable Functions
The purpose of the paper is to prove that the Lp spaces, p ≥ 1, of equivalence classes of functions are Banach spaces.
Metric Half-Spaces
This paper is a study of some of the basic properties of the metric half-space topology, a topology on a set which is derived from a metric on the set. In the first it is found that in a complete inner product space, the metric half-space topology is the same as one defined in terms of linear functionals on the space. In the second it is proven that in Rn the metric half-space topology is the same as the usual metric topology. In the third theorem it is shown that in a certain sense the nature of the metric halfspace topology generated by a norm on the space determines whether the norm is quadratic, that is to say, whether or not there exists an inner product on the space with the property that |x|^2=(x,x) for all x in the space.
Near-Rings
The primary objective of this work is to discuss some of the elementary properties of near-rings as they are related to rings. This study is divided into three subdivisions: (1) Basic Properties and Concepts of Near-Rings; (2) The Ideal Structure of Near-Rings; and (3) Homomorphism and Isomorphism of Near-Rings.
On the Stielitjes Integral
This paper is a study of the Stieltjes integral, a generalization of the Riemann integral normally studied in introductory calculus courses. The purpose of the paper is to investigate many of the basic manipulative properties of the integral.
Properties of Limit Functions
The purpose of this paper is to investigate properties of functions which are limits of functions with prescribed properties. Chapter II asks the question "Does a function which is the limit of a sequence of functions each of which is endowed with a certain property necessarily have that property?"
The Riemann-Complete Integral
The problem with which this paper is concerned is that of defining the Riemann-Complete Integral and comparing it with the Riemann and the Lebesgue Integrals.
Topological Groups
In the study of groups and topological spaces, the properties of both are often encountered in one system. The following are common examples: groups with discrete topologies, the complex numbers with the usual topology, and matrix groups with metric topologies. The need for a study of how algebraic properties and topological properties affect one another when united and interrelated in one system soon becomes evident. Thus the purpose of this thesis is to study the interrelated group and topological space, the topological group.
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