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**Degree Discipline:**Mathematics

**Degree Level:**Master's

**Collection:**UNT Theses and Dissertations

### Absolute Continuity and the Integration of Bounded Set Functions

**Date:**May 1975

**Creator:**Allen, John Houston

**Description:**The first chapter gives basic definitions and theorems concerning set functions and set function integrals. The lemmas and theorems are presented without proof in this chapter. The second chapter deals with absolute continuity and Lipschitz condition. Particular emphasis is placed on the properties of max and min integrals. The third chapter deals with approximating absolutely continuous functions with bounded functions. It also deals with the existence of the integrals composed of various combinations of bounded functions and finitely additive functions. The concluding theorem states if the integral of the product of a bounded function and a non-negative finitely additive function exists, then the integral of the product of the bounded function with an absolutely continuous function exists over any element in a field of subsets of a set U.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc663440/

### Abstract Measure

**Date:**1957

**Creator:**Bridges, Robert Miller

**Description:**This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107950/

### Abstract Vector Spaces and Certain Related Systems

**Date:**August 1961

**Creator:**Goddard, Alton Ray

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

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130465/

### Additive Functions

**Date:**June 1963

**Creator:**McNeir, Ridge W.

**Description:**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).

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108204/

### Algebraic Integers

**Date:**August 1969

**Creator:**Black, Alvin M.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131119/

### Algebraic Properties of Semigroups

**Date:**May 1971

**Creator:**Lumley, Robert Don

**Description:**This paper is an algebraic study of selected properties of semigroups. Since a semigroup is a result of weakening the group axioms, all groups are semigroups. One facet of the paper is to demonstrate various semigroup properties that induce the group axioms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131373/

### A*-algebras and Minimal Ideals in Topological Rings

**Date:**May 1973

**Creator:**Wei, Jui-Hung

**Description:**The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131622/

### The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity

**Date:**1941

**Creator:**Parrish, Herbert C.

**Description:**This study investigates the relationship between ordinary continuity and t-continuity.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc75274/

### The Analytical Development of the Trigonometric Functions

**Date:**1951

**Creator:**Mackey, Pearl Cherrington

**Description:**This thesis is a study of the analytical development of the trigonometric functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc97030/

### Applications in Fixed Point Theory

**Date:**December 2005

**Creator:**Farmer, Matthew Ray

**Description:**Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc4971/

### An Approximate Solution to the Dirichlet Problem

**Date:**August 1964

**Creator:**Redwine, Edward William

**Description:**In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130550/

### Atmospheric Gusts and Their Effect on Aircraft

**Date:**August 1958

**Creator:**Walling, Waunnetta Keene

**Description:**This thesis investigates atmospheric gusts and their effect on aircraft.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108016/

### Automorphism Groups

**Date:**August 1965

**Creator:**Edwards, Donald Eugene

**Description:**This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130613/

### Basic Fourier Transforms

**Date:**January 1962

**Creator:**Cumbie, James Randolph

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130482/

### Borel Determinacy and Metamathematics

**Date:**December 2001

**Creator:**Bryant, Ross

**Description:**Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3061/

### Borel Sets and Baire Functions

**Date:**January 1970

**Creator:**Wemple, Fred W.

**Description:**This paper examines the relationship between Borel sets and Baire functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc163964/

### The Buckling of a Uniformly Compressed Plate with Intermediate Supports

**Date:**1949

**Creator:**Dean, Thomas S.

**Description:**This problem has been selected from the mathematical theory of elasticity. We consider a rectangular plate of thickness h, length a, and width b. The plate is subjected to compressive forces. These forces act in the neutral plane and give the plate a tendency to buckle. However, this problem differs from other plate problems in that it is assumed that there are two intermediate supports located on the edges of the plate parallel to the compressive forces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc83479/

### The Cantor Ternary Set and Certain of its Generalizations and Applications

**Date:**1942

**Creator:**Hembree, Gwendolyn

**Description:**This thesis covers the Cantor Ternary Set and generalizations of the Cantor Set, and gives a complete existential theory for three set properties: denumerability, exhaustibility, and zero measure.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc70344/

### Certain Properties of Functions Related to Exhaustibility

**Date:**1952

**Creator:**Bradford, James C.

**Description:**In this thesis, we shall attempt to present a study of certain properties of real functions related to the set property exhaustible.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107821/

### Characterizations of Continua of Finite Degree

**Date:**August 2006

**Creator:**Irwin, Shana

**Description:**In this thesis, some characterizations of continua of finite degree are given. It turns out that being of finite degree (by formal definition) can be described by saying there exists an equivalent metric in which Hausdorff linear measure of the continuum is finite. I discuss this result in detail.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5367/

### Chebyshev Subsets in Smooth Normed Linear Spaces

**Date:**December 1974

**Creator:**Svrcek, Frank J.

**Description:**This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc663499/

### A Classification of Regular Planar Graphs

**Date:**December 1972

**Creator:**McCalla, Linda F.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc164029/

### Compact Convex Sets in Linear Topological Spaces

**Date:**May 1964

**Creator:**Read, David R.

**Description:**The purpose of this paper is to examine properties of convex sets in linear topological spaces with special emphasis on compact convex sets.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130516/

### Compact Operators and the Schrödinger Equation

**Date:**December 2006

**Creator:**Kazemi, Parimah

**Description:**In this thesis I look at the theory of compact operators in a general Hilbert space, as well as the inverse of the Hamiltonian operator in the specific case of L2[a,b]. I show that this inverse is a compact, positive, and bounded linear operator. Also the eigenfunctions of this operator form a basis for the space of continuous functions as a subspace of L2[a,b]. A numerical method is proposed to solve for these eigenfunctions when the Hamiltonian is considered as an operator on Rn. The paper finishes with a discussion of examples of Schrödinger equations and the solutions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5453/