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**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### Euclidean N-space

**Date:**August 1962

**Creator:**Horner, Donald R.

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

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108157/

### Euclidean Rings

**Date:**May 1974

**Creator:**Fecke, Ralph Michael

**Description:**The cardinality of the set of units, and of the set of equivalence classes of primes in non-trivial Euclidean domains is discussed with reference to the categories "finite" and "infinite." It is shown that no Euclidean domains exist for which both of these sets are finite. The other three combinations are possible and examples are given. For the more general Euclidean rings, the first combination is possible and examples are likewise given. Prime factorization is also discussed in both Euclidean rings and Euclidean domains. For Euclidean rings, an alternative definition of prime elements in terms of associates is compared and contrasted to the usual definitions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc663654/

### Examples and Applications of Infinite Iterated Function Systems

**Date:**August 2000

**Creator:**Hanus, Pawel Grzegorz

**Description:**The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc2642/

### Exhaustibility and Related Set Properties

**Date:**1950

**Creator:**Cargal, Buchanan

**Description:**The purpose of this paper is to develop certain fundamental properties of exhaustible sets and their complements and to examine various set properties which are generalizations, with respect to exhaustible neglect, or well-known set properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130236/

### Exhaustivity, continuity, and strong additivity in topological Riesz spaces.

**Access:**Use of this item is restricted to the UNT Community.

**Date:**May 2004

**Creator:**Muller, Kimberly O.

**Description:**In this paper, exhaustivity, continuity, and strong additivity are studied in the setting of topological Riesz spaces. Of particular interest is the link between strong additivity and exhaustive elements of Dedekind s-complete Banach lattices. There is a strong connection between the Diestel-Faires Theorem and the Meyer-Nieberg Lemma in this setting. Also, embedding properties of Banach lattices are linked to the notion of strong additivity. The Meyer-Nieberg Lemma is extended to the setting of topological Riesz spaces and uniform absolute continuity and uniformly exhaustive elements are studied in this setting. Counterexamples are provided to show that the Vitali-Hahn-Saks Theorem and the Brooks-Jewett Theorem cannot be extended to submeasures or to the setting of Banach lattices.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc4455/

### Existence and Uniqueness Theorems for Nth Order Linear and Nonlinear Integral Equations

**Date:**May 1969

**Creator:**Hurlbert, Gayle Jene Shultz

**Description:**The purpose of this paper is to study nth order integral equations. The integrals studied in this paper are of the Riemann type.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131097/

### Existence of a Sign-Changing Solution to a Superlinear Dirichlet Problem

**Date:**August 1995

**Creator:**Neuberger, John M. (John Michael)

**Description:**We study the existence, multiplicity, and nodal structure of solutions to a superlinear elliptic boundary value problem. Under specific hypotheses on the superlinearity, we show that there exist at least three nontrivial solutions. A pair of solutions are of one sign (positive and negative respectively), and the third solution changes sign exactly once. Our technique is variational, i.e., we study the critical points of the associated action functional to find solutions. First, we define a codimension 1 submanifold of a Sobolev space . This submanifold contains all weak solutions to our problem, and in our case, weak solutions are also classical solutions. We find nontrivial solutions which are local minimizers of our action functional restricted to various subsets of this submanifold. Additionally, if nondegenerate, the one-sign solutions are of Morse index 1 and the sign-changing solution has Morse index 2. We also establish that the action level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. Our results extend and complement the findings of Z. Q. Wang ([W]). We include a small sample of earlier works in the general area of superlinear elliptic boundary value problems.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278179/

### Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem

**Date:**May 1988

**Creator:**Sumalee Unsurangsie

**Description:**In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc331780/

### Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains

**Date:**August 1998

**Creator:**Finan, Marcel Basil

**Description:**The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278251/

### An Existence Theorem for an Integral Equation

**Date:**May 1985

**Creator:**Hunt, Cynthia Young

**Description:**The principal theorem of this thesis is a theorem by Peano on the existence of a solution to a certain integral equation. The two primary notions underlying this theorem are uniform convergence and equi-continuity. Theorems related to these two topics are proved in Chapter II. In Chapter III we state and prove a classical existence and uniqueness theorem for an integral equation. In Chapter IV we consider the approximation on certain functions by means of elementary expressions involving "bent line" functions. The last chapter, Chapter V, is the proof of the theorem by Peano mentioned above. Also included in this chapter is an example in which the integral equation has more than one solution. The first chapter sets forth basic definitions and theorems with which the reader should be acquainted.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc503874/

### An Exploration of the Word2vec Algorithm: Creating a Vector Representation of a Language Vocabulary that Encodes Meaning and Usage Patterns in the Vector Space Structure

**Date:**May 2016

**Creator:**Le, Thu Anh

**Description:**This thesis is an exloration and exposition of a highly efficient shallow neural network algorithm called word2vec, which was developed by T. Mikolov et al. in order to create vector representations of a language vocabulary such that information about the meaning and usage of the vocabulary words is encoded in the vector space structure. Chapter 1 introduces natural language processing, vector representations of language vocabularies, and the word2vec algorithm. Chapter 2 reviews the basic mathematical theory of deterministic convex optimization. Chapter 3 provides background on some concepts from computer science that are used in the word2vec algorithm: Huffman trees, neural networks, and binary cross-entropy. Chapter 4 provides a detailed discussion of the word2vec algorithm itself and includes a discussion of continuous bag of words, skip-gram, hierarchical softmax, and negative sampling. Finally, Chapter 5 explores some applications of vector representations: word categorization, analogy completion, and language translation assistance.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc849728/

### The Exponential Functions

**Date:**1947

**Creator:**Sloan, Robert S.

**Description:**The purpose of this paper is to clarify the problems of exponents by introducing notation not customarily used and by demonstrating certain theorems in regard to the properties of the exponential functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc75644/

### Extensions of Modules

**Date:**August 1967

**Creator:**Chen, Paulina Tsui-Chu

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

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130830/

### Field Extensions and Galois Theory

**Date:**August 1967

**Creator:**Votaw, Charles I.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130881/

### Finite Difference Methods for Approximating Solutions to the Heat Equation

**Date:**August 1982

**Creator:**Neuberger, Barbara O. (Barbara Osher)

**Description:**This paper is concerned with finite difference methods for approximating solutions to the partial differential heat equation. The first chapel gives some introductory background into the physical problem, then motivates three finite difference methods. Chapters II through IV provide statements and proofs for the theorems used in the methods of Chapter I. The final Chapter, V, provides conclusions and an indication of future work. An appendix includes the computer codes written by the author with numerical results.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc504382/

### Finite Dimensional Vector Space

**Date:**August 1960

**Creator:**Power, Billy Joe

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108107/

### Fourier Transforms of Functions on a Finite Abelian Group

**Date:**August 1982

**Creator:**Currey, Bradley Norton

**Description:**This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform of functions on the group Z/n of integers module n leads to a proof of the quadratic reciprocity law in Chapter IV. Chapter V addresses the, use of the Fourier transform in computing.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc503871/

### T-Functions

**Date:**June 1960

**Creator:**Barlow, John Rice

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108077/

### The Fundamental Group of Certain Toplogical Spaces

**Date:**December 1971

**Creator:**Hopkins, Billy L.

**Description:**The problem confronted in this thesis is that of determining direct calculations of the fundamental group of certain topological spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131469/

### Fundamental Issues in Support Vector Machines

**Date:**May 2014

**Creator:**McWhorter, Samuel P.

**Description:**This dissertation considers certain issues in support vector machines (SVMs), including a description of their construction, aspects of certain exponential kernels used in some SVMs, and a presentation of an algorithm that computes the necessary elements of their operation with proof of convergence. In its first section, this dissertation provides a reasonably complete description of SVMs and their theoretical basis, along with a few motivating examples and counterexamples. This section may be used as an accessible, stand-alone introduction to the subject of SVMs for the advanced undergraduate. Its second section provides a proof of the positive-definiteness of a certain useful function here called E and dened as follows: Let V be a complex inner product space. Let N be a function that maps a vector from V to its norm. Let p be a real number between 0 and 2 inclusive and for any in V , let ( be N() raised to the p-th power. Finally, let a be a positive real number. Then E() is exp(()). Although the result is not new (other proofs are known but involve deep properties of stochastic processes) this proof is accessible to advanced undergraduates with a decent grasp of linear algebra. Its ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc500155/

### Fundamental Properties of Fourier Series

**Date:**August 1954

**Creator:**Hubbard, Geogre U.

**Description:**This thesis is intended as an introduction to the study of one type of trigonometric series, the Fourier series.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130383/

### Fundamental Properties of the Contingent

**Date:**August 1960

**Creator:**Haggard, Paul W.

**Description:**This thesis explores the fundamental properties of the contingent.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108099/

### A Fundamental Study of Cardinal and Ordinal Numbers

**Date:**August 1966

**Creator:**Thornton, Robert Leslie

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130761/

### Fundamentals of Partially Ordered Sets

**Date:**August 1968

**Creator:**Compton, Lewis W.

**Description:**Gives the basic definitions and theorems of similar partially ordered sets; studies finite partially ordered sets, including the problem of combinatorial analysis; and includes the ideas of complete, dense, and continuous partially ordered sets, including proofs.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130984/