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**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### 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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### 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 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 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/

### 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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### 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/

### G-domains, G-ideals, and Hilbert Rings

**Date:**August 1972

**Creator:**Draper, Ruben P.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131539/

### A Generalization of Newton's Method

**Date:**1948

**Creator:**LeBouf, Billy Ruth

**Description:**It is our purpose here to investigate the method of solving equations for real roots by Newton's Method and to indicate a generalization arising from this method.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc75705/

### A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence

**Date:**August 1998

**Creator:**Risley, Rebecca N.

**Description:**We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.

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### A Generalization of the Weierstrass Approximation Theorem

**Date:**August 1972

**Creator:**Murchison, Jo Denton

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131551/

### Generalized Function Solutions to Nonlinear Wave Equations with Distribution Initial Data

**Date:**August 1996

**Creator:**Kim, Jongchul

**Description:**In this study, we consider the generalized function solutions to nonlinear wave equation with distribution initial data. J. F. Colombeau shows that the initial value problem u_tt - Δu = F(u); m(x,0) = U_0; u_t (x,0) = i_1 where the initial data u_0 and u_1 are generalized functions, has a unique generalized function solution u. Here we take a specific F and specific distributions u_0, u_1 then inspect the generalized function representatives for the initial value problem solution to see if the generalized function solution is a distribution or is more singular. Using the numerical technics, we show for specific F and specific distribution initial data u_0, u_1, there is no distribution solution.

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### A Generalized Study of the Conjugate and Inner-Product Functions

**Date:**June 1967

**Creator:**Wright, Dorothy P.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130818/

### Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups

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

**Date:**May 2006

**Creator:**Alhaddad, Shemsi I.

**Description:**The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5235/

### A Genesis for Compact Convex Sets

**Date:**May 1969

**Creator:**Ferguson, Ronald D.

**Description:**This paper was written in response to the following question: what conditions are sufficient to guarantee that if a compact subset A of a topological linear space L^3 is not convex, then for every point x belonging to the complement of A relative to the convex hull of A there exists a line segment yz such that x belongs to yz and y belongs to A and z belongs to A? Restated in the terminology of this paper the question bay be given as follow: what conditions may be imposed upon a compact subset A of L^3 to insure that A is braced?

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131089/

### Gibbs/Equilibrium Measures for Functions of Multidimensional Shifts with Countable Alphabets

**Date:**May 2011

**Creator:**Muir, Stephen R.

**Description:**Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc68021/