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**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### 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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### Some Properties of Metric Spaces

**Date:**August 1964

**Creator:**Brazile, Robert P.

**Description:**The study of metric spaces is closely related to the study of topology in that the study of metric spaces concerns itself, also, with sets of points and with a limit point concept based on a function which gives a "distance" between two points. In some topological spaces it is possible to define a distance function between points in such a way that a limit point of a set in the topological sense is also a limit point of the same set in a metric sense. In such a case the topological space is "metrizable". The real numbers with its usual topology is an example of a topological space which is metrizable, the distance function being the absolute value of the difference of two real numbers. Chapters II and III of this thesis attempt to classify, to a certain extent, what type of topological space is metrizable. Chapters IV and V deal with several properties of metric spaces and certain functions of metric spaces, respectively.

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### Inverse Limit Spaces

**Date:**December 1974

**Creator:**Williams, Stephen Boyd

**Description:**Inverse systems, inverse limit spaces, and bonding maps are defined. An investigation of the properties that an inverse limit space inherits, depending on the conditions placed on the factor spaces and bonding maps is made. Conditions necessary to ensure that the inverse limit space is compact, connected, locally connected, and semi-locally connected are examined. A mapping from one inverse system to another is defined and the nature of the function between the respective inverse limits, induced by this mapping, is investigated. Certain restrictions guarantee that the induced function is continuous, onto, monotone, periodic, or open. It is also shown that any compact metric space is the continuous image of the cantor set. Finally, any compact Hausdorff space is characterized as the inverse limit of an inverse system of polyhedra.

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### Properties of Some Classical Integral Domains

**Date:**May 1975

**Creator:**Crawford, Timothy B.

**Description:**Greatest common divisor domains, Bezout domains, valuation rings, and Prüfer domains are studied. Chapter One gives a brief introduction, statements of definitions, and statements of theorems without proof. In Chapter Two theorems about greatest common divisor domains and characterizations of Bezout domains, valuation rings, and Prüfer domains are proved. Also included are characterizations of a flat overring. Some of the results are that an integral domain is a Prüfer domain if and only if every overring is flat and that every overring of a Prüfer domain is a Prüfer domain.

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### 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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### Topics in Category Theory

**Date:**August 1974

**Creator:**Miller, Robert Patrick

**Description:**The purpose of this paper is to examine some basic topics in category theory. A category consists of a class of mathematical objects along with a morphism class having an associative composition. The paper is divided into two chapters. Chapter I deals with intrinsic properties of categories. Various "sub-objects" and properties of morphisms are defined and examples are given. Chapter II deals with morphisms between categories called functors and the natural transformations between functors. Special types of functors are defined and examples are given.

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### Spaces of Closed Subsets of a Topological Space

**Date:**August 1974

**Creator:**Leslie, Patricia J.

**Description:**The purpose of this paper is to examine selected topologies, the Vietoris topology in particular, on S(X), the collection of nonempty, closed subsets of a topological space X. Characteristics of open and closed subsets of S(X), with the Vietoris topology, are noted. The relationships between the space X and the space S(X), with the Vietoris topology, concerning the properties of countability, compactness, and connectedness and the separation properties are investigated. Additional topologies are defined on S(X), and each is compared to the Vietoris topology on S(X). Finally, topological convergence of nets of subsets of X is considered. It is found that topological convergence induces a topology on S(X), and that this topology is the Vietoris topology on S(X) when X is a compact, Hausdorff space.

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### Wiener's Approximation Theorem for Locally Compact Abelian Groups

**Date:**August 1974

**Creator:**Shu, Ven-shion

**Description:**This study of classical and modern harmonic analysis extends the classical Wiener's approximation theorem to locally compact abelian groups. The first chapter deals with harmonic analysis on the n-dimensional Euclidean space. Included in this chapter are some properties of functions in L1(Rn) and T1(Rn), the Wiener-Levy theorem, and Wiener's approximation theorem. The second chapter introduces the notion of standard function algebra, cospectrum, and Wiener algebra. An abstract form of Wiener's approximation theorem and its generalization is obtained. The third chapter introduces the dual group of a locally compact abelian group, defines the Fourier transform of functions in L1(G), and establishes several properties of functions in L1(G) and T1(G). Wiener's approximation theorem and its generalization for L1(G) is established.

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### Topologies on Complete Lattices

**Date:**December 1973

**Creator:**Dwyer, William Karl

**Description:**One of the more important concepts in mathematics is the concept of order, that is, the description or comparison of two elements of a set in terms of one preceding or being smaller than or equal to the other. If the elements of a set, as pairs, exhibit certain order-type characteristics, the set is said to be a partially ordered set. The purpose of this paper is to investigate a special class of partially ordered sets, called lattices, and to investigate topologies induced on these lattices by specially defined order related properties called order-convergence and star-convergence.

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### Duals and Weak Completeness in Certain Sequence Spaces

**Date:**August 1980

**Creator:**Leavelle, Tommy L. (Tommy Lee)

**Description:**In this paper the weak completeness of certain sequence spaces is examined. In particular, we show that each of the sequence spaces c0 and 9, 1 < p < c, is a Banach space. A Riesz representation for the dual space of each of these sequence spaces is given. A Riesz representation theorem for Hilbert space is also proven. In the third chapter we conclude that any reflexive space is weakly (sequentially) complete. We give 01 as an example of a non-reflexive space that is weakly complete. Two examples, c0 and YJ, are given of spaces that fail to be weakly complete.

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### Interpolation and Approximation

**Date:**May 1977

**Creator:**Lal, Ram

**Description:**In this paper, there are three chapters. The first chapter discusses interpolation. Here a theorem about the uniqueness of the solution to the general interpolation problem is proven. Then the problem of how to represent this unique solution is discussed. Finally, the error involved in the interpolation and the convergence of the interpolation process is developed. In the second chapter a theorem about the uniform approximation to continuous functions is proven. Then the best approximation and the least squares approximation (a special case of best approximation) is discussed. In the third chapter orthogonal polynomials as discussed as well as bounded linear functionals in Hilbert spaces, interpolation and approximation and approximation in Hilbert space.

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### Complete Ordered Fields

**Date:**August 1977

**Creator:**Arnold, Thompson Sharon

**Description:**The purpose of this thesis is to study the concept of completeness in an ordered field. Several conditions which are necessary and sufficient for completeness in an ordered field are examined. In Chapter I the definitions of a field and an ordered field are presented and several properties of fields and ordered fields are noted. Chapter II defines an Archimedean field and presents several conditions equivalent to the Archimedean property. Definitions of a complete ordered field (in terms of a least upper bound) and the set of real numbers are also stated. Chapter III presents eight conditions which are equivalent to completeness in an ordered field. These conditions include the concepts of nested intervals, Dedekind cuts, bounded monotonic sequences, convergent subsequences, open coverings, cluster points, Cauchy sequences, and continuous functions.

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### The Wallman Spaces and Compactifications

**Date:**December 1976

**Creator:**Liu, Wei-kong

**Description:**If X is a topological space and Y is a ring of closed sets, then a necessary and sufficient condition for the Wallman space W(X,F) to be a compactification of X is that X be T1 andYF separating. A necessary and sufficient condition for a Wallman compactification to be Hausdoff is that F be a normal base. As a result, not all T, compactifications can be of Wallman type. One point and finite Hausdorff compactifications are of Wallman type.

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### Valuations on Fields

**Date:**May 1977

**Creator:**Walker, Catherine A.

**Description:**This thesis investigates some properties of valuations on fields. Basic definitions and theorems assumed are stated in Capter I. Chapter II introduces the concept of a valuation on a field. Real valuations and non-Archimedean valuations are presented. Chapter III generalizes non-Archimedean valuations. Examples are described in Chapters I and II. A result is the theorem stating that a real valuation of a field K is non-Archimedean if and only if $(a+b) < max4# (a), (b) for all a and b in K. Chapter III generally defines a non-Archimedean valuation as an ordered abelian group. Real non-Archimedean valuations are either discrete or nondiscrete. Chapter III shows that every valuation ring identifies a non-Archimedean valuation and every non-Archimedean valuation identifies a valuation ring.

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### Subdirectly Irreducible Semigroups

**Date:**December 1978

**Creator:**Winton, Richard Alan

**Description:**Definition 1.1. The ordered pair (S,*) is a semi-group iff S is a set and * is an associative binary operation (multiplication) on S. Notation. A semigroup (S,*) will ordinarily be referred to by the set S, with the multiplication understood. In other words, if (a,b)e SX , then *[(a,b)] = a*b = ab. The proof of the following proposition is found on p. 4 of Introduction to Semigroups, by Mario Petrich. Proposition 1.2. Every semigroup S satisfies the general associative law.

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### Integrability, Measurability, and Summability of Certain Set Functions

**Date:**December 1977

**Creator:**Dawson, Dan Paul

**Description:**The purpose of this paper is to investigate the integrability, measurability, and summability of certain set functions. The paper is divided into four chapters. The first chapter contains basic definitions and preliminary remarks about set functions and absolute continuity. In Chapter i, the integrability of bounded set functions is investigated. The chapter culminates with a theorem that characterizes the transmission of the integrability of a real function of n bounded set functions. In Chapter III, measurability is defined and a characterization of the transmission of measurability by a function of n variables is provided, In Chapter IV, summability is defined and the summability of set functions is investigated, Included is a characterization of the transmission of summability by a function of n variables.

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### Hyperspaces

**Date:**December 1976

**Creator:**Voas, Charles H.

**Description:**This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.

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### 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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### Product Measure

**Date:**August 1983

**Creator:**Race, David M. (David Michael)

**Description:**In this paper we will present two different approaches to the development of product measures. In the second chapter we follow the lead of H. L. Royden in his book Real Analysis and develop product measure in the context of outer measure. The approach in the third and fourth chapters will be the one taken by N. Dunford and J. Schwartz in their book Linear Operators Part I. Specifically, in the fourth chapter, product measures arise almost entirely as a consequence of integration theory. Both developments culminate with proofs of well known theorems due to Fubini and Tonelli.

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### Dimension Theory

**Date:**August 1986

**Creator:**Frere, Scot M. (Scot Martin)

**Description:**This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.

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### The Mean Integral

**Date:**December 1985

**Creator:**Spear, Donald W.

**Description:**The purpose of this paper is to examine properties of the mean integral. The mean integral is compared with the regular integral. If [a;b] is an interval, f is quasicontinuous on [a;b] and g has bounded variation on [a;b], then the man integral of f with respect to g exists on [a;b]. The following theorem is proved. If [a*;b*] and [a;b] each is an interval and h is a function from [a*;b*] into R, then the following two statements are equivalent: 1) If f is a function from [a;b] into [a*;b*], gi is a function from [a;b] into R with bounded variation and (m)∫^b_afdg exists then (m)∫^b_ah(f)dg exists. 2) h is continuous.

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### Some Properties of Noetherian Rings

**Date:**May 1986

**Creator:**Vaughan, Stephen N. (Stephen Nick)

**Description:**This paper is an investigation of several basic properties of noetherian rings. Chapter I gives a brief introduction, statements of definitions, and statements of theorems without proof. Some of the main results in the study of noetherian rings are proved in Chapter II. These results include proofs of the equivalence of the maximal condition, the ascending chain condition, and that every ideal is finitely generated. Some other results are that if a ring R is noetherian, then R[x] is noetherian, and that if every prime ideal of a ring R is finitely generated, then R is noetherian.

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### 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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### Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models

**Date:**December 2013

**Creator:**Weng, Yu

**Description:**We consider the problem of maximum likelihood estimation of logistic sinusoidal regression models and develop some asymptotic theory including the consistency and joint rates of convergence for the maximum likelihood estimators. The key techniques build upon a synthesis of the results of Walker and Song and Li for the widely studied sinusoidal regression model and on making a connection to a result of Radchenko. Monte Carlo simulations are also presented to demonstrate the finite-sample performance of the estimators

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