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**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### 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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### 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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### 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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### 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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### Polynomial Isomorphisms of Cayley Objects Over a Finite Field

**Date:**December 1989

**Creator:**Park, Hong Goo

**Description:**In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.

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### Hausdorff, Packing and Capacity Dimensions

**Date:**August 1989

**Creator:**Spear, Donald W.

**Description:**In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation. A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that ...

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### The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors

**Date:**August 1989

**Creator:**Hipp, James W. (James William), 1956-

**Description:**We show that the maximum size of a geometry of rank n excluding the (q + 2)-point line, the 3-wheel W_3, and the 3-whirl W^3 as minor is (n - 1)q + 1, and geometries of maximum size are parallel connections of (q + 1)-point lines. We show that the maximum size of a geometry of rank n excluding the 5-point line, the 4-wheel W_4, and the 4-whirl W^4 as minors is 6n - 5, for n ≥ 3. Examples of geometries having rank n and size 6n - 5 include parallel connections of the geometries V_19 and PG(2,3).

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### Minimization of a Nonlinear Elasticity Functional Using Steepest Descent

**Date:**August 1988

**Creator:**McCabe, Terence W. (Terence William)

**Description:**The method of steepest descent is used to minimize typical functionals from elasticity.

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### Bounded, Finitely Additive, but Not Absolutely Continuous Set Functions

**Date:**May 1989

**Creator:**Gurney, David R. (David Robert)

**Description:**In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters.

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### Applications of Graph Theory and Topology to Combinatorial Designs

**Date:**December 1988

**Creator:**Somporn Sutinuntopas

**Description:**This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, ...

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### Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension

**Date:**May 1988

**Creator:**Brucks, Karen M. (Karen Marie), 1957-

**Description:**This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function, f_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-e,1]. We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps. Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, as counted by P.J. Myrberg, is shown to equal the number of MSS sequences of length n. For an MSS sequence P, let H_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the ...

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### Operators on Continuous Function Spaces and Weak Precompactness

**Date:**August 1988

**Creator:**Abbott, Catherine Ann

**Description:**If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly ...

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