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Axiom of Choice Equivalences and Some Applications

Description: In this paper several equivalences of the axiom of choice are examined. In particular, the axiom of choice, Zorn's lemma, Tukey's lemma, the Hausdorff maximal principle, and the well-ordering theorem are shown to be equivalent. Cardinal and ordinal number theory is also studied. The Schroder-Bernstein theorem is proven and used in establishing order results for cardinal numbers. It is also demonstrated that the first uncountable ordinal space is unique up to order isomorphism. We conclude by en… more
Date: August 1983
Creator: Race, Denise T. (Denise Tatsch)
Partner: UNT Libraries
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Homotopies and Deformation Retracts

Description: This paper introduces the background concepts necessary to develop a detailed proof of a theorem by Ralph H. Fox which states that two topological spaces are the same homotopy type if and only if both are deformation retracts of a third space, the mapping cylinder. The concepts of homotopy and deformation are introduced in chapter 2, and retraction and deformation retract are defined in chapter 3. Chapter 4 develops the idea of the mapping cylinder, and the proof is completed. Three special cas… more
Date: December 1990
Creator: Stark, William D. (William David)
Partner: UNT Libraries
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Duals and Weak Completeness in Certain Sequence Spaces

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 examp… more
Date: August 1980
Creator: Leavelle, Tommy L. (Tommy Lee)
Partner: UNT Libraries
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The Riesz Representation Theorem

Description: In 1909, F. Riesz succeeded in giving an integral represntation for continuous linear functionals on C[0,1]. Although other authors, notably Hadamard and Frechet, had given representations for continuous linear functionals on C[0,1], their results lacked the clarity, elegance, and some of the substance (uniqueness) of Riesz's theorem. Subsequently, the integral representation of continuous linear functionals has been known as the Riesz Representation Theorem. In this paper, three different pro… more
Date: August 1980
Creator: Williams, Stanley C. (Stanley Carl)
Partner: UNT Libraries
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The Wallman Spaces and Compactifications

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.
Date: December 1976
Creator: Liu, Wei-kong
Partner: UNT Libraries
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Continua and Related Topics

Description: This paper is a study of continue and related metric spaces, Chapter I is an introductory chapter. Irreducible continua and noncut points are the main topics in Chapter II. The third chapter begins with a few results on locally connected spaces. These results are then used to prove results in locally connected continua. Decomposable and indecomposable continua are dealt with in Chapter IV. Totally disconnected metric spaces are studied in the beginning of Chapter V. Then we see that every compa… more
Date: August 1982
Creator: Brucks, Karen M. (Karen Marie), 1957-
Partner: UNT Libraries
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The Use of the Power Method to Find Dominant Eigenvalues of Matrices

Description: This paper is the result of a study of the power method to find dominant eigenvalues of square matrices. It introduces ideas basic to the study and shows the development of the power method for the most well-behaved matrices possible, and it explores exactly which other types of matrices yield to the power method. The paper also discusses a type of matrix typically considered impossible for the power method, along with a modification of the power method which works for this type of matrix. It g… more
Date: July 1992
Creator: Cavender, Terri A.
Partner: UNT Libraries
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Algebraic Number Fields

Description: This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.
Date: August 1991
Creator: Hartsell, Melanie Lynne
Partner: UNT Libraries
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Tauberian Theorems for Certain Regular Processes

Description: In 1943 R. C. Buck showed that a sequence x is convergent if some regular matrix sums every subsequence of x. Thus, for example, if every subsequence of x is Cesaro summable, then x is actually convergent. Buck's result was quite surprising, since research in summability theory up to that time gave no hint of such a remarkable theorem. The appearance of Buck's result in the Bulletin of the American Mathematical Society (3) created immediate interest and has prompted considerable research which … more
Date: August 1975
Creator: Keagy, Thomas A.
Partner: UNT Libraries
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Conway's Link Polynomial: a Generalization of the Classic Alexander's Knot Polynomial

Description: The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define … more
Date: December 1986
Creator: Woodard, Mary Kay
Partner: UNT Libraries
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Banach Spaces and Weak and Weak* Topologies

Description: This paper examines several questions regarding Banach spaces, completeness and compactness of Banach spaces, dual spaces and weak and weak* topologies. Examples of completeness and isometries are given using the c₀ and 𝓁ᴰ spaces. The Hahn-Banach extension theorem is presented, along with some applications. General theory about finite and infinite dimensional normed linear spaces is the bulk of the second chapter. A proof of the uniform boundedness principle is also given. Chapter three talks i… more
Date: August 1989
Creator: Kirk, Andrew F. (Andrew Fitzgerald)
Partner: UNT Libraries
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A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema

Description: In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function 𝑓 which is the uniform limit of a sequence of sawtooth functions 𝑓ₙ : [0, 1] → [0, 1] with increasingly sharp teeth. Let 𝑋 = [0, 1] x [0, 1] and 𝐹(𝑋) be the Hausdorff metric space determined by 𝑋. We define contraction maps 𝑤₁ , 𝑤₂ , 𝑤₃ on 𝑋. These maps define a contraction map 𝑤 on 𝐹(𝑋) via 𝑤(𝐴) = 𝑤₁(𝐴) ⋃ 𝑤₂(𝐴) ⋃ 𝑤₃(𝐴). The iteration under 𝑤 of the diagonal in 𝑋 defines a sequence of graphs of… more
Date: December 1993
Creator: Huggins, Mark C. (Mark Christopher)
Partner: UNT Libraries
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The Mean Integral

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… more
Date: December 1985
Creator: Spear, Donald W.
Partner: UNT Libraries
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Generalized C-sets

Description: The problem undertaken in this paper is to determine what the algebraic structure of the class of C-sets is, when the notion of sum is to be the "set sum. " While the preliminary work done by Appling took place in the space of additive and bounded real valued functions, the results here are found in the more general setting of a complete lattice ordered group. As a conseque n c e , G . Birkhof f' s book, Lattice Theory, is used as the standard reference for most of the terminology used in the p… more
Date: August 1974
Creator: Keisler, D. Michael
Partner: UNT Libraries
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Properties of R-Modules

Description: This thesis investigates some of the properties of R-modules. The material is presented in three chapters. Definitions and theorems which are assumed are stated in Chapter I. Proofs of these theorems may be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1958. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Properties of R-modules are developed in Chapter II. The most important results presented in this chapter include exis… more
Date: August 1989
Creator: Granger, Ginger Thibodeaux
Partner: UNT Libraries
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On the Development of Descriptive Set Theory

Description: In the thesis, the author traces the historical development of descriptive set theory from the work of H. Lebesgue to the introduction of projective descriptive set theory. Proofs of most of the major results are given. Topics covered include Corel lattices, universal sets, the operation A, analytic sets, coanalytic sets, and the continuum hypothesis The appendix contains a translation of the famous letters exchanged between R. Baire, E. Borel, J. Hadamard and H. Lebesgue concerning Zermelo's a… more
Date: August 1988
Creator: Schlee, Glen A. (Glen Alan)
Partner: UNT Libraries
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Duals and Reflexivity of Certain Banach Spaces

Description: The purpose of this paper is to explore certain properties of Banach spaces. The first chapter begins with basic definitions, includes examples of Banach spaces, and concludes with some properties of continuous linear functionals. In the second chapter, dimension is discussed; then one version of the Hahn-Banach Theorem is presented. The third chapter focuses on dual spaces and includes an example using co, RI, and e'. The role of locally convex spaces is also explored in this chapter. In the f… more
Date: August 1991
Creator: Dahler, Cheryl L. (Cheryl Lewis)
Partner: UNT Libraries
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The Eulerian Functions of Cyclic Groups, Dihedral Groups, and P-Groups

Description: In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he defined a generalized Eulerian function. This thesis uses Hall's generalized Eulerian function to calculate generalized Eulerian functions for specific groups, namely: cyclic groups, dihedral groups, and p- groups.
Date: August 1992
Creator: Sewell, Cynthia M. (Cynthia Marie)
Partner: UNT Libraries
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Dimension Theory

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 … more
Date: August 1986
Creator: Frere, Scot M. (Scot Martin)
Partner: UNT Libraries
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Weak and Norm Convergence of Sequences in Banach Spaces

Description: We study weak convergence of sequences in Banach spaces. In particular, we compare the notions of weak and norm convergence. Although these modes of convergence usually differ, we show that in ℓ¹ they coincide. We then show a theorem of Rosenthal's which states that if {𝓍ₙ} is a bounded sequence in a Banach space, then {𝓍ₙ} has a subsequence {𝓍'ₙ} satisfying one of the following two mutually exclusive alternatives; (i) {𝓍'ₙ} is weakly Cauchy, or (ii) {𝓍'ₙ} is equivalent to the unit vector basis… more
Date: December 1993
Creator: Hymel, Arthur J. (Arthur Joseph)
Partner: UNT Libraries
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Some Properties of Noetherian Rings

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… more
Date: May 1986
Creator: Vaughan, Stephen N. (Stephen Nick)
Partner: UNT Libraries
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The Cohomology for the Nil Radical of a Complex Semisimple Lie Algebra

Description: Let g be a complex semisimple Lie algebra, Vλ an irreducible g-module with high weight λ, pI a standard parabolic subalgebra of g with Levi factor £I and nil radical nI, and H*(nI, Vλ) the cohomology group of Λn'I ⊗Vλ. We describe the decomposition of H*(nI, Vλ) into irreducible £1-modules.
Date: May 1994
Creator: Sawyer, Cameron C. (Cameron Cunningham)
Partner: UNT Libraries
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The Torus Does Not Have a Hyperbolic Structure

Description: Several basic topics from Algebraic Topology, including fundamental group and universal covering space are shown. The hyperbolic plane is defined, including its metric and show what the "straight" lines are in the plane and what the isometries are on the plane. A hyperbolic surface is defined, and shows that the two hole torus is a hyperbolic surface, the hyperbolic plane is a universal cover for any hyperbolic surface, and the quotient space of the universal cover of a surface to the group of… more
Date: August 1992
Creator: Butler, Joe R.
Partner: UNT Libraries
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Fundamental Issues in Support Vector Machines

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 … more
Date: May 2014
Creator: McWhorter, Samuel P.
Partner: UNT Libraries
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