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Polish Spaces and Analytic Sets
A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z โ> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.
Date:
August 1997
Creator:
Muller, Kimberly (Kimberly Orisja)
System:
The UNT Digital Library
Hyperspace Topologies
In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.
Date:
August 2001
Creator:
Freeman, Jeannette Broad
System:
The UNT Digital Library
Borel Determinacy and Metamathematics
Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ฯ). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.
Date:
December 2001
Creator:
Bryant, Ross
System:
The UNT Digital Library
Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis
A "complex" system typically has a relatively large number of dynamically interacting components and tends to exhibit emergent behavior that cannot be explained by analyzing each component separately. A biological neural network is one example of such a system. A multi-agent model of such a network is developed to study the relationships between a network's structure and its spike train output. Using this model, inferences are made about the synaptic structure of networks through cluster analysis of spike train summary statistics A complexity measure for the network structure is also presented which has a one-to-one correspondence with the standard time series complexity measure sample entropy.
Date:
May 2007
Creator:
Brooks, Evan
System:
The UNT Digital Library
Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World
Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the โdivineโ proportion is given.
Date:
August 2002
Creator:
Arthur, Christopher
System:
The UNT Digital Library
A Presentation of Current Research on Partitions of Lines and Space
We present the results from three papers concerning partitions of vector spaces V over the set R of reals and of the set of lines in V.
Date:
December 1999
Creator:
Nugen, Frederick T.
System:
The UNT Digital Library
The Use of the Power Method to Find Dominant Eigenvalues of Matrices
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 gives an overview of common extensions of the power method. The appendices contain BASIC versions of the power method and its modification.
Date:
July 1992
Creator:
Cavender, Terri A.
System:
The UNT Digital Library
Algebraic Number Fields
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
System:
The UNT Digital Library
Conway's Link Polynomial: a Generalization of the Classic Alexander's Knot Polynomial
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 the Conway polynomial, the Axioms for Computation are given and many examples of their use are included. A major result of this investigation is the proof of topological invariance of these polynomials and the proof that the axioms are sufficient for the calculation of the knot polynomial for any given knot or link.
Date:
December 1986
Creator:
Woodard, Mary Kay
System:
The UNT Digital Library
Banach Spaces and Weak and Weak* Topologies
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 cโ and ๐แดฐ 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 in detail about dual spaces and weak and weak* topologies. An embedding proof and proofs involving weak and weak compactness are also given. The Cauchy-Bunyakowski-Schwarz inequality and Alaoglu's theorem are also proven.
Date:
August 1989
Creator:
Kirk, Andrew F. (Andrew Fitzgerald)
System:
The UNT Digital Library
A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema
In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function ๐ which is the uniform limit of a sequence of sawtooth functions ๐โ : [0, 1] โ [0, 1] with increasingly sharp teeth. Let ๐ = [0, 1] x [0, 1] and ๐น(๐) be the Hausdorff metric space determined by ๐. We define contraction maps ๐คโ , ๐คโ , ๐คโ on ๐. These maps define a contraction map ๐ค on ๐น(๐) via ๐ค(๐ด) = ๐คโ(๐ด) โ ๐คโ(๐ด) โ ๐คโ(๐ด). The iteration under ๐ค of the diagonal in ๐ defines a sequence of graphs of continuous functions ๐โ. Since ๐ค is a contraction map in the compact metric space ๐น(๐), ๐ค has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function ๐. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in ๐ถ[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set โฆ
Date:
December 1993
Creator:
Huggins, Mark C. (Mark Christopher)
System:
The UNT Digital Library
The Mean Integral
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)โซ^b_afdg exists then (m)โซ^b_ah(f)dg exists. 2) h is continuous.
Date:
December 1985
Creator:
Spear, Donald W.
System:
The UNT Digital Library
Hyperspaces
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.
Date:
December 1976
Creator:
Voas, Charles H.
System:
The UNT Digital Library
Haar Measure on the Cantor Ternary Set
The purpose of this thesis is to examine certain questions concerning the Cantor ternary set. The second chapter deals with proving that the Cantor ternary set is equivalent to the middle thirds set of [0,1], closed, compact, and has Lebesgue measure zero. Further a proof that the Cantor ternary set is a locally compact, Hausdorff topological group is given. The third chapter is concerned with establishing the existence of a Haar integral on certain topological groups. In particular if G is a locally compact and Hausdorff topological group, then there is a non-zero translation invariant positive linear form on G. The fourth chapter deals with proving that for any Haar integral I on G there exists a unique Haar measure on G that represents I.
Date:
August 1990
Creator:
Naughton, Gerard P. (Gerard Peter)
System:
The UNT Digital Library
Properties of Power Series Rings
This thesis investigates some of the properties of power series rings. The material is divided into three chapters. In Chapter I, some of the basic concepts of rings which are a prerequisite to an understanding of the definitions and theorems which follow are stated. Simple properties of power series rings are developed in Chapter II. Many properties of a ring R are preserved when we attach the indeterminant x to form the power series ring R[[x]]. Further results of power series rings are examined in Chapter III. An important result illustrated in this chapter is that power series rings possess some of the properties of rings of polynomials.
Date:
August 1990
Creator:
O'Brien, Rita Marie
System:
The UNT Digital Library
Integrability, Measurability, and Summability of Certain Set Functions
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.
Date:
December 1977
Creator:
Dawson, Dan Paul
System:
The UNT Digital Library
Valuations on Fields
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.
Date:
May 1977
Creator:
Walker, Catherine A.
System:
The UNT Digital Library
Properties of R-Modules
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 existence theorems for R-modules and properties of submodules in R-modules. The third and final chapter presents an example which illustrates how a ring R, may be regarded as an R-module and speaks of the direct sum of ideals of a ring as a direct sum of submodules.
Date:
August 1989
Creator:
Granger, Ginger Thibodeaux
System:
The UNT Digital Library
On the Development of Descriptive Set Theory
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 axiom of choice.
Date:
August 1988
Creator:
Schlee, Glen A. (Glen Alan)
System:
The UNT Digital Library
Duals and Reflexivity of Certain Banach Spaces
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 fourth chapter, several more theorems concerning dual spaces and related topologies are presented. The final chapter focuses on reflexive spaces. In the main theorem, the relation between compactness and reflexivity is examined. The paper concludes with an example of a non-reflexive space.
Date:
August 1991
Creator:
Dahler, Cheryl L. (Cheryl Lewis)
System:
The UNT Digital Library
The Eulerian Functions of Cyclic Groups, Dihedral Groups, and P-Groups
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)
System:
The UNT Digital Library
Dimension Theory
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.
Date:
August 1986
Creator:
Frere, Scot M. (Scot Martin)
System:
The UNT Digital Library
Weak and Norm Convergence of Sequences in Banach Spaces
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 โยน they coincide. We then show a theorem of Rosenthal's which states that if {๐โ} is a bounded sequence in a Banach space, then {๐โ} has a subsequence {๐'โ} satisfying one of the following two mutually exclusive alternatives; (i) {๐'โ} is weakly Cauchy, or (ii) {๐'โ} is equivalent to the unit vector basis of โยน.
Date:
December 1993
Creator:
Hymel, Arthur J. (Arthur Joseph)
System:
The UNT Digital Library
On Groups of Positive Type
We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.
Date:
August 1995
Creator:
Moore, Monty L.
System:
The UNT Digital Library