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**Partner:**UNT Libraries

**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### The Riemann-Complete Integral

**Date:**May 1972

**Creator:**Boyd, Eddie

**Description:**The problem with which this paper is concerned is that of defining the Riemann-Complete Integral and comparing it with the Riemann and the Lebesgue Integrals.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131504/

### Some Results of Two Topological Spaces

**Date:**August 1955

**Creator:**Boyd, James Robert

**Description:**This thesis explores some results of two topological spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130437/

### The Study of Translation Equivalence on Integer Lattices

**Date:**August 2003

**Creator:**Boykin, Charles Martin

**Description:**This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc4345/

### Weakly Dense Subsets of Homogeneous Complete Boolean Algebras

**Date:**August 1990

**Creator:**Bozeman, Alan Kyle

**Description:**The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B. Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity. In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B). In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B\{0} such that {x∗b: x ∈ X} is dense in B|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras. In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc330803/

### Certain Properties of Functions Related to Exhaustibility

**Date:**1952

**Creator:**Bradford, James C.

**Description:**In this thesis, we shall attempt to present a study of certain properties of real functions related to the set property exhaustible.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107821/

### A Study of Functions on Metric Spaces

**Date:**January 1968

**Creator:**Brice, Richard S.

**Description:**This thesis describes various forms of metric spaces and establishes some of the properties of functions defined on metric spaces. No attempt is made in this paper to examine a particular type of function in detail. Instead, some of properties of several kinds of functions will be observed as the functions are defined on various forms of metric spaces such as connected spaces, compact spaces, complete spaces, etc.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130890/

### Abstract Measure

**Date:**1957

**Creator:**Bridges, Robert Miller

**Description:**This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107950/

### Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis

**Date:**May 2007

**Creator:**Brooks, Evan

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3702/

### 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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**Permallink:**digital.library.unt.edu/ark:/67531/metadc332102/

### The Structure of a Boolean Algebra

**Date:**August 1965

**Creator:**Bryant, June Anne

**Description:**The purpose of this chapter is to develop a form of a "free" Boolean algebra with Σ as a base, by imposing the usual Boolean operations on the set Σ and thus generating new elements freely within explicitly prescribed restrictions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130610/

### Borel Determinacy and Metamathematics

**Date:**December 2001

**Creator:**Bryant, Ross

**Description:**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 ω). 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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3061/

### A Computation of Partial Isomorphism Rank on Ordinal Structures

**Date:**August 2006

**Creator:**Bryant, Ross

**Description:**We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an Ehrenfeucht-Fraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5387/

### Some Properties of Negligible Sets

**Date:**1948

**Creator:**Butts, Hubert S.

**Description:**In the study of sets of points certain sets are found to be negligible, especially when applied to the theory of functions. The purpose of this paper is to discuss three of these "negligible" types, namely, exhaustible sets, denumerable sets, and sets of Lebesgue measure zero. We will present a complete existential theory in q-space for the three set properties mentioned above, followed by a more restricted discussion in the linear continuum by use of interval properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc83358/

### Multifractal Analysis of Parabolic Rational Maps

**Date:**August 1998

**Creator:**Byrne, Jesse William

**Description:**The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies sup φ < P(φ) x∈J(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278398/

### Quadratic Forms

**Date:**June 1959

**Creator:**Cadenhead, Clarence Tandy

**Description:**This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with n-ary quadratic polynomials, called forms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108034/

### Polynomial Curve and Surface Fitting

**Date:**January 1968

**Creator:**Capps, Ann Dowdy

**Description:**The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130895/

### Exhaustibility and Related Set Properties

**Date:**1950

**Creator:**Cargal, Buchanan

**Description:**The purpose of this paper is to develop certain fundamental properties of exhaustible sets and their complements and to examine various set properties which are generalizations, with respect to exhaustible neglect, or well-known set properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130236/

### The Wave Equation in One Dimension

**Date:**January 1961

**Creator:**Carlson, Kenneth Emil

**Description:**It is intended that this paper present an acceptable proof of the existence of a solution for the wave equation.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108114/

### Linear Spaces

**Date:**August 1960

**Creator:**Carroll, Nelva Dain

**Description:**The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108094/

### Topological Groups

**Date:**May 1960

**Creator:**Carry, Laroy Ray

**Description:**The notion of a topological group follows naturally from a combination of the properties of a group and a topological space. Since a group consists of a set G of elements which may be either finite or infinite and since this is also common to a topological space, a question is opened as to whether or not it is possible to assign a topology to a set of elements which form a group under a certain operation. Now it is possible to assign a topology to any set of elements if no restriction is placed on the topology assigned and hence this study would be of little value from the standpoint of the group itself. If however it is required that the group operation be continuous in the topological space then a very interesting theory is developed.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108079/

### Completeness Axioms in an Ordered Field

**Date:**December 1971

**Creator:**Carter, Louis Marie

**Description:**The purpose of this paper was to prove the equivalence of the following completeness axioms. This purpose was carried out by first defining an ordered field and developing some basic theorems relative to it, then proving that lim [(u+u)*]^n = z (where u is the multiplicative identity, z is the additive identity, and * indicates the multiplicative inverse of an element), and finally proving the equivalence of the five axioms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131462/

### Random Variables of One Dimension

**Date:**August 1958

**Creator:**Casler, Burtis Griffin

**Description:**This thesis examines random variables of one dimension.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130444/

### Properties of Extended and Contracted Ideals

**Date:**August 1966

**Creator:**Chan, George

**Description:**This paper presents an introduction to the theory of ideals in a ring with emphasis on ideals in a commutative ring with identity.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130706/

### The Relative Complexity of Various Classification Problems among Compact Metric Spaces

**Date:**May 2016

**Creator:**Chang, Cheng

**Description:**In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc849626/