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  Partner: UNT Libraries
 Degree Discipline: Mathematics
 Collection: UNT Theses and Dissertations
A Study of Functions on Metric Spaces

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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Abstract Measure

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

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

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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The Structure of a Boolean Algebra

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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Borel Determinacy and Metamathematics

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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A Computation of Partial Isomorphism Rank on Ordinal Structures

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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Some Properties of Negligible Sets

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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Multifractal Analysis of Parabolic Rational Maps

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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Quadratic Forms

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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