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Some Results of Two Topological Spaces
This thesis explores some results of two topological spaces.
Date:
August 1955
Creator:
Boyd, James Robert
System:
The UNT Digital Library
The Study of Translation Equivalence on Integer Lattices
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.
Date:
August 2003
Creator:
Boykin, Charles Martin
System:
The UNT Digital Library
Weakly Dense Subsets of Homogeneous Complete Boolean Algebras
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), …
Date:
August 1990
Creator:
Bozeman, Alan Kyle
System:
The UNT Digital Library
Certain Properties of Functions Related to Exhaustibility
In this thesis, we shall attempt to present a study of certain properties of real functions related to the set property exhaustible.
Date:
1952
Creator:
Bradford, James C.
System:
The UNT Digital Library
Some Properties of Metric Spaces
The study of metric spaces is closely related to the study of topology in that the study of metric spaces concerns itself, also, with sets of points and with a limit point concept based on a function which gives a "distance" between two points. In some topological spaces it is possible to define a distance function between points in such a way that a limit point of a set in the topological sense is also a limit point of the same set in a metric sense. In such a case the topological space is "metrizable". The real numbers with its usual topology is an example of a topological space which is metrizable, the distance function being the absolute value of the difference of two real numbers. Chapters II and III of this thesis attempt to classify, to a certain extent, what type of topological space is metrizable. Chapters IV and V deal with several properties of metric spaces and certain functions of metric spaces, respectively.
Date:
August 1964
Creator:
Brazile, Robert P.
System:
The UNT Digital Library
A Study of Functions on Metric Spaces
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.
Date:
January 1968
Creator:
Brice, Richard S.
System:
The UNT Digital Library
Abstract Measure
This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.
Date:
1957
Creator:
Bridges, Robert Miller
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
Prophet Inequalities for Multivariate Random Variables with Cost for Observations
In prophet problems, two players with different levels of information make decisions to optimize their return from an underlying optimal stopping problem. The player with more information is called the "prophet" while the player with less information is known as the "gambler." In this thesis, as in the majority of the literature on such problems, we assume that the prophet is omniscient, and the gambler does not know future outcomes when making his decisions. Certainly, the prophet will get a better return than the gambler. But how much better? The goal of a prophet problem is to find the least upper bound on the difference (or ratio) between the prophet's return, M, and the gambler's return, V. In this thesis, we present new prophet problems where we seek the least upper bound on M-V when there is a fixed cost per observations. Most prophet problems in the literature compare M and V when prophet and gambler buy (or sell) one asset. The new prophet problems presented in Chapters 3 and 4 treat a scenario where prophet and gambler optimize their return from selling two assets, when there is a fixed cost per observation. Sharp bounds for the problems on small …
Date:
August 2019
Creator:
Brophy, Edmond M.
System:
The UNT Digital Library
Continua and Related Topics
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 compact metric space is a continuous image of the Cantor set. A continuous map from the Cantor set onto [0,1] is constructed. Also, a continuous map from [0,1] onto [0,1]x[0,1] is built, Then an order preserving homeomorphism is constructed from a metric arc onto [0,1],
Date:
August 1982
Creator:
Brucks, Karen M. (Karen Marie), 1957-
System:
The UNT Digital Library
Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension
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 …
Date:
May 1988
Creator:
Brucks, Karen M. (Karen Marie), 1957-
System:
The UNT Digital Library
The Structure of a Boolean Algebra
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.
Date:
August 1965
Creator:
Bryant, June Anne
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 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.
Date:
December 2001
Creator:
Bryant, Ross
System:
The UNT Digital Library
A Computation of Partial Isomorphism Rank on Ordinal Structures
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.
Date:
August 2006
Creator:
Bryant, Ross
System:
The UNT Digital Library
The Torus Does Not Have a Hyperbolic Structure
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 automorphisms on the covering space is equivalent to the original surface.
Date:
August 1992
Creator:
Butler, Joe R.
System:
The UNT Digital Library
Some Properties of Negligible Sets
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.
Date:
1948
Creator:
Butts, Hubert S.
System:
The UNT Digital Library
Multifractal Analysis of Parabolic Rational Maps
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.
Date:
August 1998
Creator:
Byrne, Jesse William
System:
The UNT Digital Library
Quadratic Forms
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.
Date:
June 1959
Creator:
Cadenhead, Clarence Tandy
System:
The UNT Digital Library
Polynomial Curve and Surface Fitting
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.
Date:
January 1968
Creator:
Capps, Ann Dowdy
System:
The UNT Digital Library
Exhaustibility and Related Set Properties
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.
Date:
1950
Creator:
Cargal, Buchanan
System:
The UNT Digital Library
The Wave Equation in One Dimension
It is intended that this paper present an acceptable proof of the existence of a solution for the wave equation.
Date:
January 1961
Creator:
Carlson, Kenneth Emil
System:
The UNT Digital Library
Linear Spaces
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.
Date:
August 1960
Creator:
Carroll, Nelva Dain
System:
The UNT Digital Library
Topological Groups
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.
Date:
May 1960
Creator:
Carry, Laroy Ray
System:
The UNT Digital Library
Completeness Axioms in an Ordered Field
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.
Date:
December 1971
Creator:
Carter, Louis Marie
System:
The UNT Digital Library