## You limited your search to:

**Partner:**UNT Libraries

**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### On Sets and Functions in a Metric Space

**Date:**December 1971

**Creator:**Beeman, Anne L.

**Description:**The purpose of this thesis is to study some of the properties of metric spaces. An effort is made to show that many of the properties of a metric space are generalized properties of R, the set of real numbers, or Euclidean n--space, and are specific cases of the properties of a general topological space.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc131457/

### Lebesgue Linear Measure

**Date:**1940

**Creator:**Beeman, William Edwin

**Description:**This paper discusses the concept of a general definition of measure, and shows that the Lebesgue measure satisfies the requirements set forth for the ideal definition.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc75587/

### Compactness and Equivalent Notions

**Date:**August 1967

**Creator:**Bell, Wayne Charles

**Description:**One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc130821/

### Restricting Invariants and Arrangements of Finite Complex Reflection Groups

**Date:**August 2015

**Creator:**Berardinelli, Angela

**Description:**Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc804919/

### Dimensions in Random Constructions.

**Date:**May 2002

**Creator:**Berlinkov, Artemi

**Description:**We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc3160/

### Some Theorems and Product Spaces

**Date:**1957

**Creator:**Bethel, Edward Lee

**Description:**This thesis is a study of some axioms and theorems, and product spaces.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc107937/

### Metric Spaces

**Date:**1957

**Creator:**Bilyeu, Russell Gene

**Description:**This thesis covers fundamental properties of metric spaces, as well as completeness, compactness, and separability of metric spaces.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc107938/

### Algebraic Integers

**Date:**August 1969

**Creator:**Black, Alvin M.

**Description:**The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc131119/

### Means and Mean Value Theorems

**Date:**1951

**Creator:**Blummer, Raymond O.

**Description:**This study covers means, mean value theorems of the differential calculus, and mean value theorems of integral calculus.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc96987/

### Random Sampling

**Date:**January 1957

**Creator:**Booker, Aaron Hicks

**Description:**The purpose of this study is to show the use of random sampling in solving certain mathematical problems. The origin of random numbers to be used in sampling is discussed and methods of sampling from known distributions are then given together with an indication that the sampling procedures are unbiased.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc130442/

### 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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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), ...

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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 ...

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

**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.

**Contributing Partner:**UNT Libraries

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