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Geometric Problems in Measure Theory and Parametrizations
This dissertation explores geometric measure theory; the first part explores a question posed by Paul Erdös -- Is there a number c > 0 such that if E is a Lebesgue measurable subset of the plane with λ²(E) (planar measure)> c, then E contains the vertices of a triangle with area equal to one? -- other related geometric questions that arise from the topic. In the second part, "we parametrize the theorems from general topology characterizing the continuous images and the homeomorphic images of the Cantor set, C" (abstract, para. 5).
Date:
August 1981
Creator:
Ingram, John M. (John Michael)
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
T-Sets of Normed Linear Spaces
This paper is a study of T-sets of normed linear spaces. Geometrical properties of normed linear spaces are developed in terms of intersection properties shared by a subcollection of T-sets of the space and in terms of special spanning properties shared by each T-set of a subcollection of T-sets of the space. A characterization of the extreme points of the unit ball of the dual of a normed linear space is given in terms of the T-sets of the space. Conditions on the collection of T-sets of a normed linear space are determined so that the normed linear space has the property that extreme points of the unit ball of the dual space map canonically to extreme points of the unit ball of the third dual space.
Date:
December 1976
Creator:
McCormick, Robert E.
System:
The UNT Digital Library
Convergence of Conditional Expectation Operators and the Compact Range Property
The interplay between generalizations of Riezs' famous representation theorem and Radon-Nikodým type theorems has a long history. This paper will explore certain aspects of the theory of bounded linear operators on continuous function spaces, Radon-Nikodým type properties, and their connections.
Date:
August 1992
Creator:
Dawson, C. Bryan (Charles Bryan)
System:
The UNT Digital Library
Existence and Multiplicity of Solutions for Semilinear Elliptic Boundary Value Problems
This thesis studies the existence, multiplicity, bifurcation and the stability of the solutions to semilinear elliptic boundary value problems. These problems are motivated both by the mathematical structure and the numerous applications in fluid mechanics chemical reactions, nuclear reactors, Riemannian geometry and elasticity theory. This study considers the problem for different classes of nonlinearities and obtain the existence and multiplicity of positive solutions.
Date:
August 1992
Creator:
Gadam, Sudhasree
System:
The UNT Digital Library
Concerning Integral Approximations of Bounded Finitely Additive Set Functions
The purpose of this paper is to generalize a theorem that characterizes absolute continuity of bounded finitely additive set functions in the form of an integral approximation. We show that his integral exists if the condition of absolute continuity is removed.
Date:
August 1992
Creator:
Dawson, Dan Paul
System:
The UNT Digital Library
Overrings of an Integral Domain
This dissertation focuses on the properties of a domain which has the property that each ideal is a finite intersection of a π-ideal, the properties of a domain which have the property that each ideal is a finite product of π-ideal, and the containment relations of the resulting classes of ideals. Chapter 1 states definitions which are needed in later chapters. Chapters 2 and 3 focuses on domains which have the property that each ideal in D is a finite intersection of π-ideals while Chapter 4 focuses on domains with the property that each ideal is a finite product of π-ideals. Chapter 5 discusses the containment relations which occur as a result of Chapters 2 and 3.
Date:
August 1992
Creator:
Emerson, Sharon Sue
System:
The UNT Digital Library
Steepest Descent for Partial Differential Equations of Mixed Type
The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u) = 1/2 II T(u) II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type). Finally, the graphical outputs on some test boundary conditions are presented in the table of illustrations.
Date:
August 1992
Creator:
Kim, Keehwan
System:
The UNT Digital Library
A Comparison of the Group and the Individual Techniques of Teaching Algebra
This thesis presents findings made during an examination and comparison of the individualized unit and traditional group methods used to teach ninth grade algebra in Hillsboro, Texas.
Date:
August 1940
Creator:
Knox, Beulah
System:
The UNT Digital Library
The Steepest Descent Method Using Finite Elements for Systems of Nonlinear Partial Differential Equations
The purpose of this paper is to develop a general method for using Finite Elements in the Steepest Descent Method. The main application is to a partial differential equation for a Transonic Flow Problem. It is also applied to Burger's equation, Laplace's equation and the minimal surface equation. The entire method is tested by computer runs which give satisfactory results. The validity of certain of the procedures used are proved theoretically. The way that the writer handles finite elements is quite different from traditional finite element methods. The variational principle is not needed. The theory is based upon the calculation of a matrix representation of operators in the gradient of a certain functional. Systematic use is made of local interpolation functions.
Date:
August 1981
Creator:
Liaw, Mou-yung Morris
System:
The UNT Digital Library
Bounded, Finitely Additive, but Not Absolutely Continuous Set Functions
In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters.
Date:
May 1989
Creator:
Gurney, David R. (David Robert)
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 “divine” proportion is given.
Date:
August 2002
Creator:
Arthur, Christopher
System:
The UNT Digital Library
To Develop and to Evaluate a Mathematics Curriculum for the Mentally Retarded on the Junior High School Level
This thesis presents the results of a study conducted to determine the math capabilities of mentally challenged students attending Reagan Junior High School in Wichita Falls, Texas. Survey results are used to determine needed curriculum changes.
Date:
August 1940
Creator:
Miller, Zola Catheryn
System:
The UNT Digital Library
Examples and Applications of Infinite Iterated Function Systems
The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.
Date:
August 2000
Creator:
Hanus, Pawel Grzegorz
System:
The UNT Digital Library
Maximum-Sized Matroids with no Minors Isomorphic to U2,5, F7, F7¯, OR P7
Let M be the class of simple matroids which do not contain the 5-point line U2,5 , the Fano plane F7 , the non-Fano plane F7- , or the matroid P7 , as minors. Let h(n) be the maximum number of points in a rank-n matroid in M. We show that h(2)=4, h(3)=7, and h(n)=n(n+1)/2 for n>3, and we also find all the maximum-sized matroids for each rank.
Date:
May 2000
Creator:
Mecay, Stefan Terence
System:
The UNT Digital Library
The Computation of Ultrapowers by Supercompactness Measures
The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.
Date:
August 1999
Creator:
Smith, John C.
System:
The UNT Digital Library
On the Cohomology of the Complement of a Toral Arrangement
The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.
Date:
August 1999
Creator:
Sawyer, Cameron Cunningham
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
Infinite Planar Graphs
How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.
Date:
May 2000
Creator:
Aurand, Eric William
System:
The UNT Digital Library
Maximum Likelihood Estimation of Logistic Sinusoidal Regression Models
We consider the problem of maximum likelihood estimation of logistic sinusoidal regression models and develop some asymptotic theory including the consistency and joint rates of convergence for the maximum likelihood estimators. The key techniques build upon a synthesis of the results of Walker and Song and Li for the widely studied sinusoidal regression model and on making a connection to a result of Radchenko. Monte Carlo simulations are also presented to demonstrate the finite-sample performance of the estimators
Date:
December 2013
Creator:
Weng, Yu
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
Tauberian Theorems for Certain Regular Processes
In 1943 R. C. Buck showed that a sequence x is convergent if some regular matrix sums every subsequence of x. Thus, for example, if every subsequence of x is Cesaro summable, then x is actually convergent. Buck's result was quite surprising, since research in summability theory up to that time gave no hint of such a remarkable theorem. The appearance of Buck's result in the Bulletin of the American Mathematical Society (3) created immediate interest and has prompted considerable research which has taken the following directions: (i) to study regular matrix transformations in order to shed light on Buck's theorem, (ii) to extend Buck's theorem, (iii) to obtain analogs of Buck's theorem for sequence spaces other than the space of convergent sequences, and (iv) to obtain analogs of Buck's theorem involving processes other than subsequencing, such as stretching. The purpose of the present paper is to contribute to all facets of the problem, particularly to (i), (iii), and (iv).
Date:
August 1975
Creator:
Keagy, Thomas A.
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