UNT Libraries - Browse

A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities
Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1
Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems
In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.
Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains
The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.
Polish Spaces and Analytic Sets
A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.
Algebraically Determined Rings of Functions
Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and &#966;:H &#8594;R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.
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.
Two Axiomatic Definitions of the Natural Numbers
The purpose of this thesis is to present an axiomatic foundation for the development of the natural numbers from two points of view. It makes no claim at originality other than at the point of organization and presentation of previously developed works.
An Invariant Integral Over a Compact Topological Group
The purpose of this paper is to develop an invariant integral for a compact topological group and, then to use that integral to prove the fundamental Peter-Weyl Theorem.
Primary Abelian Groups and Height
This thesis is a study of primary Abelian groups and height.
Partially Ordered Groups and Rings
This report presents both the most essential known results and new results in the theory of partially ordered groups and rings. This report deals with partially ordered groups and rings in an algebraic aspect because it is more important than partially ordered, fully ordered and lattice-ordered semigroup theory.
Some Fundamental Properties of Valuations Defined on a Field
The purpose of this thesis is to develop some properties of a special class of functions called valuations. The study begins with and examination of the properties of valuations defined on an arbitrary field, F, and later, consideration is given to valuations defined on the field of rational numbers. The concept of a pseud-valuation is introduced and an investigation is made of the properties of pseudo-valuations.
Integrals Defined on a Field of Sets
The purpose of this paper is to define an integral for real-valued functions which are defined on a field of sets and to demonstrate several properties of such an integral.
Some Properties of Topological Spaces
This thesis presents a development of some useful concepts concerning topological spaces. Most of the theorems given apply to the most general form of topological space.
A Genesis for Compact Convex Sets
This paper was written in response to the following question: what conditions are sufficient to guarantee that if a compact subset A of a topological linear space L^3 is not convex, then for every point x belonging to the complement of A relative to the convex hull of A there exists a line segment yz such that x belongs to yz and y belongs to A and z belongs to A? Restated in the terminology of this paper the question bay be given as follow: what conditions may be imposed upon a compact subset A of L^3 to insure that A is braced?
Existence and Uniqueness Theorems for Nth Order Linear and Nonlinear Integral Equations
The purpose of this paper is to study nth order integral equations. The integrals studied in this paper are of the Riemann type.
Helly-Type Theorems
The purpose of this paper is to present two proofs of Helly's Theorem and to use it in the proofs of several theorems classified in a group called Helly-type theorems.
Some Fundamental Properties of Categories
This paper establishes a basis for abelian categories, then gives the statement and proof of two equivalent definitions of an abelian category, the development of the basic theory of such categories, and the proof of some theorems involving this basic theory.
Valuations and Places
The purpose of this paper is devoted to investigating some fundamental properties of valuations and places.
Convex Sets in the Plane
The purpose of this paper is to investigate some of the properties of convex sets in the plane through synthetic geometry.
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.
Development of a Geometry from a Set of Axioms
The purpose of this paper is to develop a geometry based on fourteen axioms and four undefined terms.
Lebesgue-Stieltjes Measure and Integration
The purpose of the thesis is to investigate an approach to Lebesgue-Stieltjes measure and integration.
Simplicial Homology
The purpose of this thesis is to construct the homology groups of a complex over an R-module. The thesis begins with hyperplanes in Euclidean n-space. Simplexes and complexes are defined, and orientations are given to each simplex of a complex. The chains of a complex are defined, and each chain is assigned a boundary. The function which assigns to each chain a boundary defines the set of r-dimensional cycles and the set of r—dimensional bounding cycles. The quotient of those two submodules is the r-dimensional homology group.
A*-algebras and Minimal Ideals in Topological Rings
The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.
Some Properties of Commutative Rings Without a Unity
This thesis investigates some of the properties of commutative rings which do not necessarily contain a multiplicative identity (unity).
Set Function Integrals and Absolute Continuity
The purpose of this thesis is to investigate a theory of integration of real-valued functions defined on fields of sets.
Ideals in Quadratic Number Fields
The purpose of this thesis is to investigate the properties of ideals in quadratic number fields, A field F is said to be an algebraic number field if F is a finite extension of R, the field of rational numbers. A field F is said to be a quadratic number field if F is an extension of degree 2 over R. The set 1 of integers of R will be called the rational integers.
Product and Function Spaces
In this paper the Cartesian product topology for an arbitrary family of topological spaces and some of its basic properties are defined. The space is investigated to determine which of the separation properties of the component spaces are invariant.
Separation Properties
The problem with which this paper is concerned is that of investigating a class of topological properties commonly called separation properties. A topological space which satisfies only the definition may be very limited in open sets. By use of the separation properties, specific families of open sets can be guaranteed.
Radicals of a Ring
The problem with which this investigation is concerned is that of determining the properties of three radicals defined on an arbitrary ring and determining when these radicals coincide. The three radicals discussed are the nil radical, the Jacobsson radical, and the Brown-McCoy radical.
Some Properties of Ideals in a Commutative Ring
This thesis exhibits a collection of proofs of theorems on ideals in a commutative ring with and without a unity. Theorems treated involve properties of ideals under certain operations (sum, product, quotient, intersection, and union); properties of homomorphic mappings of ideals; contraction and extension theorems concerning ideals and quotient rings of domains with respect to multiplicative systems; properties of maximal, minimal, prime, semi-prime, and primary ideals; properties of radicals of ideals with relations to quotient rings, semi-prime, and primary ideals.
Properties of an Integral of E.J. McShane
The problem with which this paper is concerned is that of investigating the properties of an integral which was first defined by E. J. McShane in lecture notes presented at the Conference on Modern Theories of Integration, held at the University of Oklahoma in June, 1969.
Algebraic Properties of Semigroups
This paper is an algebraic study of selected properties of semigroups. Since a semigroup is a result of weakening the group axioms, all groups are semigroups. One facet of the paper is to demonstrate various semigroup properties that induce the group axioms.
Connectedness and Some Concepts Related to Connectedness of a Topological Space
The purpose of this thesis is to investigate the idea of topological "connectedness" by presenting some of the basic ideas concerning connectedness along with several related concepts.
Borel Sets and Baire Functions
This paper examines the relationship between Borel sets and Baire functions.
Regular Semigroups
This thesis describes semigroups and the properties of both regular and inverse semigroups.
Rings of Continuous Functions
The purpose of this paper is to examine properties of the ring C(X) of all complex or real-valued continuous functions on an arbitrary topological space X.
On Lane's Integral
The problem and purpose of this paper is to develop Lane's Integral in two-space, and then to expand these concepts into three-space and n-space. Lane's Integral can be used by both mathematicians and statisticians as one of the tools in the calculation of certain probabilities and expectations. The method of presentation is straightforward with the basic concepts of integration theory and Stieltjes Integral assumed.
The Fundamental Group of Certain Toplogical Spaces
The problem confronted in this thesis is that of determining direct calculations of the fundamental group of certain topological spaces.
Inequalities and Set Function Integrals
This thesis investigates some inequalities and some relationships between function properties and integral properties.
Some Properties of Valuation Rings
This thesis investigates some of the properties of valuation rings. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Unless otherwise stated, all rings considered in this thesis are commutative rings with a unity.
Algebraic Integers
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.
On Sets and Functions in a Metric Space
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.
Semitopological Groups
This thesis is a study of semitopological groups, a similar but weaker notion than that of topological groups. It is shown that all topological groups are semitopological groups but that the converse is not true. This thesis investigates some of the conditions under which semitopological groups are, in fact, topological groups. It is assumed that the reader is familiar with basic group theory and topology.
Uniform Locally Compact Spaces
The purpose of this paper is to develop some properties of uniformly locally compact spaces. The terminology and symbology used are the same as those used in General Topology, by J. L. Kelley.
Linear First-Order Differential-Difference Equations of Retarded Type with Constant Coefficients
This paper is concerned with equations in which all derivatives are ordinary rather than partial derivatives. The customary meanings of differential order and difference order of an equation are observed.
Peripherally Continuous Functions, Graph Maps and Connectivity Maps
The purpose of this paper is to investigate some of the more basic properties of peripherally continuous functions, graph maps and connectivity maps.
Extensions of Modules
This thesis discusses groups, modules, the module of homomorphisms, and extension of modules.
Some Properties of the Fibonacci Numbers
This thesis is presented as an introduction to the Fibonacci sequence of integers. It is hoped that this thesis will create in the reader more interest in this type of sequence and especially the Fibonacci sequence. It seems that this particular area of mathematics is often ignored in the classroom or touched upon far too briefly to stimulate curiosity and develop further interest in this field.
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.