### A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities

**Date:**May 2001

**Creator:**May, Russell J.

**Description:**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

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### Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems

**Date:**December 1998

**Creator:**Richardson, Peter A. (Peter Adolph), 1955-

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278917/

### Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains

**Date:**August 1998

**Creator:**Finan, Marcel Basil

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278251/

### Polish Spaces and Analytic Sets

**Date:**August 1997

**Creator:**Muller, Kimberly (Kimberly Orisja)

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc277605/

### Algebraically Determined Rings of Functions

**Date:**August 2010

**Creator:**McLinden, Alexander Patrick

**Description:**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 φ:H →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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc31543/

### Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World

**Date:**August 2002

**Creator:**Arthur, Christopher

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3269/

### Two Axiomatic Definitions of the Natural Numbers

**Date:**June 1970

**Creator:**Rhoads, Lana Sue

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131260/

### An Invariant Integral Over a Compact Topological Group

**Date:**August 1970

**Creator:**Nelson, John D.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131297/

### Primary Abelian Groups and Height

**Date:**June 1969

**Creator:**Ingram, Lana J.

**Description:**This thesis is a study of primary Abelian groups and height.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131098/

### Partially Ordered Groups and Rings

**Date:**August 1968

**Creator:**Lott, Kenneth L.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131010/

### Some Fundamental Properties of Valuations Defined on a Field

**Date:**January 1969

**Creator:**Doerr, James C.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131051/

### Integrals Defined on a Field of Sets

**Date:**August 1968

**Creator:**Troute, Grady W.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131034/

### Some Properties of Topological Spaces

**Date:**August 1968

**Creator:**Smith, Bayard M., Jr.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131028/

### A Genesis for Compact Convex Sets

**Date:**May 1969

**Creator:**Ferguson, Ronald D.

**Description:**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?

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### Existence and Uniqueness Theorems for Nth Order Linear and Nonlinear Integral Equations

**Date:**May 1969

**Creator:**Hurlbert, Gayle Jene Shultz

**Description:**The purpose of this paper is to study nth order integral equations. The integrals studied in this paper are of the Riemann type.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131097/

### Helly-Type Theorems

**Date:**August 1968

**Creator:**Davenport, Edward W.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130986/

### Some Fundamental Properties of Categories

**Date:**June 1968

**Creator:**Gardner, Harold L.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130942/

### Valuations and Places

**Date:**January 1970

**Creator:**Lo, Louis M. T.

**Description:**The purpose of this paper is devoted to investigating some fundamental properties of valuations and places.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131208/

### Convex Sets in the Plane

**Date:**June 1970

**Creator:**McPherson, Janie L.

**Description:**The purpose of this paper is to investigate some of the properties of convex sets in the plane through synthetic geometry.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131256/

### Polynomial Curve and Surface Fitting

**Date:**January 1968

**Creator:**Capps, Ann Dowdy

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130895/

### Development of a Geometry from a Set of Axioms

**Date:**May 1973

**Creator:**Glasscock, Anita Louise

**Description:**The purpose of this paper is to develop a geometry based on fourteen axioms and four undefined terms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131608/

### Lebesgue-Stieltjes Measure and Integration

**Date:**May 1973

**Creator:**Seale, Laura S.

**Description:**The purpose of the thesis is to investigate an approach to Lebesgue-Stieltjes measure and integration.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131612/

### Simplicial Homology

**Date:**August 1973

**Creator:**Chang, Chih-Chen

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131634/

### A*-algebras and Minimal Ideals in Topological Rings

**Date:**May 1973

**Creator:**Wei, Jui-Hung

**Description:**The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131622/