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**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Decade:**1990-1999

**Collection:**UNT Theses and Dissertations

### 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.

**Contributing Partner:**UNT Libraries

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

### Minimality of the Special Linear Groups

**Date:**December 1997

**Creator:**Hayes, Diana Margaret

**Description:**Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of Sā.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc279280/

### Properties of Bicentric Circles for Three-Sided Polygons

**Date:**August 1998

**Creator:**Heinlein, David J. (David John)

**Description:**We define and construct bicentric circles with respect to three-sided polygons. Then using inherent properties of these circles, we explore both tangent properties, and areas generated from bicentric circles.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278727/

### Aspects of Universality in Function Iteration

**Date:**December 1991

**Creator:**Taylor, John (John Allen)

**Description:**This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278799/

### Topics in Fractal Geometry

**Date:**August 1994

**Creator:**Wang, JingLing

**Description:**In this dissertation, we study fractal sets and their properties, especially the open set condition, Hausdorff dimensions and Hausdorff measures for certain fractal constructions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc279332/

### Multifractal Measures

**Date:**May 1994

**Creator:**Olsen, Lars

**Description:**The purpose of this dissertation is to introduce a natural and unifying multifractal formalism which contains the above mentioned multifractal parameters, and gives interesting results for a large class of natural measures. In Part 2 we introduce the proposed multifractal formalism and study it properties. We also show that this multifractal formalism gives natural and interesting results when applied to (nonrandom) graph directed self-similar measures in Rd and "cookie-cutter" measures in R. In Part 3 we use the multifractal formalism introduced in Part 2 to give a detailed discussion of the multifractal structure of random (and hence, as a special case, non-random) graph directed self-similar measures in R^d.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc279084/

### 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 ā> 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/

### Physical Motivation and Methods of Solution of Classical Partial Differential Equations

**Date:**August 1995

**Creator:**Thompson, Jeremy R. (Jeremy Ray)

**Description:**We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc277898/

### On Groups of Positive Type

**Date:**August 1995

**Creator:**Moore, Monty L.

**Description:**We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc277804/