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**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Degree Level:**Doctoral

**Collection:**UNT Theses and Dissertations

### Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank

**Date:**August 2013

**Creator:**Dahal, Rabin

**Description:**Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.

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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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### 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 S∞.

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### 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/

### 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/

### 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.

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

### A Topological Uniqueness Result for the Special Linear Groups

**Date:**August 1997

**Creator:**Opalecky, Robert Vincent

**Description:**The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.

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### Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices

**Date:**August 1999

**Creator:**Huff, Cheryl Rae

**Description:**The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and ...

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### Traveling Wave Solutions of the Porous Medium Equation

**Date:**May 2013

**Creator:**Paudel, Laxmi P.

**Description:**We prove the existence of a one-parameter family of solutions of the porous medium equation, a nonlinear heat equation. In our work, with space dimension 3, the interface is a half line whose end point advances at constant speed. We prove, by using maximum principle, that the solutions are stable under a suitable class of perturbations. We discuss the relevance of our solutions, when restricted to two dimensions, to gravity driven flows of thin films. Here we extend the results of J. Iaia and S. Betelu in the paper "Solutions of the porous medium equation with degenerate interfaces" to a higher dimension.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc271876/

### Graev Metrics and Isometry Groups of Polish Ultrametric Spaces

**Date:**May 2013

**Creator:**Shi, Xiaohui

**Description:**This dissertation presents results about computations of Graev metrics on free groups and characterizes isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces. In Chapter 2, computations of Graev metrics are performed on free groups. One of the related results answers an open question of Van Den Dries and Gao. In Chapter 3, isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces are characterized. The notion of generalized tree is defined and a correspondence between the isomorphism group of a generalized tree and the isometry group of a Heine-Borel Polish ultrametric space is established. The concept of a weak inverse limit is introduced to capture the characterization of isomorphism groups of generalized trees. In Chapter 4, partial results of isometry groups of uncountable compact ultrametric spaces are given. It turns out that every compact ultrametric space has a unique countable orbital decomposition. An orbital space consists of disjoint orbits. An orbit subspace of an orbital space is actually a compact homogeneous ultrametric subspace.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc271898/