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**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Degree Level:**Doctoral

### 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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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278561/

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

### 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/

### 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/

### The Computation of Ultrapowers by Supercompactness Measures

**Date:**August 1999

**Creator:**Smith, John C.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc2201/

### 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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**Permallink:**digital.library.unt.edu/ark:/67531/metadc278330/

### On the Cohomology of the Complement of a Toral Arrangement

**Date:**August 1999

**Creator:**Sawyer, Cameron Cunningham

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc2198/

### Infinite Planar Graphs

**Date:**May 2000

**Creator:**Aurand, Eric William

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc2545/

### Maximum-Sized Matroids with no Minors Isomorphic to U2,5, F7, F7¯, OR P7

**Date:**May 2000

**Creator:**Mecay, Stefan Terence

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc2514/