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**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### On Steinhaus Sets, Orbit Trees and Universal Properties of Various Subgroups in the Permutation Group of Natural Numbers

**Date:**August 2012

**Creator:**Xuan, Mingzhi

**Description:**In the first chapter, we define Steinhaus set as a set that meets every isometric copy of another set at exactly one point. We show that there is no Steinhaus set for any four-point subset in a plane.In the second chapter, we define the orbit tree of a permutation group of natural numbers, and further introduce compressed orbit trees. We show that any rooted finite tree can be realized as a compressed orbit tree of some permutation group. In the third chapter, we investigate certain classes of closed permutation groups of natural numbers with respect to their universal and surjectively universal groups. We characterize two-sided invariant groups, and prove that there is no universal group for countable groups, nor universal group for two-sided invariant groups in permutation groups of natural numbers.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc149691/

### Hochschild Cohomology and Complex Reflection Groups

**Date:**August 2012

**Creator:**Foster-Greenwood, Briana A.

**Description:**A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc149591/

### Kleinian Groups in Hilbert Spaces

**Date:**August 2012

**Creator:**Das, Tushar

**Description:**The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and PoincarĂ©. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the PoincarĂ© exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc149579/

### 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/