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**Partner:**UNT Libraries

**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### A Classification of Regular Planar Graphs

**Date:**December 1972

**Creator:**McCalla, Linda F.

**Description:**The purpose of this paper is the investigation and classification of regular planar graphs. The motive behind this investigation was a desire to better understand those properties which allow a graph to be represented in the plane in such a manner that no two edges cross except perhaps at vertices.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc164029/

### Continuous Multifunctions

**Date:**August 1972

**Creator:**Rulon, Susan Ree

**Description:**This paper is a discussion of multifunctions, various types of continuity defined on multifunctions, and implications of continuity for the range and domain sets of the multifunctions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc164025/

### On the Stielitjes Integral

**Date:**August 1972

**Creator:**Keagy, Thomas A.

**Description:**This paper is a study of the Stieltjes integral, a generalization of the Riemann integral normally studied in introductory calculus courses. The purpose of the paper is to investigate many of the basic manipulative properties of the integral.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc164008/

### Completing the Space of Step Functions

**Date:**August 1972

**Creator:**Massey, Linda K.

**Description:**In this thesis a study is made of the space X of all step functions on [0,1]. This investigation includes determining a completion space, X*, for the incomplete space X, defining integration for X*, and proving some theorems about integration in X*.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc164014/

### Measure Functions

**Date:**August 1965

**Creator:**Ottwell, Otho F.

**Description:**This thesis examines measure functions. A measure function has as its domain of definition a class of sets. It also must satisfy a certain additive condition. To state a concise definition of a measure function, it is convenient to define set function and completely additive set function.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc163875/

### Elliptic Geometry

**Date:**January 1966

**Creator:**Robertson, Barbara McKinzie

**Description:**This thesis discusses elliptic geometry including the order and incidence properties, projective properties and congruence properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc163883/

### Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the Line

**Date:**August 2012

**Creator:**Larsen, Jeannette M.

**Description:**Certain subquotients of Vec(R)-modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)-representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc149627/

### 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/

### 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/

### 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/