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**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### Uniqueness Results for the Infinite Unitary, Orthogonal and Associated Groups

**Date:**May 2008

**Creator:**Atim, Alexandru Gabriel

**Description:**Let H be a separable infinite dimensional complex Hilbert space, let U(H) be the Polish topological group of unitary operators on H, let G be a Polish topological group and φ:G→U(H) an algebraic isomorphism. Then φ is a topological isomorphism. The same theorem holds for the projective unitary group, for the group of *-automorphisms of L(H) and for the complex isometry group. If H is a separable real Hilbert space with dim(H)≥3, the theorem is also true for the orthogonal group O(H), for the projective orthogonal group and for the real isometry group. The theorem fails for U(H) if H is finite dimensional complex Hilbert space.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc6136/

### Around the Fibonacci Numeration System

**Date:**May 2007

**Creator:**Edson, Marcia Ruth

**Description:**Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3676/

### Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis

**Date:**May 2007

**Creator:**Brooks, Evan

**Description:**A "complex" system typically has a relatively large number of dynamically interacting components and tends to exhibit emergent behavior that cannot be explained by analyzing each component separately. A biological neural network is one example of such a system. A multi-agent model of such a network is developed to study the relationships between a network's structure and its spike train output. Using this model, inferences are made about the synaptic structure of networks through cluster analysis of spike train summary statistics A complexity measure for the network structure is also presented which has a one-to-one correspondence with the standard time series complexity measure sample entropy.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3702/

### Compact Operators and the Schrödinger Equation

**Date:**December 2006

**Creator:**Kazemi, Parimah

**Description:**In this thesis I look at the theory of compact operators in a general Hilbert space, as well as the inverse of the Hamiltonian operator in the specific case of L2[a,b]. I show that this inverse is a compact, positive, and bounded linear operator. Also the eigenfunctions of this operator form a basis for the space of continuous functions as a subspace of L2[a,b]. A numerical method is proposed to solve for these eigenfunctions when the Hamiltonian is considered as an operator on Rn. The paper finishes with a discussion of examples of Schrödinger equations and the solutions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5453/

### A characterization of homeomorphic Bernoulli trial measures.

**Date:**August 2006

**Creator:**Yingst, Andrew Q.

**Description:**We give conditions which, given two Bernoulli trial measures, determine whether there exists a homeomorphism of Cantor space which sends one measure to the other, answering a question of Oxtoby. We then provide examples, relating these results to the notions of good and refinable measures on Cantor space.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5331/

### Characterizations of Continua of Finite Degree

**Date:**August 2006

**Creator:**Irwin, Shana

**Description:**In this thesis, some characterizations of continua of finite degree are given. It turns out that being of finite degree (by formal definition) can be described by saying there exists an equivalent metric in which Hausdorff linear measure of the continuum is finite. I discuss this result in detail.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5367/

### A Computation of Partial Isomorphism Rank on Ordinal Structures

**Date:**August 2006

**Creator:**Bryant, Ross

**Description:**We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an Ehrenfeucht-Fraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5387/

### Hyperbolic Monge-Ampère Equation

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**Date:**August 2006

**Creator:**Howard, Tamani M.

**Description:**In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5322/

### Dimension spectrum and graph directed Markov systems.

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**Date:**May 2006

**Creator:**Ghenciu, Eugen Andrei

**Description:**In this dissertation we study graph directed Markov systems (GDMS) and limit sets associated with these systems. Given a GDMS S, by the Hausdorff dimension spectrum of S we mean the set of all positive real numbers which are the Hausdorff dimension of the limit set generated by a subsystem of S. We say that S has full Hausdorff dimension spectrum (full HD spectrum), if the dimension spectrum is the interval [0, h], where h is the Hausdorff dimension of the limit set of S. We give necessary conditions for a finitely primitive conformal GDMS to have full HD spectrum. A GDMS is said to be regular if the Hausdorff dimension of its limit set is also the zero of the topological pressure function. We show that every number in the Hausdorff dimension spectrum is the Hausdorff dimension of a regular subsystem. In the particular case of a conformal iterated function system we show that the Hausdorff dimension spectrum is compact. We introduce several new systems: the nearest integer GDMS, the Gauss-like continued fraction system, and the Renyi-like continued fraction system. We prove that these systems have full HD spectrum. A special attention is given to the backward continued fraction ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5226/

### Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups

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**Date:**May 2006

**Creator:**Alhaddad, Shemsi I.

**Description:**The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc5235/