### The Analytical Development of the Trigonometric Functions

**Date:**1951

**Creator:**Mackey, Pearl Cherrington

**Description:**This thesis is a study of the analytical development of the trigonometric functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc97030/

### Applications in Fixed Point Theory

**Date:**December 2005

**Creator:**Farmer, Matthew Ray

**Description:**Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc4971/

### An Approximate Solution to the Dirichlet Problem

**Date:**August 1964

**Creator:**Redwine, Edward William

**Description:**In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130550/

### 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/

### 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/

### Atmospheric Gusts and Their Effect on Aircraft

**Date:**August 1958

**Creator:**Walling, Waunnetta Keene

**Description:**This thesis investigates atmospheric gusts and their effect on aircraft.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108016/

### Automorphism Groups

**Date:**August 1965

**Creator:**Edwards, Donald Eugene

**Description:**This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130613/

### Basic Fourier Transforms

**Date:**January 1962

**Creator:**Cumbie, James Randolph

**Description:**The purpose of this paper is to develop some of the more basic Fourier transforms which are the outgrowth of the Fourier theorem. Although often approached from the stand-point of the series, this paper will approach the theorem from the standpoint of the integral.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130482/

### Borel Determinacy and Metamathematics

**Date:**December 2001

**Creator:**Bryant, Ross

**Description:**Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3061/

### Borel Sets and Baire Functions

**Date:**January 1970

**Creator:**Wemple, Fred W.

**Description:**This paper examines the relationship between Borel sets and Baire functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc163964/