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**Partner:**UNT Libraries

**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### 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.

**Contributing Partner:**UNT Libraries

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

### 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/

### Some Properties of Negligible Sets

**Date:**1948

**Creator:**Butts, Hubert S.

**Description:**In the study of sets of points certain sets are found to be negligible, especially when applied to the theory of functions. The purpose of this paper is to discuss three of these "negligible" types, namely, exhaustible sets, denumerable sets, and sets of Lebesgue measure zero. We will present a complete existential theory in q-space for the three set properties mentioned above, followed by a more restricted discussion in the linear continuum by use of interval properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc83358/

### Quadratic Forms

**Date:**June 1959

**Creator:**Cadenhead, Clarence Tandy

**Description:**This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with n-ary quadratic polynomials, called forms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108034/

### Polynomial Curve and Surface Fitting

**Date:**January 1968

**Creator:**Capps, Ann Dowdy

**Description:**The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130895/

### Exhaustibility and Related Set Properties

**Date:**1950

**Creator:**Cargal, Buchanan

**Description:**The purpose of this paper is to develop certain fundamental properties of exhaustible sets and their complements and to examine various set properties which are generalizations, with respect to exhaustible neglect, or well-known set properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130236/

### The Wave Equation in One Dimension

**Date:**January 1961

**Creator:**Carlson, Kenneth Emil

**Description:**It is intended that this paper present an acceptable proof of the existence of a solution for the wave equation.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108114/

### Linear Spaces

**Date:**August 1960

**Creator:**Carroll, Nelva Dain

**Description:**The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108094/

### Topological Groups

**Date:**May 1960

**Creator:**Carry, Laroy Ray

**Description:**The notion of a topological group follows naturally from a combination of the properties of a group and a topological space. Since a group consists of a set G of elements which may be either finite or infinite and since this is also common to a topological space, a question is opened as to whether or not it is possible to assign a topology to a set of elements which form a group under a certain operation. Now it is possible to assign a topology to any set of elements if no restriction is placed on the topology assigned and hence this study would be of little value from the standpoint of the group itself. If however it is required that the group operation be continuous in the topological space then a very interesting theory is developed.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108079/

### Completeness Axioms in an Ordered Field

**Date:**December 1971

**Creator:**Carter, Louis Marie

**Description:**The purpose of this paper was to prove the equivalence of the following completeness axioms. This purpose was carried out by first defining an ordered field and developing some basic theorems relative to it, then proving that lim [(u+u)*]^n = z (where u is the multiplicative identity, z is the additive identity, and * indicates the multiplicative inverse of an element), and finally proving the equivalence of the five axioms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131462/