### 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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### Hyperspace Topologies

**Date:**August 2001

**Creator:**Freeman, Jeannette Broad

**Description:**In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.

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### Ideals and Boolean Rings: Some Properties

**Date:**May 1968

**Creator:**Hu, Grace Min-Ying Chin

**Description:**The purpose of this thesis is to investigate certain properties of rings, ideals, and a special type of ring called a Boolean ring.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130947/

### Ideals in Quadratic Number Fields

**Date:**May 1971

**Creator:**Hamilton, James C.

**Description:**The purpose of this thesis is to investigate the properties of ideals in quadratic number fields, A field F is said to be an algebraic number field if F is a finite extension of R, the field of rational numbers. A field F is said to be a quadratic number field if F is an extension of degree 2 over R. The set 1 of integers of R will be called the rational integers.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131365/

### Ideals in Semigroups

**Date:**January 1965

**Creator:**Rodgers, Samuel A.

**Description:**This thesis investigates ideals in semigroups.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130573/

### Improper Integrals

**Date:**1957

**Creator:**Hildebrand, Shelby K.

**Description:**In this paper a definitions shall be given for different types of improper integrals and several theorems concerning them shall be proved.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107963/

### Inequalities and Set Function Integrals

**Date:**December 1971

**Creator:**Milligan, Kenneth Wayne

**Description:**This thesis investigates some inequalities and some relationships between function properties and integral properties.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131479/

### Infinite Matrices

**Date:**August 1957

**Creator:**Smallwood, James D.

**Description:**This paper will be mostly concerned with matrices of infinite order with elements which lie in Hilbert Space. All the properties of real and complex numbers and all the properties of infinite series and infinite sequences that are not listed will be assumed.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107976/

### Infinite Planar Graphs

**Date:**May 2000

**Creator:**Aurand, Eric William

**Description:**How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc2545/

### 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/