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**Partner:**UNT Libraries

**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### Mycielski-Regular Measures

**Date:**August 2011

**Creator:**Bass, Jeremiah Joseph

**Description:**Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc84171/

### A Set of Axioms for a Topological Space

**Date:**August 1960

**Creator:**Batcha, Joseph Patrick

**Description:**Axioms for a topological space are generally based on neighborhoods where "neighborhood" is an undefined term. Then, limit points are defined in terms of neighborhoods. However, limit points seem to be the basic concept of a topological space, rather than neighborhoods. For this reason, it will be attempted to state a set of axioms for a topological space, using limit point as the undefined concept, and to delete the idea of neighborhoods from the theory.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108089/

### The Development of the Natural Numbers by Means of the Peano Postulates

**Date:**1951

**Creator:**Baugh, Orvil Lee

**Description:**This thesis covers the development of the natural numbers by means of the peano postulates.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc96984/

### On Sets and Functions in a Metric Space

**Date:**December 1971

**Creator:**Beeman, Anne L.

**Description:**The purpose of this thesis is to study some of the properties of metric spaces. An effort is made to show that many of the properties of a metric space are generalized properties of R, the set of real numbers, or Euclidean n--space, and are specific cases of the properties of a general topological space.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131457/

### Lebesgue Linear Measure

**Date:**1940

**Creator:**Beeman, William Edwin

**Description:**This paper discusses the concept of a general definition of measure, and shows that the Lebesgue measure satisfies the requirements set forth for the ideal definition.

**Contributing Partner:**UNT Libraries

**Permallink:**digital.library.unt.edu/ark:/67531/metadc75587/

### Compactness and Equivalent Notions

**Date:**August 1967

**Creator:**Bell, Wayne Charles

**Description:**One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130821/

### Dimensions in random constructions.

**Date:**May 2002

**Creator:**Berlinkov, Artemi

**Description:**We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3160/

### Some Theorems and Product Spaces

**Date:**1957

**Creator:**Bethel, Edward Lee

**Description:**This thesis is a study of some axioms and theorems, and product spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107937/

### Metric Spaces

**Date:**1957

**Creator:**Bilyeu, Russell Gene

**Description:**This thesis covers fundamental properties of metric spaces, as well as completeness, compactness, and separability of metric spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107938/

### Algebraic Integers

**Date:**August 1969

**Creator:**Black, Alvin M.

**Description:**The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131119/