### Abstract Measure

**Date:**1957

**Creator:**Bridges, Robert Miller

**Description:**This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107950/

### Abstract Vector Spaces and Certain Related Systems

**Date:**August 1961

**Creator:**Goddard, Alton Ray

**Description:**The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130465/

### Additive Functions

**Date:**June 1963

**Creator:**McNeir, Ridge W.

**Description:**The purpose of this paper is the analysis of functions of real numbers which have a special additive property, namely, f(x+y) = f(x)+f(y).

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108204/

### 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/

### Algebraic Properties of Semigroups

**Date:**May 1971

**Creator:**Lumley, Robert Don

**Description:**This paper is an algebraic study of selected properties of semigroups. Since a semigroup is a result of weakening the group axioms, all groups are semigroups. One facet of the paper is to demonstrate various semigroup properties that induce the group axioms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131373/

### Algebraically Determined Rings of Functions

**Date:**August 2010

**Creator:**McLinden, Alexander Patrick

**Description:**Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and φ:H →R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc31543/

### Algebraically Determined Semidirect Products

**Date:**May 2011

**Creator:**Jasim, We'am Muhammad

**Description:**Let G be a Polish group. We say that G is an algebraically determined Polish group if given any Polish group L and any algebraic isomorphism from L to G, then the algebraic isomorphism is a topological isomorphism. We will prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This will smooth the way for the proofs for some special groups. For example, let H be a separable Hilbert space and let G be a subset of the unitary group U(H) acting transitively on the unit sphere. Assume that -I in G and G is a Polish topological group in some topology such that H x G to H, (x,U) to U(x) is continuous, then H x G is a Polish topological group. Hence H x G is an algebraically determined Polish group. In addition, we apply the above the above result on the unitary group U(A) of a separable irreducible C*-algebra A with identity acting transitively on the unit sphere in a separable Hilbert space H and proved that the natural semidirect product H x U(A) is an algebraically determined Polish group. A similar theorem is true ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc67993/

### A*-algebras and Minimal Ideals in Topological Rings

**Date:**May 1973

**Creator:**Wei, Jui-Hung

**Description:**The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131622/

### The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity

**Date:**1941

**Creator:**Parrish, Herbert C.

**Description:**This study investigates the relationship between ordinary continuity and t-continuity.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc75274/

### Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions

**Date:**December 2002

**Creator:**Valdes, LeRoy I.

**Description:**Hill and Monticino (1998) introduced a constructive method for generating random probability measures with a prescribed mean or distribution on the mean. The method involves sequentially generating an array of barycenters that uniquely defines a probability measure. This work analyzes statistical properties of the measures generated by sequential barycenter array constructions. Specifically, this work addresses how changing the base measures of the construction affects the statististics of measures generated by the SBA construction. A relationship between statistics associated with a finite level version of the SBA construction and the full construction is developed. Monte Carlo statistical experiments are used to simulate the effect changing base measures has on the statistics associated with the finite level construction.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3304/