Latest content added for Digital Library Partner: UNT Librarieshttps://digital.library.unt.edu/explore/partners/UNT/browse/?sort=title&fq=str_degree_department:Department+of+Mathematics&fq=untl_collection:UNTETD2015-06-24T09:39:17-05:00UNT LibrariesThis is a custom feed for browsing Digital Library Partner: UNT LibrariesAbsolute Continuity and the Integration of Bounded Set Functions2015-06-24T09:39:17-05:00https://digital.library.unt.edu/ark:/67531/metadc663440/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc663440/"><img alt="Absolute Continuity and the Integration of Bounded Set Functions" title="Absolute Continuity and the Integration of Bounded Set Functions" src="https://digital.library.unt.edu/ark:/67531/metadc663440/small/"/></a></p><p>The first chapter gives basic definitions and theorems concerning set functions and set function integrals. The lemmas and theorems are presented without proof in this chapter. The second chapter deals with absolute continuity and Lipschitz condition. Particular emphasis is placed on the properties of max and min integrals. The third chapter deals with approximating absolutely continuous functions with bounded functions. It also deals with the existence of the integrals composed of various combinations of bounded functions and finitely additive functions. The concluding theorem states if the integral of the product of a bounded function and a non-negative finitely additive function exists, then the integral of the product of the bounded function with an absolutely continuous function exists over any element in a field of subsets of a set U.</p>Abstract Measure2012-10-12T10:26:10-05:00https://digital.library.unt.edu/ark:/67531/metadc107950/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc107950/"><img alt="Abstract Measure" title="Abstract Measure" src="https://digital.library.unt.edu/ark:/67531/metadc107950/small/"/></a></p><p>This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.</p>Abstract Vector Spaces and Certain Related Systems2012-12-27T22:03:54-06:00https://digital.library.unt.edu/ark:/67531/metadc130465/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc130465/"><img alt="Abstract Vector Spaces and Certain Related Systems" title="Abstract Vector Spaces and Certain Related Systems" src="https://digital.library.unt.edu/ark:/67531/metadc130465/small/"/></a></p><p>The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.</p>Additive Functions2012-10-12T10:26:10-05:00https://digital.library.unt.edu/ark:/67531/metadc108204/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc108204/"><img alt="Additive Functions" title="Additive Functions" src="https://digital.library.unt.edu/ark:/67531/metadc108204/small/"/></a></p><p>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).</p>Algebraic Integers2012-12-27T22:03:54-06:00https://digital.library.unt.edu/ark:/67531/metadc131119/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc131119/"><img alt="Algebraic Integers" title="Algebraic Integers" src="https://digital.library.unt.edu/ark:/67531/metadc131119/small/"/></a></p><p>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.</p>Algebraic Number Fields2015-03-09T08:15:06-05:00https://digital.library.unt.edu/ark:/67531/metadc501201/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc501201/"><img alt="Algebraic Number Fields" title="Algebraic Number Fields" src="https://digital.library.unt.edu/ark:/67531/metadc501201/small/"/></a></p><p>This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.</p>Algebraic Properties of Semigroups2012-12-27T22:03:54-06:00https://digital.library.unt.edu/ark:/67531/metadc131373/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc131373/"><img alt="Algebraic Properties of Semigroups" title="Algebraic Properties of Semigroups" src="https://digital.library.unt.edu/ark:/67531/metadc131373/small/"/></a></p><p>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.</p>Algebraically Determined Rings of Functions2011-03-30T21:15:03-05:00https://digital.library.unt.edu/ark:/67531/metadc31543/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc31543/"><img alt="Algebraically Determined Rings of Functions" title="Algebraically Determined Rings of Functions" src="https://digital.library.unt.edu/ark:/67531/metadc31543/small/"/></a></p><p>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.</p>Algebraically Determined Semidirect Products2012-01-09T21:53:51-06:00https://digital.library.unt.edu/ark:/67531/metadc67993/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc67993/"><img alt="Algebraically Determined Semidirect Products" title="Algebraically Determined Semidirect Products" src="https://digital.library.unt.edu/ark:/67531/metadc67993/small/"/></a></p><p>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 for the natural semidirect product R^{n} x G(n), where G(n) = GL(n,R), or GL^{+}(n,R), or SL(n,R), or |SL(n,R)|={A in GL(n,R) : |det(A)|=1}. On the other hand, it is known that the Heisenberg group H_{3}(R) , (R, +), (R{0}, x), and GL^{+}(n,R) are not algebraically determined Polish groups.</p>A*-algebras and Minimal Ideals in Topological Rings2012-12-27T22:03:54-06:00https://digital.library.unt.edu/ark:/67531/metadc131622/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc131622/"><img alt="A*-algebras and Minimal Ideals in Topological Rings" title="A*-algebras and Minimal Ideals in Topological Rings" src="https://digital.library.unt.edu/ark:/67531/metadc131622/small/"/></a></p><p>The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.</p>