Latest content added for Digital Library Collection: UNT Theses and Dissertationshttps://digital.library.unt.edu/explore/collections/UNTETD/browse/?fq=untl_institution:UNT&sort=creator&fq=str_degree_discipline:Mathematics2017-10-09T11:44:47-05:00UNT LibrariesThis is a custom feed for browsing Digital Library Collection: UNT Theses and DissertationsConvergence of Infinite Series2015-05-10T06:16:59-05:00https://digital.library.unt.edu/ark:/67531/metadc503870/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc503870/"><img alt="Convergence of Infinite Series" title="Convergence of Infinite Series" src="https://digital.library.unt.edu/ark:/67531/metadc503870/small/"/></a></p><p>The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.</p>Operators on Continuous Function Spaces and Weak Precompactness2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc331171/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc331171/"><img alt="Operators on Continuous Function Spaces and Weak Precompactness" title="Operators on Continuous Function Spaces and Weak Precompactness" src="https://digital.library.unt.edu/ark:/67531/metadc331171/small/"/></a></p><p>If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly bounded is given. In chapter IV, weakly precompact subsets of L1(μ,X) are examined. For a Banach space X whose dual has the Radon-Nikodym property, it is shown that the weakly precompact subsets of L1(μ,X) are exactly the uniformly integrable subsets of L1(μ,X). Furthermore, it is shown that this characterization does not hold in Banach spaces X for which X* does not have the weak Radon-Nikodym property.</p>Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials2014-02-01T18:14:03-06:00https://digital.library.unt.edu/ark:/67531/metadc271768/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc271768/"><img alt="Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials" title="Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials" src="https://digital.library.unt.edu/ark:/67531/metadc271768/small/"/></a></p><p>Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .</p>The Moore-Smith Limit2012-10-12T10:26:10-05:00https://digital.library.unt.edu/ark:/67531/metadc107833/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc107833/"><img alt="The Moore-Smith Limit" title="The Moore-Smith Limit" src="https://digital.library.unt.edu/ark:/67531/metadc107833/small/"/></a></p><p>It is the purpose of this thesis to indicate in more detail how various limits defined in analysis, as well as other concepts not ordinarily defined as limits, may be obtained as special cases of the Moore-Smith limit.</p>Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups2008-05-05T14:14:01-05:00https://digital.library.unt.edu/ark:/67531/metadc5235/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc5235/"><img alt="Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups" title="Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups" src="https://digital.library.unt.edu/ark:/67531/metadc5235/small/"/></a></p><p>The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.</p>Uniqueness of Positive Solutions for Elliptic Dirichlet Problems2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc330654/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc330654/"><img alt="Uniqueness of Positive Solutions for Elliptic Dirichlet Problems" title="Uniqueness of Positive Solutions for Elliptic Dirichlet Problems" src="https://digital.library.unt.edu/ark:/67531/metadc330654/small/"/></a></p><p>In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - Δ u(x)= g(λ,u(x)) in B, u(x) = 0 on ϑB,
where A is the Laplace operator, B is the unit ball in RˆN, and A>0. We show that if g(λ,u)=uˆ(N+2)/(N-2) + λ, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(λ,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order.
We also present a regularity result on linear elliptic equation where a coefficient has critical growth.</p>A Development of a Set of Functions Analogous to the Trigonometric and the Hyperbolic Functions2012-12-27T22:03:54-06:00https://digital.library.unt.edu/ark:/67531/metadc130370/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc130370/"><img alt="A Development of a Set of Functions Analogous to the Trigonometric and the Hyperbolic Functions" title="A Development of a Set of Functions Analogous to the Trigonometric and the Hyperbolic Functions" src="https://digital.library.unt.edu/ark:/67531/metadc130370/small/"/></a></p><p>The purpose of this paper is to define and develop a set of functions of an area in such a manner as to be analogous to the trigonometric and the hyperbolic functions.</p>A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers2017-10-09T11:44:47-05:00https://digital.library.unt.edu/ark:/67531/metadc1011753/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc1011753/"><img alt="A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers" title="A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers" src="https://digital.library.unt.edu/ark:/67531/metadc1011753/small/"/></a></p><p>Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And finally we show that for a Borel subset X of real numbers the product is CDH iff X is a G-delta zero-dimensional set or an interval.</p>Absolute 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>Integration of Vector Valued Functions2012-12-27T22:03:54-06:00https://digital.library.unt.edu/ark:/67531/metadc131526/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc131526/"><img alt="Integration of Vector Valued Functions" title="Integration of Vector Valued Functions" src="https://digital.library.unt.edu/ark:/67531/metadc131526/small/"/></a></p><p>This paper develops an integral for Lebesgue measurable functions mapping from the interval [0, 1] into a Banach space.</p>