Latest content added for Digital Library Partner: UNT Librarieshttps://digital.library.unt.edu/explore/partners/UNT/browse/?fq=dc_rights_access:public&fq=untl_collection:UNTETD&display=grid&fq=str_year:1988&fq=str_degree_discipline:Mathematics2015-03-09T08:15:06-05:00UNT LibrariesThis is a custom feed for browsing Digital Library Partner: UNT LibrariesOn the Development of Descriptive Set Theory2015-03-09T08:15:06-05:00https://digital.library.unt.edu/ark:/67531/metadc500836/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc500836/"><img alt="On the Development of Descriptive Set Theory" title="On the Development of Descriptive Set Theory" src="https://digital.library.unt.edu/ark:/67531/metadc500836/small/"/></a></p><p>In the thesis, the author traces the historical development of descriptive set theory from the work of H. Lebesgue to the introduction of projective descriptive set theory. Proofs of most of the major results are given. Topics covered include Corel lattices, universal sets, the operation A, analytic sets, coanalytic sets, and the continuum hypothesis The appendix contains a translation of the famous letters exchanged between R. Baire, E. Borel, J. Hadamard and H. Lebesgue concerning Zermelo's axiom of choice.</p>Dually Semimodular Consistent Lattices2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc330641/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc330641/"><img alt="Dually Semimodular Consistent Lattices" title="Dually Semimodular Consistent Lattices" src="https://digital.library.unt.edu/ark:/67531/metadc330641/small/"/></a></p><p>A lattice L is said to be dually semimodular if for all elements a and b in L, a ∨ b covers b implies that a covers a ∧ b. L is consistent if for every join-irreducible j and every element x in L, the element x ∨ j is a join-irreducible in the upper interval [x,l]. In this paper, finite dually semimodular consistent lattices are investigated. Examples of these lattices are the lattices of subnormal subgroups of a finite group. In 1954, R. P. Dilworth proved that in a finite modular lattice, the number of elements covering exactly k elements is equal to the number of elements covered by exactly k elements. Here, it is established that if a finite dually semimodular consistent lattice has the same number of join-irreducibles as meet-irreducibles, then it is modular. Hence, a converse of Dilworth's theorem, in the case when k equals 1, is obtained for finite dually semimodular consistent lattices. Several combinatorial results are shown for finite consistent lattices similar to those already established for finite geometric lattices. The reach of an element x in a lattice L is the difference between the rank of x*, the join of x and all the elements covering x, and the rank of x; the maximum reach of all elements in L is the reach of L. Sharp lower bounds for the total number of elements and the number of elements of a given reach in a semimodular consistent lattice given the rank, the reach, and the number of join-irreducibles are found. Extremal lattices attaining these bounds are described. Similar results are then obtained for finite dually semimodular consistent lattices.</p>Minimization of a Nonlinear Elasticity Functional Using Steepest Descent2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc331296/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc331296/"><img alt="Minimization of a Nonlinear Elasticity Functional Using Steepest Descent" title="Minimization of a Nonlinear Elasticity Functional Using Steepest Descent" src="https://digital.library.unt.edu/ark:/67531/metadc331296/small/"/></a></p><p>The method of steepest descent is used to minimize typical functionals from elasticity.</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>Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc332102/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc332102/"><img alt="Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension" title="Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension" src="https://digital.library.unt.edu/ark:/67531/metadc332102/small/"/></a></p><p>This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function,
f_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-e,1]. We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps. Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, as counted by P.J. Myrberg, is shown to equal the number of MSS sequences of length n. For an MSS sequence P, let H_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the itinerary of λ(P) under the map is λ(P)f_e is H_ϖ(P).
In Chapter four it is shown that only period doubling or period halving bifurcations can occur for the family λf_e, λΣ[0,1]. Results concerning how the size of a stable orbit changes as bifurcations of the family λf_e occur are given.
Let λΣ[0,1] be such that 1/2 is a periodic point of λf_e. In this case 1/2 is superstable. Chapter five investigates the boundary of the basin of attraction of this stable orbit. An algorithm is given that yields a graph directed construction such that the object constructed is the basin boundary. From this we analyze the Hausdorff dimension and measure in that dimension of the boundary. The dimension is related to the simple β-numbers, as defined by Parry.</p>Applications of Graph Theory and Topology to Combinatorial Designs2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc331968/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc331968/"><img alt="Applications of Graph Theory and Topology to Combinatorial Designs" title="Applications of Graph Theory and Topology to Combinatorial Designs" src="https://digital.library.unt.edu/ark:/67531/metadc331968/small/"/></a></p><p>This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, then it must have a 1-factor which contains A. The second part of the dissertation is concerned with determining if two designs are isomorphic. Here the v-set is any group G and translation by any element in G gives a design automorphism. Given a design B and its difference family D, two topological spaces, B and D, are constructed. We give topological conditions which imply that a design isomorphism is a group isomorphism.</p>Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem2014-08-22T18:00:56-05:00https://digital.library.unt.edu/ark:/67531/metadc331780/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc331780/"><img alt="Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem" title="Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem" src="https://digital.library.unt.edu/ark:/67531/metadc331780/small/"/></a></p><p>In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.</p>