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  Partner: UNT Libraries
 Decade: 1990-1999
 Degree Discipline: Mathematics
 Collection: UNT Theses and Dissertations
A Presentation of Current Research on Partitions of Lines and Space

A Presentation of Current Research on Partitions of Lines and Space

Date: December 1999
Creator: Nugen, Frederick T.
Description: We present the results from three papers concerning partitions of vector spaces V over the set R of reals and of the set of lines in V.
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The Computation of Ultrapowers by Supercompactness Measures

The Computation of Ultrapowers by Supercompactness Measures

Date: August 1999
Creator: Smith, John C.
Description: The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.
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Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices

Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices

Date: August 1999
Creator: Huff, Cheryl Rae
Description: The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and ...
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On the Cohomology of the Complement of a Toral Arrangement

On the Cohomology of the Complement of a Toral Arrangement

Date: August 1999
Creator: Sawyer, Cameron Cunningham
Description: The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.
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Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems

Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems

Date: December 1998
Creator: Richardson, Peter A. (Peter Adolph), 1955-
Description: In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.
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Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains

Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains

Date: August 1998
Creator: Finan, Marcel Basil
Description: The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.
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A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence

A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence

Date: August 1998
Creator: Risley, Rebecca N.
Description: We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.
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Multifractal Analysis of Parabolic Rational Maps

Multifractal Analysis of Parabolic Rational Maps

Date: August 1998
Creator: Byrne, Jesse William
Description: The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies sup φ < P(φ) x∈J(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.
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Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation

Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation

Date: August 1998
Creator: Debrecht, Johanna M.
Description: We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.
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Primitive Substitutive Numbers are Closed under Rational Multiplication

Primitive Substitutive Numbers are Closed under Rational Multiplication

Date: August 1998
Creator: Ketkar, Pallavi S. (Pallavi Subhash)
Description: Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
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