Physical Sciences and Mathematics Commons^™

Discovering Regularity In Point Clouds Of Urban Scenes, Sam Friedman

Dissertations, Theses, and Capstone Projects

Despite the apparent chaos of the urban environment, cities are actually replete with regularity. From the grid of streets laid out over the earth, to the lattice of windows thrown up into the sky, periodic regularity abounds in the urban scene. Just as salient, though less uniform, are the self-similar branching patterns of trees and vegetation that line streets and fill parks. We propose novel methods for discovering these regularities in 3D range scans acquired by a time-of-flight laser sensor. The applications of this regularity information are broad, and we present two original algorithms. The first exploits the efficiency of …

Reducibility, Degree Spectra, And Lowness In Algebraic Structures, Rebecca M. Steiner

Dissertations, Theses, and Capstone Projects

This dissertation addresses questions in computable structure theory, which is a branch of mathematical logic hybridizing computability theory and the study of familiar mathematical structures. We focus on algebraic structures, which are standard topics of discussion among model theorists. The structures examined here are fields, graphs, trees under a predecessor function, and Boolean algebras.

For a computable field F, the splitting set S_F of F is the set of polynomials in F[X] which factor over F, and the root set R_F of F is the set of polynomials in F[X] which have a root in F …

Holomorphic Motions And Extremal Annuli, Zhe Wang

Dissertations, Theses, and Capstone Projects

Holomorphic motions, soon after they were introduced, became an important subject in complex analysis. It is now an important tool in the study of complex dynamical systems and in the study of Teichmuller theory. This thesis serves on two purposes: an expository of the past developments and a discovery of new theories.

First, I give an expository account of Slodkowski's theorem based on the proof given by Chirka. Then I present a result about infinitesimal holomorphic motions. I prove the |ε log ε| modulus of continuity for any infinitesimal holomorphic motion. This proof is a very well application of Schwarz's …

On The Dynamics Of Quasi-Self-Matings Of Generalized Starlike Complex Quadratics And The Structure Of The Mated Julia Sets, Ross Flek

Dissertations, Theses, and Capstone Projects

It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between …

On The Dynamics Of Quasi-Self-Matings Of Generalized Starlike Complex Quadratics And The Structure Of The Mated Julia Sets, Ross Flek

Dissertations, Theses, and Capstone Projects

It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between …

Rigidity And Stability For Isometry Groups In Hyperbolic 4-Space, Youngju Kim

Dissertations, Theses, and Capstone Projects

It is known that a geometrically finite Kleinian group is quasiconformally stable. We prove that this quasiconformal stability cannot be generalized in 4-dimensional hyperbolic space. This is due to the presence of screw parabolic isometries in dimension 4. These isometries are topologically conjugate to strictly parabolic isometries. However, we show that screw parabolic isometries are not quasiconformally conjugate to strictly parabolic isometries. In addition, we show that two screw parabolic isometries are generically not quasiconformally conjugate to each other. We also give some geometric properties of a hyperbolic 4-manifold related to screw parabolic isometries.

A Fuchsian thrice-punctured sphere group has …

Countable Short Recursively Saturated Models Of Arithmetic, Erez Shochat

Dissertations, Theses, and Capstone Projects

Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated.

Infinitely Often Dense Bases And Geometric Structure Of Sumsets, Jaewoo Lee

Dissertations, Theses, and Capstone Projects

We'll discuss two problems related to sumsets.

Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.

In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see …

Splitting Of Vector Bundles On Punctured Spectrum Of Regular Local Rings, Mahdi Majidi-Zolbanin

Dissertations, Theses, and Capstone Projects

In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings. We give a splitting criterion for vector bundles of small rank in terms of vanishing of their intermediate cohomology modules Hⁱ(U, E)_{2_i_n−3}, where n is the dimension of the regular local ring. This is the local analog of a result by N. Mohan Kumar, C. Peterson, and A. Prabhakar Rao for splitting of vector bundles of small rank on projective spaces.

As an application we give a positive answer (in a special case) to a conjecture …

On String Topology Of Three Manifolds, Hossein Abbaspour

Dissertations, Theses, and Capstone Projects

In this dissertation we establish a connection between some aspects of the string topology of three dimensional manifolds and their topology and geometry using the theory of the prime decomposition and characteristic surfaces.

Class Groups Of Real Quadratic Number Fields, Paul B. Massell

Dissertations, Theses, and Capstone Projects

No abstract provided.