Geometry and Topology Commons^™

A General Elliptic Nonlinear System Of Multiple Functions With Application, Timothy Robertson, Joon Hyuk Kang

Faculty Publications

The purpose of this paper is to give a sufficient condition for the existence, nonexistence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain Ω in Rn . We also investigate the effects of perturbation on the positive solutions to the system. The techniques used in this paper are upper-lower solutions, eigenvalues of operators, the maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations. This result yields an algebraically computable criterion for the positive coexistence of competing species …

A General Elliptic Nonlinear System Of Two Functions With Application, Timothy Robertson, Joon Hyuk Kang

Faculty Publications

The purpose of this paper is to give a sufficient condition for the existence and nonexistence of positive solutions to a rather general type of elliptic system of the Dirichlet problem on the bounded domain Ω in Rn. Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are upper-lower solutions, eigenvalues of operators, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of competing species of animals in many …

On The Derivative Of 2-Holonomy For A Non-Abelian Gerbe, Cheyne J. Miller

Dissertations, Theses, and Capstone Projects

The local 2-holonomy for a non abelian gerbe with connection is first studied via a local zig-zag Hochschild complex. Next, by locally integrating the cocycle data for our gerbe with connection, and then glueing this data together, an explicit definition is offered for a global version of 2-holonomy. After showing this definition satisfies the desired properties for 2-holonomy, its derivative is calculated whereby the only interior information added is the integration of the 3-curvature. Finally, for the case when the surface being mapped into the manifold is a sphere, the derivative of 2-holonomy is extended to an equivariant closed form …

Some 2-Categorical Aspects In Physics, Arthur Parzygnat

Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description …

A Geometric Model Of Twisted Differential K-Theory, Byung Do Park

Dissertations, Theses, and Capstone Projects

We construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion class. We use smooth U(1)-gerbes with connection as differential twists and twisted vector bundles with connection as cycles. The model we construct satisfies the axioms of Kahle and Valentino, including functoriality, naturality of twists, and the hexagon diagram. We also construct an odd twisted Chern character of a twisted vector bundle with an automorphism. In addition to our geometric model of twisted differential K-theory, we introduce a smooth variant of the Hopkins-Singer model of differential K-theory. We prove that our model is naturally …

Hyperplanes That Intersect Each Ray Of A Cone Once And A Banach Space Counterexample, Chris Mccarthy

Publications and Research

Suppose � is a cone contained in real vector space �. When does � contain a hyperplane � that intersects each of the 0-rays in �\{0} exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question.We also present an example of a salient, closed Banach space cone � for which there does not exist a hyperplane that intersects each 0-ray in � \ {0} exactly once.

The Fourth Movement Of György Ligeti's Piano Concerto: Investigating The Musical-Mathematical Connection, Cynthia L. Wong

Dissertations, Theses, and Capstone Projects

This interdisciplinary study explores musical-mathematical analogies in the fourth movement of Ligeti’s Piano Concerto. Its aim is to connect musical analysis with the piece’s mathematical inspiration. For this purpose, the dissertation is divided into two sections. Part I (Chapters 1-2) provides musical and mathematical context, including an explanation of ideas related to Ligeti’s mathematical inspiration. Part II (Chapters 3-5) delves into an analysis of the rhythm, form, melody / motive, and harmony. Appendix A is a reduced score of the entire movement, labeled according to my analysis.

Projective Geometry Hidden Inside: Can You Spot It?, Tom Clark

Faculty Work Comprehensive List

In this talk Dr. Clark shared about a Math Teachers’ Circle session he recently ran centered around the children’s game Spot it! This game has some very interesting mathematics behind it and naturally begs to be explored with inquiry. He described the way he led teachers to ask questions about the game, the way the teachers then explored the topic, and the mathematics behind it all.

Ε-Kernel Coresets For Stochastic Points, Haitao Wang, Lingxiao Huang, Jian Li, Jeff Mark Phillips

Computer Science Faculty and Staff Publications

With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the …

Bridge Spectra Of Cables Of 2-Bridge Knots, Nicholas John Owad

Department of Mathematics: Dissertations, Theses, and Student Research

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.

Advisors: Mark Brittenham and Susan Hermiller

Patterns Formed By Coins, Andrey M. Mishchenko

Journal of Humanistic Mathematics

This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non- overlapping circles. The first half of the article is an exposition of the two most important facts about circle packings, (1) that essentially whatever pattern we ask for, we may always arrange circles in that pattern, and (2) that under simple conditions on the pattern, there is an essentially unique arrangement of circles in that pattern. In the second half of the article, we consider related questions, but where we …

Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, Joseph O'Rourke

Computer Science: Faculty Publications

A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does …

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces (Corrected), Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Two Riemann surfaces S₁ and S₂ with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S₁ -> S₂ which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials H^q(S₁) and H^q(S₂). In this note we study the differences between topological equivalence and H^q equivalence of prime cyclic actions, where S₁/G and S₂/G have genus zero.

The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder

Geometry

No abstract provided.

Pythagorean Combinations For Lego Robot Building., Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

This paper provides tips for LEGO robot construction involving bracing or gear meshing along a diagonal using standard Botball kits.

Cohomology Of Certain Polyhedral Product Spaces, Elizabeth A. Vidaurre

Dissertations, Theses, and Capstone Projects

The study of torus actions led to the discovery of moment-angle complexes and their generalization, polyhedral product spaces. Polyhedral products are constructed from a simplicial complex. This thesis focuses on computing the cohomology of polyhedral products given by two different classes of simplicial complexes: polyhedral joins (composed simplicial complexes) and $n$-gons. A homological decomposition of a polyhedral product developed by Bahri, Bendersky, Cohen and Gitler is used to derive a formula for the case of polyhedral joins. Moreover, methods from and results by Cai will be used to give a full description of the non-trivial cup products in a real …

Quaternion Algebras And Hyperbolic 3-Manifolds, Joseph Quinn

Dissertations, Theses, and Capstone Projects

I use a classical idea of Macfarlane to obtain a complex quaternion model for hyperbolic 3-space and its group of orientation-preserving isometries, analogous to Hamilton’s famous result on Euclidean rotations. I generalize this to quaternion models over number fields for the action of Kleinian groups on hyperbolic 3-space, using arithmetic invariants of the corresponding hyperbolic 3-manifolds. The class of manifolds to which this technique applies includes all cusped arithmetic manifolds and infinitely many commensurability classes of cusped non-arithmetic, compact arithmetic, and compact non-arithmetic manifolds. I obtain analogous results for actions of Fuchsian groups on the hyperbolic plane. I develop new …

An Investigation Of Minimal Surfaces In So(3), Luke Bohn

Rose-Hulman Undergraduate Research Publications

Classical minimal surface theory can be thought of as dealing with the shapes of soap films stretched across wires in Euclidean space R³. This article will examine such structures in an abstract three-dimensional space, the Lie Group SO(3). This is the space of possible rotations in R³, where each rotation is expressed as three angles: two to indicate the axis of rotation and one to indicate the amount of rotation. The properties of the space SO(3) may result in minimal surfaces that behave differently than they do in R³.

Sequences Of Spiral Knot Determinants, Ryan Stees

Senior Honors Projects, 2010-2019

Spiral knots are a generalization of the well-known class of torus knots indexed by strand number and base word repetition. By fixing the strand number and varying the repetition index we obtain integer sequences of spiral knot determinants. In this paper we examine such sequences for spiral knots of up to four strands using a new periodic crossing matrix method. Surprisingly, the resulting sequences vary widely in character and, even more surprisingly, nearly every one of them is a known integer sequence in the Online Encyclopedia of Integer Sequences. We also develop a general form for these sequences in terms …

Model Behavior: The Mathematics Behind Three-Dimensional Modeling And Animation, Kathryn Duff, Vivian Cyrus

Celebration of Student Scholarship Poster Sessions Archive

No abstract provided.

Mathematics And Origami; Unfolding Mathematical "Impossibilities", Dustin Tyler Adams

Celebration of Student Scholarship Poster Sessions Archive

No abstract provided.

Drawing Numbers And Listening To Patterns, Loren Zo Haynes

Honors College Theses

The triangular numbers is a series of number that add the natural numbers. Parabolic shapes emerge when this series is placed on a lattice, or imposed with a limited number of columns that causes the sequence to continue on the next row when it has reached the kth column. We examine these patterns and construct proofs that explain their behavior. We build off of this to see what happens to the patterns when there is not a limited number of columns, and we formulate the graphs as musical patterns on a staff, using each column as a line or space …

The Cantor Set Before Cantor, Nicholas A. Scoville

Topology

A special construction used in both analysis and topology today is known as the Cantor set. Cantor used this set in a paper in the 1880s. Yet it appeared as early as 1875 in a paper by the Irish mathematician Henry John Stephen Smith (1826 - 1883). Smith, who is best known for the Smith normal form of a matrix, was a professor at Oxford who made great contributions in matrix theory and number theory. In this project, we will explore parts of a paper he wrote titled On the Integration of Discontinuous Functions.

Topology From Analysis, Nicholas A. Scoville

Topology

Topology is often described as having no notion of distance, but a notion of nearness. How can such a thing be possible? Isn't this just a distinction without a difference? In this project, we will discover the notion of nearness without distance by studying the work of Georg Cantor and a problem he was investigating involving Fourier series. We will see that it is the relationship of points to each other, and not their distances per se, that is a proper view. We will see the roots of topology organically springing from analysis.

Connecting Connectedness, Nicholas A. Scoville

Topology

No abstract provided.

The Exigency Of The Euclidean Parallel Postulate And The Pythagorean Theorem, Jerry Lodder

Geometry

No abstract provided.

Conjugacy Geodesics In Coxeter Groups, Aaron Calderon

UCARE: Research Products

Take a square and flip it over the vertical axis, rotate it 90 degrees counterclockwise and then flip it again over the vertical axis. This sequence is the same as a 90 degree clockwise rotation but takes more steps to demonstrate the same symmetry. In general, the question of when a sequence of symmetries has minimal length is hard to answer and is dependent on the chosen generating set (in our toy example, rotation by 90 degrees and reflection). By realizing sequences of symmetries as paths in a group's Cayley graph, the problem becomes one about the set of shortest …

Coding Strategies, The Choquet Game And Domain Representability, Lynne Yengulalp

Mathematics Faculty Publications

We prove that if the NONEMPTY player has a winning strategy in the strong Choquet game on a regular space X then NONEMPTY has a winning coding strategy in that game (a strategy that only depends on the previous 2 moves). We also prove that any regular domain representable space is generalized subcompact.

Domain Representability And Topological Completeness, Matthew D. Devilbiss

Honors Theses

Topological completeness properties seek to generalize the definition of complete metric space to the context of topologies. Chapter 1 gives an overview of some of these properties. Chapter 2 introduces domain theory, a field originally intended for use in theoretical computer science. Finally, Chapter 3 examines how this computer-scientific notion can be employed in the study of topological completeness in the form of domain representability. The connections between domain representability and other topological completeness properties are subsequently examined.

Area And Volume Where Do The Formulas Come From?, Roger Yarnell

Masters Essays

No abstract provided.