Geometry and Topology Commons^™

Construction Of Weavings In The Plane, Eden Delight Miro, Aliw-Iw Zambrano, Agnes Garciano

Mathematics Faculty Publications

This work develops, in graph-theoretic terms, a methodology for systematically constructing weavings of overlapping nets derived from 2-colorings of the plane. From a 2-coloring, two disjoint simple, connected graphs called nets are constructed. The union of these nets forms an overlapping net, and a weaving map is defined on the intersection points of the overlapping net to form a weaving. Furthermore, a procedure is given for the construction of mixed overlapping nets and for deriving weavings from them.

Series Solutions Of Polarized Gowdy Universes, Doniray Brusaferro

Theses and Dissertations

Einstein's field equations are a system of ten partial differential equations. For a special class of spacetimes known as Gowdy spacetimes, the number of equations is reduced due to additional structure of two dimensional isometry groups with mutually orthogonal Killing vectors. In this thesis, we focus on a particular model of Gowdy spacetimes known as the polarized T³ model, and provide an explicit solution to Einstein's equations.

A Survey Of Butterfly Diagrams For Knots And Links, Mark Ronnenberg

Dissertations and Theses @ UNI

A “butterfly diagram” is a representation of a knot as a kind of graph on the sphere. This generalization of Thurston’s construction of the Borromean rings was introduced by Hilden, Montesinos, Tejada, and Toro to study the bridge number of knots. In this paper, we study various properties of butterfly diagrams for knots and links. We prove basic some combinatorial results about butterflies and explore properties of butterflies for classes of links, especially torus links. The Wirtinger presentation for the knot group will be adapted to butterfly diagrams, and we translate the Reidemeister moves for knot diagrams into so-called “butterfly …

Properties Of Left-Separated Spaces And Their Unions, Eric Scheidecker

Dissertations and Theses @ UNI

Left-separated spaces are topological spaces which can be well ordered such that every initial segment is closed. In this paper, we examine what topological properties imply left-separation, and under what circumstances left-separation is preserved by unions. We also introduce several known theorems regarding elementary submodels as they are one of the primary tools that we use. We prove that for a topological space X;

1. If X has a point-countable base, then X is left-separated if and only if X has closed intersection with any elementary submodel M such that X ∈ M.

2. If every elementary submodel …

The Programmatic Manipulation Of Planar Diagram Codes To Find An Upper Bound On The Bridge Index Of Prime Knots, Genevieve R. Johnson

Dissertations and Theses @ UNI

The “bridge index” of a knot is the least number of maximal overpasses taken over all diagrams of the knot. A naïve method to determine the bridge index of a knot is to perform Reidemeister moves on diagrams of the knot, and this method quickly becomes tedious to implement by hand. In this paper, we introduce a sequence of Reidemeister moves which we call a “drag the underpass” move and prove how planar diagram codes change as Reidemeister moves are performed. We then use these results to programatically perform Reidemeister moves using Python 2.7 to calculate an upper bound on …

Characterization Of Rectifying And Sphere Curves In R^3, Yun Myung Oh, Julie Logan

Faculty Publications

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extracondition on the constant curvature, Y.L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This …

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 …

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 …

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 …

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.