Geometry and Topology Commons^™

Sudoku Variants On The Torus, Kira A. Wyld

HMC Senior Theses

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Pattern Recognition In Stock Data, Kathryn Dover

HMC Senior Theses

Finding patterns in high dimensional data can be difficult because it cannot be easily visualized. There are many different machine learning methods to fit data in order to predict and classify future data but there is typically a large expense on having the machine learn the fit for a certain part of a dataset. We propose a geometric way of defining different patterns in data that is invariant under size and rotation. Using a Gaussian Process, we find that pattern within stock datasets and make predictions from it.

Introduction To Axiomatic Geometry, Mark Barsamian

OHIO Open Faculty Textbooks

This book presents Euclidean Geometry and was designed for a one-semester course preparing junior and senior level college students to teach high school Geometry. The book could also serve as a text for a junior level Introduction to Proofs course.

Martin Gardner Puzzle-Games, Stephen Bloom, Lacey Echols, Jeremiah Farrell, Shannon Lieb

Scholarship and Professional Work - LAS

No abstract provided.

Euler Entertainments, Jeremiah Farrell, Karen Farrell

Scholarship and Professional Work - LAS

No abstract provided.

A Gathering For Gardner Puzzle-Game, Jeremiah Farrell, Chris Morgan

Scholarship and Professional Work - LAS

Each different letter of "GATHERING FOR GARDNER" is used exactly three times in the following words: DIE, FAD, FIT, FOG, GIN, HAG, HER, HOD, NOR, RAT, TEN.

The Tea Party, Stephen Bloom, Jeremiah Farrell

Scholarship and Professional Work - LAS

No abstract provided.

Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap

Faculty Publications

Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds N_n and N^µ_n whose volume is approximately n and whose length spectra agree up to length n.

Both N_n and N^µ_n are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that …

Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap

Faculty Publications

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.

The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least …

On Independence, Matching, And Homomorphism Complexes, Wesley K. Hough

Theses and Dissertations--Mathematics

First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory.

In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the …

Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews

Honors Theses

This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.

Some Examples Of The Interplay Between Algebra And Topology, Joseph D. Malionek

Honors Theses

This thesis presents several undergraduate and graduate level concepts in the fields of algebraic topology and topological group theory in a manner which requires very little mathematical background of the reader. It uses non-rigorous interpretations of concepts while introducing the reader to the rigorous ideas with which they are associated. In order to give the reader an idea of how the fields of algebra and topology are closely affiliated, the paper goes over five main concepts, the fundamental group, homology, cohomology, Eilenberg-Maclane spaces, and group dimension.

The Partition Lattice In Many Guises, Dustin G. Hedmark

Theses and Dissertations--Mathematics

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m …

Long And Short-Range Air Navigation On Spherical Earth, Nihad E. Daidzic

International Journal of Aviation, Aeronautics, and Aerospace

Global range air navigation implies non-stop flight between any two airports on Earth. Such effort would require airplanes with the operational air range of at least 12,500 NM which is about 40-60% longer than anything existing in commercial air transport today. Air transportation economy requires flying shortest distance, which in the case of spherical Earth are Orthodrome arcs. Rhumb-line navigation has little practical use in long-range flights, but has been presented for historical reasons and for comparison. Database of about 50 major international airports from every corner of the world has been designed and used in testing and route validation. …

Spot It! Mathematical Structure In A Children's Game, Tom Clark

Faculty Work Comprehensive List

"Whenever I lead the Northwest Iowa Math Teachers’ Circle I want to use a problem that is easy to explain, unfolds in many directions, and encourages inquiry. Spot It! is the best session topic I have ever seen for this. All you have to do is play the game for a minute or two, and questions arise spontaneously."

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.

Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar

Honors Theses

Chirality (or handedness) is the property that a structure is “different” from its mirror image. Topology can be used to provide a rigorous framework for the notion of chirality. This project examines various types of chirality and discusses tools to detect chirality in graphs and knots. Notable theorems that are discussed in this work include ones that identify chirality using properties of link polynomials (HOMFLY polynomials), rigid vertex graphs, and knot linking numbers. Various other issues of chirality are explored, and some specially unique structures are discussed. This paper is borne out of reading Dr. Erica Flapan’s book, When Topology …

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 …