Geometry and Topology Commons^™

Evaluating The Historical Accuracy Of Blackwork Embroidery With Fractal Analysis, Rhiannon Cire

Undergraduate Theses and Capstone Projects

The intricate monochromatic embroidery that graced the collars and cuffs of Renaissance nobility and domestic materials from that era has been little studied beyond the historical costuming and crafting communities. This style, known as blackwork, for it was traditionally done in black silk on white linen, exemplifies how complex and visually-appealing designs can arise from repetition of simple forms, often demonstrating the fractal property of self-similarity. Though most blackwork patterns are not true fractals, fractal analysis offers a means of objectively quantifying their complexity and new lens through which to examine this embroidery technique. The purpose of this study was …

Visualizing Geometric Structures On Topological Surfaces, Andrea Clark

All NMU Master's Theses

We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the …

Sobolev Inequalities And Riemannian Manifolds, John Reever

Honors Scholar Theses

Sobolev inequalities, named after Sergei Lvovich Sobolev, relate norms in Sobolev spaces and give insight to how Sobolev spaces are embedded within each other. This thesis begins with an overview of Lebesgue and Sobolev spaces, leading into an introduction to Sobolev inequalities. Soon thereafter, we consider the behavior of Sobolev inequalities on Riemannian manifolds. We discuss how Sobolev inequalities are used to construct isoperimetric inequalities and bound volume growth, and how Sobolev inequalities imply families of other Sobolev inequalities. We then delve into the usefulness of Sobolev inequalities in determining the geometry of a manifold, such as how they can …

Determining Quantum Symmetry In Graphs Using Planar Algebras, Akshata Pisharody

Undergraduate Honors Theses

A graph has quantum symmetry if the algebra associated with its quantum automorphism group is non-commutative. We study what quantum symmetry means and outline one specific method for determining whether a graph has quantum symmetry, a method that involves studying planar algebras and manipulating planar tangles. Modifying a previously used method, we prove that the 5-cycle has no quantum symmetry by showing it has the generating property.

Solving Parabolic Interface Problems With A Finite Element Method, Henry Brown

Mathematics Student Work

Partial differential equations (PDEs) dominate mathematical models given their effectiveness and accuracy at modeling the physical realities which govern the world. Though we have these powerful tools, analytic solutions can only be found in the simplest of cases due to the complexity of PDE models. Thus, efficient and accurate computational methods are needed to approximate solutions to PDE models. One class of these methods are finite element methods which can be used domain to provide close approximations to the PDE model in a finite domain. In this presentation, we discuss the use of a Discontinuous Galerkin (DG) Finite Element Methods …

New Characterizations Of Reproducing Kernel Hilbert Spaces And Applications To Metric Geometry, Daniel Alpay, Palle E. T. Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

Translation Of: Isoparametrische Hyperflächen In Sphären, Math. Ann. 251, 57–71 (1980) By Hans Friedrich Münzner., Thomas E. Cecil

Mathematics Department Faculty Scholarship

This is an English translation of the article "Isoparametrische Hyperflächen in Sphären" by Hans Friedrich Münzner, which was originally published in Math. Ann. 251, 57–71 (1980).

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper. I want to thank Cristina Ballantine for her help with the translation.

Translation Of: Isoparametrische Hyperflächen In Sphären Ii. Über Die Zerlegung Der Sphäre In Ballbündel, Math. Ann. 256, 215–232 (1981) By Hans Friedrich Münzner., Thomas E. Cecil

Mathematics Department Faculty Scholarship

This is an English translation of the article "Isoparametrische Hyperflächen in Sphären II. Über die Zerlegung der Sphäre in Ballbündel" by Hans Friedrich Münzner, which was originally published in Math. Ann. 256, 215–232 (1981).

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper. I want to thank Cristina Ballantine for her help with the translation.

On The Menger And Almost Menger Properties In Locales, Tilahun Bayih, Themba Dube, Oghenetega Ighedo

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober T_D-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.

Quasipositive Braids And Ribbon Surfaces, Rachel Snyder

WWU Honors College Senior Projects

Meant to serve as an accessible exploration of knot theory for undergraduates and those without much experience in topology, this paper will start by exploring the basics of knot theory and will work through investigating the relationships between knots and surfaces, ending with an analysis of the relationship between quasipositive braids and surfaces in 4-space. We will begin by defining a knot and introducing the ways in which we are able to manipulate them. Following that, we will explore the basics of surfaces, building up to a proof that all surfaces are homeomorphic to a series of disks and bands …

Entropic Dynamics Of Networks, Felipe Xavier Costa, Pedro Pessoa

Northeast Journal of Complex Systems (NEJCS)

Here we present the entropic dynamics formalism for networks. That is, a framework for the dynamics of graphs meant to represent a network derived from the principle of maximum entropy and the rate of transition is obtained taking into account the natural information geometry of probability distributions. We apply this framework to the Gibbs distribution of random graphs obtained with constraints on the node connectivity. The information geometry for this graph ensemble is calculated and the dynamical process is obtained as a diffusion equation. We compare the steady state of this dynamics to degree distributions found on real-world networks.

Knots And Links In Overtwisted Contact Manifolds, Rima Chatterjee

LSU Doctoral Dissertations

Suppose $(\M,\xi)$ be an overtwisted contact 3-manifold. We prove that any Legendrian and transverse link in $(\M,\xi)$ having overtwisted complement can be coarsely classified by their classical invariants. Next, we defined an invariant called the support genus for transverse links and extended the definition of support genus of Legendrian knots to Legendrian links and prove that any coarse equivalence class of Legendrian and transverse loose links has support genus zero. Further, we show that the converse is not true by explicitly constructing an example. We also find a relationship between the support genus of the transverse link and its Legendrian …

Designing Efficient Algorithms For Sensor Placement, Gabriel Loos

Honors College Theses

Sensor placement has many applications and uses that can be seen everywhere you go.These include, but not limited to, monitoring the structural health of buildings and bridgesand navigating Unmanned Aerial Vehicles(UAV).We study ways that leads to efficient algorithms that will place as few as possible sen-sors to cover an entire area. We will tackle the problem from both 2-dimensional and3-dimensional points of view. Two famous related problems are discussed: the art galleryproblem and the terrain guarding problem. From the top view an area presents a 2-D im-age which will enable us to partition polygonal shapes and use graph theoretical results …

On The Surface Area Of Scalene Cones And Other Conical Bodies, Daniel J. Curtin

Euleriana

This paper first appeared in the Novi Commentarii academiae scientiarum Petropolitanae vol. 1, 1750, pp. 3-19 and is reprinted in the Opera Omnia: Series 1, Volume 27, pp. 181–199. Its Eneström number is E133. This translation and the Latin original are available from the Euler Archive.

One Straight Line Addresses Another Traveling In The Same Direction On An Infinite Plane, Daniel W. Galef

Journal of Humanistic Mathematics

No abstract provided.

Boxes, Extended Boxes And Sets Of Positive Upper Density In The Euclidean Space, Polona Durcik, Vjekoslav Kovač

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

Gordian Adjacency For Positive Braid Knots, Tolson H. Bell, David C. Luo, Luke Seaton, Samuel P. Serra

Rose-Hulman Undergraduate Mathematics Journal

A knot $K_1$ is said to be Gordian adjacent to a knot $K_2$ if $K_1$ is an intermediate knot on an unknotting sequence of $K_2$. We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between classes of positive braid knots through manipulations of braid words. In addition, we explore unknotting sequences of positive braid knots and give a proof that there are only finitely many positive braid knots for a given unknotting number.

Elementary College Geometry (2021 Ed.), Henry Africk

Open Educational Resources

This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra. The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem proving or formal methods of reasoning. However the topics …

Cayley Map Embeddings Of Complete Graphs, Miriam Scheinblum

Honors Program Theses

This paper looks at Cayley map embeddings of complete graphs on orientable surfaces. Cayley maps constrain graph embeddings to those with cyclical edge rotations, so optimal embeddings on surfaces with the minimum genus may not always be possible. We explore instances when Cayley maps succeed at optimally embedding complete graphs, and when optimal embeddings are not possible, we determine how close to optimal they can get by finding vertex rotations that result in the smallest possible genus. Many of the complete graphs we consider have prime numbers of vertices, so for each complete graph Kn we focus on mappings with …

The Adams Spectral Sequence For Topological Modular Forms, Robert Bruner, John Rognes

Mathematics Faculty Research Publications

The connective topological modular forms spectrum, 𝑡𝑚𝑓, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of 𝑡𝑚𝑓 and several 𝑡𝑚𝑓-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account …