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Visualizing Geometric Structures On Topological Surfaces, Andrea Clark 2021 Northern Michigan University

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 ...


Determining Quantum Symmetry In Graphs Using Planar Algebras, Akshata Pisharody 2021 William & Mary

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 2021 West Chester University of Pennsylvania

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 ...


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 2021 College of the Holy Cross

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.


Translation Of: Isoparametrische Hyperflächen In Sphären, Math. Ann. 251, 57–71 (1980) By Hans Friedrich Münzner., Thomas E. Cecil 2021 College of the Holy Cross

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.


Entropic Dynamics Of Networks, Felipe Xavier Costa, Pedro Pessoa 2021 Department of Physics, University at Albany, State University of New York

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.


I Principii Di Geometria Logicamente Esposti, Giuseppe Peano 2021 Livre de Lyon

I Principii Di Geometria Logicamente Esposti, Giuseppe Peano

Science and Mathematical Science

No abstract provided.


On The Surface Area Of Scalene Cones And Other Conical Bodies, Daniel J. Curtin 2021 Northern Kentucky University

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 2021 McGill University

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

Journal of Humanistic Mathematics

No abstract provided.


Gordian Adjacency For Positive Braid Knots, Tolson H. Bell, David C. Luo, Luke Seaton, Samuel P. Serra 2021 Georgia Institute of Technology

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.


Designing Efficient Algorithms For Sensor Placement, Gabriel Loos 2021 Georgia Southern University

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 ...


Filaments, Fibers, And Foliations In Frustrated Soft Materials, Daria Atkinson 2020 University of Massachusetts Amherst

Filaments, Fibers, And Foliations In Frustrated Soft Materials, Daria Atkinson

Doctoral Dissertations

Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to the distance of closest approach between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. Dual to strong dependence of inter-filament interactions on changes in the distance of closest approach is their relative insensitivity to reptations, translations along the filament backbone. In this dissertation, after briefly reviewing the mechanics and geometry ...


Introduction To Classical Field Theory, Charles G. Torre 2020 Department of Physics, Utah State University

Introduction To Classical Field Theory, Charles G. Torre

All Complete Monographs

This is an introduction to classical field theory. Topics treated include: Klein-Gordon field, electromagnetic field, scalar electrodynamics, Dirac field, Yang-Mills field, gravitational field, Noether theorems relating symmetries and conservation laws, spontaneous symmetry breaking, Lagrangian and Hamiltonian formalisms.


Notes On Lie Sphere Geometry And The Cyclides Of Dupin, Thomas E. Cecil 2020 College of the Holy Cross

Notes On Lie Sphere Geometry And The Cyclides Of Dupin, Thomas E. Cecil

Mathematics Department Faculty Scholarship

In these notes, we give a detailed account of the method for studying Dupin hypersurfaces in Rn or Sn using Lie sphere geometry, and we conclude with a classification of the cyclides of Dupin obtained by using this approach.
Specifically, an oriented hypersurface Mn−1Rn is a cyclide of Dupin of characteristic (p,q), where p +q = n − 1, if Mn−1 has two distinct principal curvatures at each point with respective multiplicities p and q, and each principal curvature function is constant along each leaf of its corresponding principal foliation. We show that ...


Nested Links, Linking Matrices, And Crushtaceans, Madeline Brown 2020 Scripps College

Nested Links, Linking Matrices, And Crushtaceans, Madeline Brown

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.


Application Of Tda Mapper To Water Data And Bird Data, Wako Bungula 2020 University of Wisconsin - La Crosse

Application Of Tda Mapper To Water Data And Bird Data, Wako Bungula

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.


Configuration Spaces For The Working Undergraduate, Lucas Williams 2020 Reed College

Configuration Spaces For The Working Undergraduate, Lucas Williams

Rose-Hulman Undergraduate Mathematics Journal

Configuration spaces form a rich class of topological objects which are not usually presented to an undergraduate audience. Our aim is to present configuration spaces in a manner accessible to the advanced undergraduate. We begin with a slight introduction to the topic before giving necessary background on algebraic topology. We then discuss configuration spaces of the euclidean plane and the braid groups they give rise to. Lastly, we discuss configuration spaces of graphs and the various techniques which have been developed to pursue their study.


Perceiving Mathematics And Art, Edmund Harriss 2020 University of Arkansas, Fayetteville

Perceiving Mathematics And Art, Edmund Harriss

Mic Lectures

Mathematics and art provide powerful lenses to perceive and understand the world, part of an ancient tradition whether it starts in the South Pacific with tapa cloth and wave maps for navigation or in Iceland with knitting patterns and sunstones. Edmund Harriss, an artist and assistant clinical professor of mathematics in the Fulbright College of Arts and Sciences, explores these connections in his Honors College Mic lecture.


Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern 2020 College of the Holy Cross

Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern

Mathematics Department Faculty Scholarship

A hypersurface Mn-1 in Euclidean space En is proper Dupin if the number of distinct principal curvatures is constant on Mn-1, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together with ...


Guide To Geometry, Karla Childs Ph.D., Maddison Webb 2020 Pittsburg State University

Guide To Geometry, Karla Childs Ph.D., Maddison Webb

Faculty Submissions

Contents: Euclid’s Postulates--Polygons--Fundamentals of Euclidean Geometry--Similar Figures--Trigonometry--Tessellations--Non-Euclidean Geometry.


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