Geometry and Topology Commons^™

An Introduction To Differential Geometry Through Computation, Mark E. Fels

Mathematics and Statistics Faculty Presentations

No abstract provided.

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

The Kretschmann Scalar, Charles G. Torre

How to... in 10 minutes or less

On a pseudo-Riemannian manifold with metric g, the "Kretschmann scalar" is a quadratic scalar invariant of the Riemann R tensor of g, defined by contracting all indices with g. In this worksheet we show how to calculate the Kretschmann scalar from a metric.

A Special Tribute To Martin Gardner, Jeremiah Farrell

Scholarship and Professional Work - LAS

There are exactly 12 different letters in the phrase GATHERING FOR MARTIN GARDNER. We use each of the 12 letters three times each in 18 different two-letter words that are to be placed on the nodes of the graph so adjoining nodes have a letter in common.

Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden

Faculty Publications

In this paper, we show that any nonarithmetic hyperbolic 2-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of 3-manifolds with nontrivial JSJ-decomposition and rank-two fundamental groups. We also show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the 6²₂ link complement. Our work requires a careful analysis of the tilings of R² that come from lifting the canonical triangulations of …

Locally Anisotropic Toposes, Jonathon Funk, Pieter Hofstra

Publications and Research

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.

Geometric Deformations Of Sodalite Frameworks, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

In mathematical crystallography and computational materials science, it is important to infer flexibility properties of framework materials from their geometric representation. We study combinatorial, geometric and kinematic properties for frameworks modeled on sodalite.

Classification Of Compact 2-Manifolds, George H. Winslow

Theses and Dissertations

It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we …

Algorithmic Foundations Of Heuristic Search Using Higher-Order Polygon Inequalities, Newton Henry Campbell Jr.

CCAC Theses and Dissertations

The shortest path problem in graphs is both a classic combinatorial optimization problem and a practical problem that admits many applications. Techniques for preprocessing a graph are useful for reducing shortest path query times. This dissertation studies the foundations of a class of algorithms that use preprocessed landmark information and the triangle inequality to guide A* search in graphs. A new heuristic is presented for solving shortest path queries that enables the use of higher order polygon inequalities. We demonstrate this capability by leveraging distance information from two landmarks when visiting a vertex as opposed to the common single landmark …

Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We give a new interpretation of the Alexander polynomial Δ₀ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, Δ₀ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.

Exploration Of Curvature Through Physical Materials, Lucinda-Joi Chu-Ketterer

Pitzer Senior Theses

Parametric equations are commonly used to describe surfaces. Looking at parametric equations does not provide tangible information about an object. Thus through the use of physical materials, an understanding of the limitations of the materials allows someone to gain a broader understanding of the surface. A M$\ddot{o}$bius strip and Figure 8 Klein bottle were created through knitting due to the precision and steady increase in curvature allowed through knitting. A more standard Klein bottle was created through crochet due to the ease in creating quick increases in curvature. Both methods demonstrate the change in curvature for both surfaces where the …

In Search Of A Class Of Representatives For Su-Cobordism Using The Witten Genus, John E. Mosley

Theses and Dissertations--Mathematics

In algebraic topology, we work to classify objects. My research aims to build a better understanding of one important notion of classification of differentiable manifolds called cobordism. Cobordism is an equivalence relation, and the equivalence classes in cobordism form a graded ring, with operations disjoint union and Cartesian product. My dissertation studies this graded ring in two ways:

1. by attempting to find preferred class representatives for each class in the ring.

2. by computing the image of the ring under an interesting ring homomorphism called the Witten Genus.

Nuclear Space Facts, Strange And Plain, Jeremy Becnel, Ambar Sengupta

Faculty Publications

We present a scenic but practical guide through nuclear spaces and their dual spaces, examining useful, unexpected, and often unfamiliar results both for nuclear spaces and their strong and weak duals.

Complements To Classic Topics Of Circles Geometry, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors’ aspiration and attraction, as a poet writes lyrics about spring according to his emotions.

Trilobic Vibrant Systems, Florentin Smarandache, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Sisteme Vibrante Trilobice, Florentin Smarandache, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Single Valued Neutrosophic Graphs: Degree, Order And Size, Florentin Smarandache, Said Broumi, Mohamed Talea, Assia Bakali

Branch Mathematics and Statistics Faculty and Staff Publications

The single valued neutrosophic graph is a new version of graph theory presented recently as a generalization of fuzzy graph and intuitionistic fuzzy graph. The single valued neutrosophic graph (SVN-graph) is used when the relation between nodes (or vertices) in problems are indeterminate. In this paper, we examine the properties of various types of degrees, order and size of single valued neutrosophic graphs and a new definition for regular single valued neutrosophic graph is given.

New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe

Williams Honors College, Honors Research Projects

The balanced minimal evolution (BME) polytope arises from the study of phylogenetic trees in biology. It is a geometric structure which has a variant for each natural number n. The main application of this polytope is that we can use linear programming with it in order to determine the most likely phylogenetic tree for a given genetic data set. In this paper, we explore the geometric and combinatorial structure of the BME polytope. Background information will be covered, highlighting some points from previous research, and a new result on the structure of the BME polytope will be given.

Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.

Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish

Mathematics, Statistics and Data Science Faculty Works

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and p-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which p the graph is p-colorable, and that a p-coloring of a graph corresponds to a representation of …

Involutory Quandles Of (2,2,R)-Montesinos Links, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

In this paper we show that Montesinos links of the form L(1/2, 1/2, p/q;e), which we call (2,2,r)-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the (2, 2, q)-pretzel links. We also describe some properties of these quandles.

Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang

Faculty Publications

The uniqueness of positive solution to the elliptic model

∆u + u[a + g(u, v)] = 0 in Ω, ∆v + v[a + h(u, v)] = 0 in Ω, u = v = 0 on ∂Ω,

were investigated.

Some Convergence Properties Of Minkowski Functionals Given By Polytopes, Jesse Moeller

Dissertations and Theses @ UNI

In this work we investigate the behavior of the Minkowski Functionals admitted by a sequence of sets which converge to the unit ball ‘from the inside’. We begin in R 2 and use this example to build intuition as we extend to the more general R n case. We prove, in the penultimate chapter, that convergence ‘from the inside’ in this setting is equivalent to two other characterizations of the convergence: a geometric characterization which has to do with the sizes of the faces of each polytope in the sequence converging to zero, and the convergence of the Minkowski functionals …

Geometric Auxetics, Ciprian Borcea, Ileana Streinu

Computer Science: Faculty Publications

We formulate a mathematical theory of auxetic behavior based on one-parameter deformations of periodic frameworks. Our approach is purely geometric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behavior to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.

On The Construction Of Simply Connected Solvable Lie Groups, Mark E. Fels

Research Vignettes

This worksheet contains the implementation of Theorems 4.2, 5.4 and 5.7 in the paper On the Construction of Solvable Lie Groups. All the examples in the paper are demonstrated here, along with one in Section 6 that was too long to include in the article.

Hypercube Unfoldings That Tile R3 And R2, Giovanna Diaz, Joseph O'Rourke

Computer Science: Faculty Publications

We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane.

Differentialgeometry In Brno, Ian M. Anderson

Presentations

This page will provide files supporting Ian Anderson's presentations in Brno, December 2015. The files can be found and downloaded from "Additional Files", below.

The files include:

(1) DifferentialGeometryUSU.mla: This is the Maple Library Archive file which provides all the DifferentialGeometry functionality. Here are Installation Instructions.

(2) DifferentialGeometry.help : this is the latest version of the DifferentialGeometry documentation. Copy this file to the same directory used for DifferentialGeometryUSU.mla (from step (1)).

Asymptotically Double Lacunary Equivalent Sequences In Topological Groups, Ayhan Esi, M. K. Ozdemir

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we study the concept of asymptotically double lacunary statistical convergent sequences in topological groups and prove some inclusion theorems.

Spiral Unfoldings Of Convex Polyhedra, Joseph O'Rourke

Computer Science: Faculty Publications

The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution.

A Fast Algorithm For Simulating Multiphase Flows Through Periodic Geometries Of Arbitrary Shape, Gary R. Marple, Alex Barnett, Adrianna Gillman, Shravan Veerapaneni

Dartmouth Scholarship

This paper presents a new boundary integral equation (BIE) method for simulating particulate and mul- tiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system—multiple vesicles suspended in a periodic channel of arbitrary shape—to describe the numerical method and test its performance. Rather than relying on the periodic Green’s function as classical BIE methods do, the method combines the free-space Green’s function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a …