Geometry and Topology Commons^™

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

Tutorials on... in 1 hour or less

This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Classification Of Minimal Separating Sets In Low Genus Surfaces, J. J. P. Veerman, William Maxwell, Victor Rielly, Austin K. Williams

Mathematics and Statistics Faculty Publications and Presentations

Consider a surface S and let M ⊂ S. If S \ M is not connected, then we say M separates S, and we refer to M as a separating set of S. If M separates S, and no proper subset of M separates S, then we say M is a minimal separating set of S. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus g = 2 and g = 3. The classification for genus 0 and 1 was done …

Graph Structures In Bipolar Neutrosophic Environment, Florentin Smarandache, Muhammad Akram, Muzzamal Sitara

Branch Mathematics and Statistics Faculty and Staff Publications

A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations.

Constructing A Square An Ancient Indian Way Activity, Cynthia J. Huffman Ph.D.

Open Educational Resources - Math

In this activity students use string to model one of the ways that was used in ancient India for constructing a square. The construction was used in building a temporary fire altar. The activity is based on a translation by Sen and Bag of the Baudhāyana-śulba-sūtra.

Euler Construction Activity, Cynthia J. Huffman Ph.D.

Open Educational Resources - Math

Original sources of mathematics provide many opportunities for students to both do mathematics and to improve their problem solving skills. It is also interesting to explore original sources in new ways with the use of technology. In this activity, students can gain experience with dynamic geometry software and enhance their geometric intuition by working through a construction given by Euler in 1783.

Constructing A Square Indian Fire Altar Activity, Cynthia J. Huffman Ph.D.

Open Educational Resources - Math

In this activity, we will model constructing a square fire altar with a method similar to one used by people in ancient India. The fire altars, which were made of bricks, had various shapes. Instructions for building the altars were in Vedic texts called Śulba-sūtras. We will follow instructions for constructing a square gārhapatya fire altar from the Baudhāyana-śulba-sūtra, which was written during the Middle Vedic period, about 800-500 BC.

Mathematical Description And Mechanistic Reasoning: A Pathway Toward Stem Integration, Paul J. Weinberg

Journal of Pre-College Engineering Education Research (J-PEER)

Because reasoning about mechanism is critical to disciplined inquiry in science, technology, engineering, and mathematics (STEM) domains, this study focuses on ways to support the development of this form of reasoning. This study attends to how mechanistic reasoning is constituted through mathematical description. This study draws upon Smith’s (2007) characterization of mathematical description of scientific phenomena as ‘‘bootstrapping,’’ where negotiating the relationship between target phenomena and represented relations is fundamental to learning. In addition, the development of mathematical representation presents a viable pathway towards STEM integration. In this study, participants responded to an assessment of mechanistic reasoning while cognitive interviews …

The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville

Topology

No abstract provided.

A Compact Introduction To A Generalized Extreme Value Theorem, Nicholas A. Scoville

Topology

In a short paper published just one year prior to his thesis, Maurice Frechet gives a simple generalization one what we might today call the Extreme value theorem. This generalization is a simple matter of coming up with ``the right" definitions in order to make this work. In this mini PSP, we work through Frechet's entire 1.5 page paper to give an extreme value theorem in more general topological spaces, ones which, to use Frechet's newly coined term, are compact.

An Introduction To Topology For The High School Student, Nathaniel Ferron

Masters Essays

No abstract provided.

Complex Neutrosophic Graphs Of Type 1, Florentin Smarandache, Said Broumi, Assia Bakali, Mohamed Talea

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we introduced a new neutrosophic graphs called complex neutrosophic graphs of type1 (CNG1) and presented a matrix representation for it and studied some properties of this new concept. The concept of CNG1 is an extension of generalized fuzzy graphs of type 1 (GFG1) and generalized single valued neutrosophic graphs of type 1 (GSVNG1).

Conference Program, University Of Dayton

Summer Conference on Topology and Its Applications

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications.

Solving Liouville-Type Problems On Manifolds With Poincar´E-Sobolev Inequality By Broadening Q-Energy From Finite To Infinite, Lina Wu

Publications and Research

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincar´e-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincar´e-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincar´e-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of q-energy from finite to infinite. At this point, a …

Turaev Surfaces And Toroidally Alternating Knots, Seungwon Kim

Dissertations, Theses, and Capstone Projects

In this thesis, we study knots and links via their alternating diagrams on closed orientable surfaces. Every knot or link has such a diagram by a construction of Turaev, which is called the Turaev surface of the link. Links that have an alternating diagram on a torus were defined by Adams as toroidally alternating. For a toroidally alternating link, the minimal genus of its Turaev surface may be greater than one. Hence, these surfaces provide different topological measures of how far a link is from being alternating.

First, we classify link diagrams with Turaev genus one and two in terms …

Manifold Convergence: Sewing Sequences Of Riemannian Manifolds With Positive Or Nonnegative Scalar Curvature, Jorge E. Basilio

Dissertations, Theses, and Capstone Projects

In this thesis, we develop a new method of performing surgery on 3-dimensional manifolds called "sewing" and use this technique to construct sequences of Riemannian manifolds with positive or nonnegative scalar curvature. The foundation of our method is a strengthening of the Gromov-Lawson tunnel construction which guarantees the existence of “tiny” and arbitrarily “short” tunnels. We study the limits of sequences of sewn spaces under the Gromov-Hausdorff (GH) and Sormani-Wenger Instrinsic-Flat (SWIF) distances and discuss to what extent the notion of scalar curvature extends to these spaces. We give three applications of the sewing technique to demonstrate that stability theorems …

Intercusp Geodesics And Cusp Shapes Of Fully Augmented Links, Rochy Flint

Dissertations, Theses, and Capstone Projects

We study the geometry of fully augmented link complements in the 3-sphere by looking at their link diagrams. We extend the method introduced by Thistlethwaite and Tsvietkova to fully augmented links and define a system of algebraic equations in terms of parameters coming from edges and crossings of the link diagrams. Combining it with the work of Purcell, we show that the solutions to these algebraic equations are related to the cusp shapes of fully augmented link complements. As an application we use the cusp shapes to study the commensurability classes of fully augmented links.

On Product Stability Of Asymptotic Property C, Gregory C. Bell, Andrzej Nagórko

Summer Conference on Topology and Its Applications

Asymptotic property C is a dimension-like large-scale invariant of metric spaces that is of interest when applied to spaces with infinite asymptotic dimension. It was first described by Dranishnikov, who based it on Haver's topological property C. Topological property C fails to be preserved by products in very striking ways and so a natural question that remained open for some 10+ years is whether asymptotic property C is preserved by products. Using a technique inspired by Rohm we show that asymptotic property C is preserved by direct products of metric spaces.

Domains And Probability Measures: A Topological Retrospective, Michael Mislove

Summer Conference on Topology and Its Applications

Domain theory has seen success as a semantic model for high-level programming languages, having devised a range of constructs to support various effects that arise in programming. One of the most interesting - and problematic - is probabilistic choice, which traditionally has been modeled using a domain-theoretic rendering of sub-probability measures as valuations. In this talk, I will place the domain-theoretic approach in context, by showing how it relates to the more traditional approaches such as functional analysis and set theory. In particular, we show how the topologies that arise in the classic approaches relate to the domain-theoretic rendering. We …

Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, Thomas Weighill, Jerzy Dydak

Summer Conference on Topology and Its Applications

Coarse geometry is the study of the large scale behaviour of spaces. The motivation for studying such behaviour comes mainly from index theory and geometric group theory. In this talk we introduce the notion of (hybrid) large scale normality for large scale spaces and prove analogues of Urysohn’s Lemma and the Tietze Extension Theorem for spaces with this property, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which …

Braid Group Actions On Rational Maps, Eriko Hironaka, Sarah Koch

Summer Conference on Topology and Its Applications

Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if …

Entropy In Topological Groups, Part 1, Dikran Dikranjan

Summer Conference on Topology and Its Applications

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another.

Roughly speaking, entropy is a non-negative real number or infinity assigned to a "selfmap" T of a "space" X, where the "space" X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The "selfmap" T can be, …

Pseudo-Contractibility, Felix Capulín, Leonardo Juarez-Villa, Fernando Orozco

Summer Conference on Topology and Its Applications

Let X, Y be topological spaces and let f, g:X→ Y be mappings, we say that f is pseudo-homotopic to g if there exist a continuum C, points a, b ∈ C and a mapping H:X ×C → Y such that H(x, a)=f(x) and H(x, b)=g(x) for each x ∈ X. The mapping H is called a pseudo-homotopy between f and g. A topological space X is said to be pseudo-contractible if the identity mapping is pseudo-homotopic to a constant mapping in X. i.e., if there exist a continuum C, points a, b ∈ C, x₀ ∈ X and …

Normal Images Of A Product And Countably Paracompact Condensation, Jila Niknejad

Summer Conference on Topology and Its Applications

In 1997, Buzjakova proved that for a pseudocompact Tychonoff space X and λ = | βX|⁺, X condenses onto a compact space if and only if X×(λ+1) condenses onto a normal space. This is a condensation form of Tamano's theorem. An interesting problem is to determine how much of Buzjakova's result will hold if "pseudocompact" is removed from the hypothesis.

In this talk, I am going to show for a Tychonoff space X, there is a cardinal λ such that if X×(λ+1) condenses onto a normal space, then X condenses onto a countably paracompact space.

On The Tightness And Long Directed Limits Of Free Topological Algebras, Gábor Lukács, Rafael Dahmen

Summer Conference on Topology and Its Applications

For a limit ordinal λ, let (A_α)_{α < λ} be a system of topological algebras (e.g., groups or vector spaces) with bonding maps that are embeddings of topological algebras, and put A = ∪_{α < λ} A_α. Let (A, T) and (A, A) denote the direct limit (colimit) of the system in the category of topological spaces and topological algebras, respectively. One always has T ⊇ A, but the inclusion may be strict; however, if the tightness of A is smaller than the cofinality of λ, then A=T.

In 1988, Tkachenko proved …

A New Class Of Dendrites Having Unique Second Symmetric Product, David Maya, José G. Anaya, Fernando Orozco Zitli

Summer Conference on Topology and Its Applications

The second symmetric product of a continuum X, F₂(X), is the hyperspace consisting of all nonempty subsets of X having at most two points. A continuum X has unique hyperspace F₂(X) provided that each continuum Y satisfying that F₂(X) and F₂(Y) are homeomorphic must be homeomorphic to X. In this talk, a new class of dendrites having unique F₂(X) will be presented.

On The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, Leonard Mdzinarishvili, Anzor Beridze

Summer Conference on Topology and Its Applications

The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J.Milnor. In Milnor, the uniqueness theorem is proved using the Eilenberg-Steenrod axioms and as well as relative homeomorphism and clusres axioms. J. Milnor constructed the homology theory on the category Top²_C of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms - the Eilenberg-Steenrod, relative homeomorphis and cluster axioms (see theorem 5 in Milnor). Besides, he proved that constructed homology theory satisfies partial continuity property on the subcategory Top²_CM (see theorem 4 in Milnor) and the …

On Roitman's Principle For Box Products, Hector Alonso Barriga-Acosta

Summer Conference on Topology and Its Applications

One of the oldest problems in box products is if the countable box product of the convergent sequence is normal. It is known that consistenly (e.g., b=d, d=c) the answer is affirmative. A recent progress is due to Judy Roitman that states a combinatorial principle which also implies the normality and holds in many models.

Although the countable box product of the convergent sequence is normal in some models of b < d < c, Roitman asked what happen with her principle in this models. We answer that Roitman's principle is true in some models of b < d < c.

A Compact Minimal Space Whose Cartesian Square Is Not Minimal, Jan P. Boronski, Alex Clark, Piotr Oprocha

Summer Conference on Topology and Its Applications

A compact metric space X is called minimal if it admits a minimal homeomorphism; i.e. a homeomorphism h:X→ X such that the forward orbit {hⁿ(x):n=1, 2, ...} is dense in X, for every x ∈ X. In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from "A compact minimal space Y such that its square YxY is not minimal" whose existence answer the following long standing problem in the negative.

Problem. Is minimality preserved under Cartesian product in the class of compact spaces?

Note that for the fixed point property this question …

Liouville Numbers And One-Sided Ergodic Hilbert Transformations, David Constantine, Joanna Furno

Summer Conference on Topology and Its Applications

We examine one-sided ergodic Hilbert transforms for irrational circle rotations and some mean-zero functions. Our approach uses continued fraction expansions to specify rotations by Liouville numbers for which the transformation has everywhere convergence or divergence.

Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), Joshua Sack

Summer Conference on Topology and Its Applications

A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C^*(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an intermediate ring if C^*(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P describes a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y) both satisfy P, …