Constructing A Square An Ancient Indian Way Activity, 2017 Pittsburg State University

#### 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, 2017 Pittsburg State University

#### 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, 2017 Pittsburg State University

#### 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, 2017 Oakland University

#### 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 ...

The Closure Operation As The Foundation Of Topology, 2017 Ursinus College

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

*Topology*

No abstract provided.

A Compact Introduction To A Generalized Extreme Value Theorem, 2017 Ursinus College

#### 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, 2017 John Carroll University

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

*Masters Essays*

No abstract provided.

Conference Program, 2017 University of Dayton

#### 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.

Turaev Surfaces And Toroidally Alternating Knots, 2017 The Graduate Center, City University of New York

#### Turaev Surfaces And Toroidally Alternating Knots, Seungwon Kim

*All Graduate Works by Year: 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 ...

Intercusp Geodesics And Cusp Shapes Of Fully Augmented Links, 2017 The Graduate Center, City University of New York

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

*All Graduate Works by Year: 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.

Enriched Topology And Asymmetry, 2017 Youngstown State University

#### Enriched Topology And Asymmetry, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton

*Summer Conference on Topology and Its Applications*

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be "variable-basis". This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces ...

On A Construction Of Some Class Of Metric Spaces, 2017 Adam Mickiewicz University of Poznan

#### On A Construction Of Some Class Of Metric Spaces, Dariusz Bugajewski

*Summer Conference on Topology and Its Applications*

In this talk we are going to describe Sz´az’s construction of some class of metric spaces. Most of all we will analyze topological properties of metric spaces obtained by using Sz´az’s construction. In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we will discuss the relation between Sz´az’s construction and the “linking construction”. A particular attention will be drawn to the “floor” metric, the analysis of which provides some interesting observations.

Liouville Numbers And One-Sided Ergodic Hilbert Transformations, 2017 Wesleyan University

#### 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.

Compactness Via Adherence Dominators, 2017 Morgan State University

#### Compactness Via Adherence Dominators, Bhamini M. P. Nayar, Terrence A. Edwards, James E. Joseph, Myung H. Kwack

*Summer Conference on Topology and Its Applications*

This talk is based on a joint work by T. A. Edwards, J. E. Joseph, M. H. Kwack and B. M. P. Nayar that apperared in the *Journal of Advanced studies in Topology,* Vol. 5 (4), 2014), 8 - 15. B

An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the family of closed subsets of X satisfying A(Ω) ⊆ π(Ω) where A(Ω) is the adherence of Ω. The notations π(Ω) and A(Ω) are used for the values of the functions π and A and π ...

Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, 2017 University of Oklahoma

#### Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, Ignat Soroko, Robert Kropholler, Ian Leary

*Summer Conference on Topology and Its Applications*

An interplay between algebra and topology goes in many ways. Given a space X, we can study its homology and homotopy groups. In the other direction, given a group G, we can form its Eilenberg-Maclane space K(G, 1). It is natural to wish that it is `small' in some sense. If K(G, 1) space has n-skeleton with finitely many cells, then G is said to have type F_{n}. Such groups act naturally on the cellular chain complex of the universal cover for K(G, 1), which has finitely generated free modules in all dimensions up to n ...

On The Tightness And Long Directed Limits Of Free Topological Algebras, 2017 Technische Universitat Darmstadt

#### 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 that the free topological group F(X) is sequential ...

Domains And Probability Measures: A Topological Retrospective, 2017 Tulane University

#### 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 also describe ...

On The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, 2017 Georgian Technical University

#### 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 Top^{2}_{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 Top^{2}_{CM} (see theorem 4 in Milnor) and the ...

Locally Compact Groups: Traditions And Trends, 2017 Technische Universitat Darmstadt

#### Locally Compact Groups: Traditions And Trends, Karl Heinrich Hofmann, Wolfgang Herfort, Francesco G. Russo

*Summer Conference on Topology and Its Applications*

For a lecture in the Topology+Algebra and Analysis section, the subject of locally compact groups appears particularly fitting: Historically and currently as well, the structure and representation theory of locally compact groups draws its methods from each of theses three fields of mathematics. Nowadays one might justifiably add combinatorics and number theory as sources. The example of a study of a class of locally compact groups called “near abelian,” undertaken by W. Herfort, K. H. Hofmann, and F. G. Russo, may be used to illustrate the liaison of topological group theory with this different areas of interest. Concepts like ...

On Di-Injective T0-Quasi-Metric Spaces, 2017 North-West University (South Africa)

#### On Di-Injective T0-Quasi-Metric Spaces, Collins Amburo Agyingi

*Summer Conference on Topology and Its Applications*

We prove that every q-hyperconvex T0-quasi-metric space (X, d) is di-injective without appealing to Zorn’s lemma. We also demonstrate that QX as constructed by Kemajou et al. and Q(X) (the space of all Katˇetov function pairs on X) are di-injective. Moreover we prove that di-injective T0-quasi-metric spaces do not contain proper essential extensions. Among other results, we state a number of ways in which the the di-injective envelope of a T0-quasi-metric space can be characterized.