DOCSLIB.ORG

Canadian Undergraduate Mathematics Conference 2019 Conférence Canadienne Des Étudiant(E)S En Mathématiques 2019

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Canadian Undergraduate Mathematics Conference 2019 Conférence Canadienne Des Étudiant(E)S En Mathématiques 2019 Canadian Undergraduate Mathematics Conference 2019 Conférence Canadienne des Étudiant(e)s en Mathématiques 2019 July 24th-28th Kingston, Ontario ii Contents Welcome to CUMC 2019! Bienvenue au CCÉM 2019! 1 From the Conference Organizers/ Des Organisateurs et Organisatrices . 1 From the CMS student committee/ Du comité des Étudiants du SMC . 1 Useful Information/Informations utiles . 2 Wi-Fi .................................... 2 Residences/Résidences . 2 Places to Eat and Drink in Kingston1 ....................... 3 On Queen’s Campus . 3 Coffee shop and bakeries . 3 Casual dining, no table service . 4 Casual dining, table service . 5 Conference Schedule/Horaire de la Conférence . 6 Maps/Cartes . 6 Events/Événements 9 Wednesday/ Mercredi July/ juillet 24th ................... 9 Academia and Industry Panel + Booths . 9 Opening Barbecue/Barbecue D’ouverture . 9 Thursday/Jeudi July/ juillet 25th ....................... 10 Math Workshop: Building the 120 cell . 10 Friday/Vendredi July/ juillet 26th ...................... 10 Diversity Panel/ Panel Diversité . 10 Poster Session/ Session des affiches . 11 Software Workshops . 11 Movie Night: Twelve Angry Mathematicians, and The Imitation Game ........................... 11 Saturday/Samedi July/ juillet 27th ...................... 11 So Long Sucker: A Game of Negotiation . 11 Biostatistics Workshop . 12 Closing Banquet/Banquet Final . 12 Plenary Talks/Conférences plénières 13 Algebra and Geometry of Polynomials: Theory and Applications (Bachir El Khadir) ................................... 13 1With thanks to Professor Francesco Cellarosi of Queen’s University for this List. iii Discrete-Event Control to Keep Secrets Secret (Karen Rudie) . 13 An invitation to Geometric Group Theory (Daniel Wise) . 14 Towers, tilings, and dynamical systems (Ayse Sahin) . 14 (Marnie Landon)................................. 14 A quarter century of percolation (Yvan Saint-Aubin) . 15 Mathematics of imperfect vaccines (Felicia Maria G. Magpantay) . 15 What is the Central Limit Theorem? (Ram Murty) . 15 Student Talks/Exposés Étudiants 17 Abstracts/Résumés des présentations . 19 Thursday/Jeudi July/ juillet 25th ....................... 19 Exploring the Shape of Hysteresis Loops in Ordinary Differential Equations (Gina Faraj Rabbah)............................ 19 Suslin’s problem and the axiomatization of the reals (Léo Lortie) . 19 Simplifying Nahm Data with Group Actions (Christopher Lang) . 20 Self-Similarity and Long-Range Dependence (Zhenyuan Zhang) . 20 A Survey of Gerstenhaber’s Problem (Wanchun Shen) . 21 The Case for Recategorification: From Counting Sheep to Representation The- ory (Calder Morton-Ferguson)....................... 21 Tensor Product Reproducing Kernel Hilbert Spaces and Learning Tensor (Junqi Liao)..................................... 22 Introduction to Algebraic Topology (Leon Yao) . 22 The Inscribed Square Problem (Amanda Petcu) . 23 Ciphers for Dummies: How They Work and How to Crack Them (Valerie Gilchrist) 23 Finite Model Theory and First Order Definability (Nikki Sigurdson) . 24 On the Structure of NTRU and BIKE Key Encapsulation Mechanisms (Katarina Spasojevic) ................................. 24 Computational Study of Optimal Control for Mathematical Models for Infec- tious Diseases (Hongruyu Chen)...................... 25 Schur Polynomials and Young Tableaux (Kevin Anderson) . 25 Lie superalgebra : a basic introduction (Ekta Tiwari) . 26 Graph Theory and Stable Sparse Systems (Rebecca Bonham-Carter) . 26 Quantum Computing 101 (Alexandra Kirillova) . 27 Approximating Chance-Constrained Knapsack Sets (Brendan Ross) . 27 Pseudolinear and Psuedocircular Arrangements (Lily Wang) . 27 An Algebraic Proof of Quadratic Reciprocity (Alex Rutar) . 28 Power of States: Electoral College (Max Sun) .................. 28 Spectral Methods and Delay Differential Equations (Reuben Rauch) . 29 Friday/Vendredi July/ juillet 26th ...................... 29 Box count de l’ensemble des fonctions schlicht (Philippe Drouin) . 29 Inca String Theory (Emma Classen-Howes) ................... 29 Connectivity in dominating graphs (Young Lim Ko Park) . 30 Paradoxes in Mathematical Logic (Sonia Knowlton) . 30 It’s not about the model; It’s all about your data (Shuo Feng) . 30 An Introduction to Mapping Class Groups and The Nielsen-Thurston Classifi- cation of Mapping Classes (Curtis Grant) . 31 iv Borel’s Proof of the Heine-Borel Theorem (Zishen Qu) . 31 Extinction of Variant spelling (Suemin Lee) ................... 32 The Bounded Moment Problem (Nour Fahmy).................. 32 Long-Term Dependencies in Neural Networks (Victor Geadah) . 32 An Introduction to the Hilbert Scheme with Applications to Number Theory (Siddharth Mahendraker).......................... 33 Visualization of Water Flow in the Columbia-Kootenay Rivers Confluence (Michelle Boham) ................................... 33 Combinatorial Straightening Algorithms of Classical and Symplectic Bideter- minants (Colin Krawchuk) ......................... 34 Analyzing Risks the Wrong-Way (Agassi Iu)................... 34 Some Applications of Mathematics in Brain Modeling (Rebecca Bonham-Carter) 35 Saturday/Samedi July/ juillet 27th ...................... 35 Fun with Cech˘ Cohomology (Liam M. Fox) ................... 35 A Partial History of the Prime Number Theorem (Sophie Kapsales) . 36 Conformal Field Theories and the Bootstrap (Jonathan Classen-Howes) . 37 Hypernetworks For Team Selection in Professional Soccer (Abdullah Zafar,Farzad Yousefian).................................. 37 Rational Approximations of Numbers (David Salwinski) . 38 Portfolio Optimization in the Multi-period Trinomial Tree Model (Hiromichi Kato) .................................... 38 Weyl’s Law or: The Asymptotic Behavior of the Laplacian Eigenvalues (Charles Senécal)................................... 39 A Graph theory approach for determining the solvability for Quantum Binary Linear System (Junqiao Lin)........................ 39 Le problème de ballottement (Julien Mayrand) . 40 An Upper Bound on Sum-Free Subsets (Nathaniel Libman) . 40 Aggregation Closures for Packing Integer Programs (Haripriya Pulyassary) . 41 A generalization of Singmaster’s conjecture (William Verreault) . 41 Resonant Oscillations in Frustums of Cones (Kevin Dembski) . 41 A bijective proof of the hook-length formula (Yuval Ohapkin) . 42 The mathematics of binary constraint system games (David Cui) . 42 Fermat’s Last Theorem by descent (Stephen Wen) . 43 Feynman diagrams: integration by pictures (Alex Karapetyan) . 43 Introduction to Machine Learning and Deep Neural Networks (Adam Gronowski) 44 The Riemann Hypothesis: A Brief Introduction (David Hoskin) . 44 Hidden Structures in Graphs with Large Chromatic Number (Farbod Yadegarian) 45 The Beauty of the Hyperbolic Plane (Isabel Beach) . 45 An Introduction to Algebraic Number Theory: Unique Factorization (Adrian Carpenter) ................................. 45 What makes a good calculus test?: a summer research project (Kate Ing & Ruo Ning Qiu).................................. 46 Dimensionality Reduction: An Approach to Better Understand Neural Networks (Stefan Horoi)................................ 46 Sunday/Dimanche July/juillet 28th ..................... 46 Introduction to the Ricci Flow (Salim Deaibes) . 47 v The Nine Dragon Tree Conjecture (Logan Grout) . 47 Student Impressions of Active Learning Spaces in First-Year Calculus (Yuveshen Mooroogen)................................. 47 From Plato to Coxeter: The History of Polytopes (Spencer Whitehead) . 48 An Introduction to Matchings (Benjamin Cook) . 48 Numerical Evidence in the Theory of 1/3-homogenous Dendroids (Edie Shillum) 48 An Introduction to Thompson’s Group, and its Connections to Topology (Luke Cooper)................................... 49 A Mathematician’s Introduction to General Relativity (Erin Crawley) . 49 This About Covers It: Broadcast Domination and Multipacking (Elizabeth McKenzie- Case) .................................... 50 On the Computation of Beta Invariants on Toric Varieties (Keenan McPhail) . 50 Localization at prime ideals, and its applications (Amar Venga) . 50 So Long Sucker, a Game of Betrayal (Marie Rose Jerade) . 51 Independent Sets in Double Vertex Graphs (Fady Abdelmalek & Esther Vander Meulen)................................... 51 Girth conditions and Rota’s basis conjecture (Benjamin Friedman) . 52 Algebra of the Hitchin system for matrix valued polynomials over a finite field (Brandon Gill)................................ 52 Sponsors/Commanditaires 53 vi Welcome to CUMC 2019! Bienvenue au CCÉM 2019! From the Conference Organizers/ Des Organisateurs et Organisatrices It is with great pleasure that we would like to welcome you to the 2019 Canadian Unde- graduate Mathematics Conference, here at Queen’s University in Kingston, Ontario. We hope that you enjoy all the sights and sounds that the summer in Kingston has to offer, and that you take this time to meet new people, learn new things, and come together in a spirit of excitement and eagerness to learn. The CUMC Team C’est avec énorme plaisir que nous vous acceuillons à la Conférence Candienne des Étudiant(e)s en Mathématique 2019, ici à l’Université Queen’s chez Kingston, Ontario. Nous espérons que vous allez appréciez tous ce que cette belle ville a à offrir, et que vous allez utiliser cette opportunité pour tisser de nouveaux liens, apprendre des nouvelles choses, et approcher cette expéreience dans un esprit de communauté et d’excitement. L’équipe du CCÉM From the CMS student committee/ Du comité des Étudiants du SMC
Load more

Recommended publications
  • A Survey on the Square Peg Problem
    A survey on the Square Peg Problem Benjamin Matschke∗ March 14, 2014 Abstract This is a short survey article on the 102 years old Square Peg Problem of Toeplitz, which is also called the Inscribed Square Problem. It asks whether every continuous simple closed curve in the plane contains the four vertices of a square. This article first appeared in a more compact form in Notices Amer. Math. Soc. 61(4), 2014. 1 Introduction In 1911 Otto Toeplitz [66] furmulated the following conjecture. Conjecture 1 (Square Peg Problem, Toeplitz 1911). Every continuous simple closed curve in the plane γ : S1 ! R2 contains four points that are the vertices of a square. Figure 1: Example for Conjecture1. A continuous simple closed curve in the plane is also called a Jordan curve, and it is the same as an injective map from the unit circle into the plane, or equivalently, a topological embedding S1 ,! R2. Figure 2: We do not require the square to lie fully inside γ, otherwise there are counter- examples. ∗Supported by Deutsche Telekom Stiftung, NSF Grant DMS-0635607, an EPDI-fellowship, and MPIM Bonn. This article is licensed under a Creative Commons BY-NC-SA 3.0 License. 1 In its full generality Toeplitz' problem is still open. So far it has been solved affirmatively for curves that are \smooth enough", by various authors for varying smoothness conditions, see the next section. All of these proofs are based on the fact that smooth curves inscribe generically an odd number of squares, which can be measured in several topological ways.
    [Show full text]
  • Single Digits
    ...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library
    [Show full text]
  • A Combinatorial Approach to the Inscribed Square Problem
    A Combinatorial Approach to the Inscribed Square Problem Elizabeth Kelley Francis Edward Su, Advisor Rob Thompson, Reader Department of Mathematics May, 2015 Copyright c 2015 Elizabeth Kelley. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract The inscribed square conjecture, also known as Toeplitz’ conjecture or the square peg problem, asserts that every Jordan curve in the Euclidean plane admits an inscribed square. Although there exists no proof of the general conjecture, there are affirmative proofs of the conjecture subject to addi- tional local smoothness conditions. The weakest of these local smooth- ness conditions include the special trapezoid criterion, due to Matschke, and local monotonicity, due to Stromquist. We develop several combinato- rial approaches to the inscribed square problem, explicitly locate inscribed squares for conic sections, and explore the existence of inscribed squares in the Koch snowflake, which we prove to be not locally monotone. Contents Abstract iii 1 Historical Development of the Inscribed Square Problem 1 1.1 Statement of the Problem . .1 1.2 Past Work . .2 1.3 Related Problems . .6 2 Conic Sections 9 3 Koch Snowflake 17 3.1 Inscribed Squares in the Koch Snowflake . 20 4 Discretizing the Parameter Space 35 4.1 Pivot vertex and side length . 36 4.2 Two pivot vertices .
    [Show full text]
  • A Survey on the Square Peg Problem
    A survey on the Square Peg Problem Benjamin Matschke∗ November 23, 2012 Abstract This is a short survey article on the 100 years old Square Peg Problem of Toeplitz, which is also called the Inscribed Square Problem. It asks whether every continuous simple closed curve in the plane contains the four vertices of a square. 1 Introduction In 1911 Otto Toeplitz [Toe11] furmulated the following conjecture. Conjecture 1.1 (Square Peg Problem, Toeplitz). Every continuous simple closed curve γ : S1 ! R2 contains four points that are the vertices of a square. Figure 1: Example for Conjecture 1.1. A continuous simple closed curve in the plane is also often called a Jordan curve, and it is the same as an injective map from the unit circle into the plane, or equivalently, an embedding S1 ,! R2. In its full generality Toeplitz' problem is still open. So far it has been solved affirmatively for curves that are \smooth enough", by various authors for varying smoothness conditions, see Section 2. All of these proofs are based on the fact that smooth curves inscribe generically an odd number of squares, which can be measured in several topological ways. None of these methods however could be made work so far for the general continuous case. One may think that the general case of the Square Peg Problem can be reduced to the case of smooth curves by approximating a given continuous curve γ by a sequence of smooth curves γn: Any γn inscribes a square Qn, and by compactness there is a converging subse- quence (Qnk )k, whose limit is an inscribed square for the given curve γ.
    [Show full text]
  • FOR GEOMETRY Barbara J
    rite” “W W e a h y T M s a t t p h e m m o r a t P ic al s Journ FOR GEOMETRY Barbara J. Dougherty and Jenny Simmons University of Hawai‘i at Manoa– Curriculum Research & Development Group providing quality educational programs and services for preschool through grade 12 © University of Hawai‘i. All rights reserved. For permission requests contact us at: UHM Curriculum Research & Development Group 1776 University Avenue | Honolulu, HI 96822 | (808) 956-4969 | crdg.hawaii.edu The “Write” Way Mathematics Journal Prompts and More FOR GEOMETRY © University of Hawai‘i. All rights reserved. For permission requests contact us at: UHM Curriculum Research & Development Group 1776 University Avenue | Honolulu, HI 96822 | (808) 956-4969 | crdg.hawaii.edu Curriculum Research & Development Group Donald B. Young, Director Kathleen Berg, Associate Director Lori Ward, Managing Editor Cecilia H. Fordham, Project Manager Book design: Darrell Asato Cover design: Byron Inouye Published by Curriculum Research & Development Group ©2006 by the University of Hawai‘i ISBN-10 1-58351-081-8 ISBN-13 978-1-58351-081-0 eISBN 978-1-58351-131-2 Printed in the United States of America All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, or by any storage or retrieval system, without permission in writing from the publisher except as indicated in the material. Distributed by the Curriculum Research & Development Group University of Hawai‘i 1776 University Avenue Honolulu, Hawai‘i 96822-2463 Email: [email protected] Website: http://www.hawaii.edu/crdg © University of Hawai‘i.
    [Show full text]
  • Tomáš Ye Rectangles Inscribed in Jordan Curves
    BACHELOR THESIS Tom´aˇsYe Rectangles inscribed in Jordan curves Mathematical Institute of Charles University Supervisor of the bachelor thesis: doc. RNDr. Zbynˇek S´ır,Ph.D.ˇ Study programme: Mathematics Study branch: General mathematics Prague 2018 I declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act. In ........ date ............ signature of the author i Title: Rectangles inscribed in Jordan curves Author: Tom´aˇsYe Institute: Mathematical Institute of Charles University Supervisor: doc. RNDr. Zbynˇek S´ır,Ph.D.,ˇ department Abstract: We will introduce quotients, which are very special kinds of continuous maps. We are going to study their nice universal properties and use them to formalize the notion of topological gluing. This concept will allow us to define interesting topological structures and analyze them. Finally, the developed theory will be used for writing down a precise proof of the existence of an inscribed rectangle in any Jordan curve. Keywords: Jordan curve, rectangle, topology ii I would like to thank my supervisor doc. RNDr. Zbynˇek S´ır,Ph.D.ˇ for helping me willingly with writing of this bachelor thesis. It was a pleasure working under his guidance. iii Contents Introduction 2 1 Topological preliminaries 3 1.1 General topology .
    [Show full text]
  • The Square Peg Problem
    GRAU DE MATEMATIQUES` Treball final de grau THE SQUARE PEG PROBLEM Autora: Raquel Rius Casado Director: Dr. Joan Carles Naranjo Del Val Realitzat a: Departament de Matem`atiques i Inform`atica Barcelona, 15 de juny de 2019 Contents 1 Introduction 1 2 Jordan Theorem 3 3 Square Peg Problem 8 3.1 History of the problem . .8 3.2 Several results . .9 3.2.1 Witty and easy results . .9 3.2.2 Arnold Emch . 12 3.2.3 Lev Schnirelmann . 17 3.3 Previous tools . 18 3.4 Walter Stromquist's proof . 23 4 Latest study 38 Bibliography 41 Abstract The Square Peg Problem, also known as Toeplitz' Conjecture, is an unsolved problem in the mathematical areas of geometry and topology which states the following: every Jordan curve in the plane inscribes a square. Although it seems like an innocent statement, many authors throughout the last century have tried, but failed, to solve it. It is proved to be true with certain \smoothness conditions" applied on the curve, but the general case is still an open problem. We intend to give a general historical view of the known approaches and, more specifically, focus on an important result that allowed the Square Peg Problem to be true for a great sort of curves: Walter Stromquist's theorem. 2010 Mathematics Subject Classification. 54A99, 53A04, 51M20 Acknowledgements Primer de tot, m'agradaria agrair al meu tutor Joan Carles Naranjo la paci`enciai el recolzament que m'ha oferit durant aquest treball. Moltes gr`aciesper acompanyar-me en aquest trajecte final. Aquesta carrera ha suposat un gran repte personal a nivell mental.
    [Show full text]
  • Single Digits: in Praise of Small Numbers
    March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. ...................................contents preface xi Chapter 1 The Number One 1 Sliced Origami 1 Fibonacci Numbers and the Golden Ratio 2 Representing Numbers Uniquely 5 Factoring Knots 6 Counting and the Stern Sequence 8 Fractals 10 Gilbreath’s Conjecture 13 Benford’s Law 13 The Brouwer Fixed-Point Theorem 16 Inverse Problems 17 Perfect Squares 19 The Bohr–Mollerup Theorem 19 The Picard Theorems 21 Chapter 2 The Number Two 24 The Jordan Curve Theorem and Parity Arguments 24 Aspect Ratio 26 How Symmetric Are You? 27 For general queries, contact [email protected] March 31, 2015 Time: 09:31am prelims.tex © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. vi contents The Pythagorean Theorem 29 Beatty Sequences 32 Euler’s Formula 33 Matters of Prime Importance 34 The Ham Sandwich Theorem 38 Power Sets and Powers of Two 39 The Sylvester–Gallai Theorem 42 Formulas for 43 Multiplication 44 The Thue–Morse Sequence 45 Duals 48 Apollonian Circle Packings 51 Perfect Numbers and Mersenne Primes 53 Pythagorean Tuning and the Square Root of 2 54 Inverse Square Laws 56 The Arithmetic-Geometric Mean Inequality 57 Positive Polynomials 59 Newton’s Method for Root
    [Show full text]
  • Widthheightdepths- Widthheightdepthheight
    What is topology... Some topological theorems that we have learned/will learn U ndergraduate S tudent's T opology(H) C ourse An Introduction Zuoqin Wang March 9, 2021 Zuoqin Wang What is Topology What is topology... Some topological theorems that we have learned/will learn Outline 1 What is topology... A: The origin of the subject B: Some abstract words C: Some concrete pictures 2 Some topological theorems that we have learned/will learn Part I: Point-Set Topology and Applications to Analysis Part II: Algebraic/Geometric Topology Zuoqin Wang What is Topology A: The origin of the subject What is topology... B: Some abstract words Some topological theorems that we have learned/will learn C: Some concrete pictures What's next 1 What is topology... A: The origin of the subject B: Some abstract words C: Some concrete pictures 2 Some topological theorems that we have learned/will learn Part I: Point-Set Topology and Applications to Analysis Part II: Algebraic/Geometric Topology Zuoqin Wang What is Topology A: The origin of the subject What is topology... B: Some abstract words Some topological theorems that we have learned/will learn C: Some concrete pictures A: The origin of the subject/name Zuoqin Wang What is Topology Euler used the name Analysis situs A: The origin of the subject What is topology... B: Some abstract words Some topological theorems that we have learned/will learn C: Some concrete pictures Euler! • The first topologist: Leonhard Euler (1707-1783) Zuoqin Wang What is Topology A: The origin of the subject What is topology..
    [Show full text]
  • Arxiv:2010.05101V1 [Math.MG] 10 Oct 2020 in This Paper, We Prove the Following
    EVERY JORDAN CURVE INSCRIBES UNCOUNTABLY MANY RHOMBI ANTONY T.H. FUNG Abstract. We prove that every Jordan curve in R2 inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve. 1. Introduction The inscribed square problem is a famous open problem due to Otto Toeplitz in 1911 [7], which asks whether all Jordan curves in R2 inscribe a square. By “inscribe”, it means that all four vertices lie on the curve. There are lots of results for the case when the Jordan curve is “nice” [2] [3] [4] [6]. However, other than Vaughan’s result stating that every Jordan curve in R2 inscribes a rectangle (see [5]), little progress has been done on the general case. One of the earliest results was Arnold Emch’s work [1]. In 1916, he proved that all piecewise analytic Jordan curves in R2 with only finitely many “bad” points in- scribe a square. Emch’s approach was that given a direction τ, construct the set of medians Mτ , defined as (roughly speaking) the set of midpoints of pairs of points on the curve that lie on the same line going in direction τ. When the Jordan curve is nice enough, Mτ is a nice path. Now consider an orthogonal direction σ and sim- ilarly construct Mσ. The paths Mτ and Mσ must intersect, and their intersections correspond to quadrilaterals with diagonals perpendicularly bisecting each other, i.e. rhombi. Then by rotating and using the intermediate value theorem, he showed that at some direction, one of those rhombi is a square. In this paper, we will follow a similar approach to show the existence of rhombi.
    [Show full text]
  • The Ambitious Horse: Ancient Chinese Mathematics Problems the Right to Reproduce Material for Use in His Or Her Own Classroom
    The AW\bitious t-lo .. se Ancient Chinese Mathematics Problems Law~ence W. Swienciki, Ph.D. ~~~~~ Key Curriculum Press II Innovators in Mathematics Education Project Editor: Anna Werner Editorial Assistants: Beth Masse, Halo Golden Reviewers: Dr. Robin D. S. Yates, McGill University, Montreal, Canada, and Tian Zaijin, Senior Editor, People's Education Press, Beijing, China Consultant: Wei Zhang Production Editor: Jennifer Strada Copy Editor: Margaret Moore Production and Manufacturing Manager: Diana Jean Parks Production Coordinator: Laurel Roth Patton Text Designer/Illustrator and Cover Illustrator: Lawrence W. Swienciki Compositor: Kirk Mills Art Editor: Jason Luz Cover Designer: Caroline Ayres Prepress and Printer: Malloy Lithographing, Inc. Executive Editor: Casey FitzSimons Publisher: Steven Rasmussen © 2001 by Key Curriculum Press. All rights reserved. Limited Reproduction Permission The publisher grants the teacher who purchases The Ambitious Horse: Ancient Chinese Mathematics Problems the right to reproduce material for use in his or her own classroom. Unauthorized copying of The Ambitious Horse: Ancient Chinese Mathematics Problems constitutes copyright infringement and is a violation of federal law. Key Curriculum Press 1150 65th Street Emeryville, CA 94608 510-595-7000 [email protected] http: I /www.keypress.com Printed in the United States of America 10 9 8 7 6 5 4 3 2 05 04 ISBN 1-55953-461-3 Cof!\tef!\ts Preface v Suggested Uses vii Chinese Writing 1 Traditional Numerals 3 Ling Symbol for Zero 4 Instruction Chart 5 The
    [Show full text]
  • | Problem Section | Problem Denoted Additively
    134 NAW 5/22 nr. 2 juni 2021 Problemen Edition 2021-1 We received solutions from Rik Bos, Pieter de Groen, Nicky Hekster, Marnix Klooster and Rutger Wessels. Problem 2021-1/A (proposed by Daan van Gent and Hendrik Lenstra) Let G and A be groups, where G is denoted multiplicatively and where A is abelian and | Problem Section | Problem denoted additively. Assume that A is 2-torsion-free, i.e. it contains no element of order 2. Suppose that qG: " A is a map satisfying the parallelogram identity: for all xy, ! G we have qx()yq+=()xy-1 22qx()+ qy(). (1) Prove that for all xy, ! G we have qx()yx--11y = 0. Solution We received solutions from Rik Bos, Nicky Hekster, Marnix Klooster, and Rutger Wessels. The solution provided uses the one by Nicky. We let 1 denote the identity element of G, and 0 the identity element of A. Substituting y = 1 yields 21q()= 0, whence q()10= since A is 2-torsion free. Substituting x = 1 in (1) gives qy()= qy()-1 forall yG! . (2) In the case xy= , formula (1) implies qx()2 = 4qx() forall xG! . (3) By switching the x and y in (1) we obtain qy()xq+=()yx-1 22qy()+ qx(), so using (2) we get qx()yq+=()xy--11qx()yq+=((yx ))--11qx()yq+=()yx qy()xq+ ()yx-1 , hence qx()yq= ()yx forall xy,.! G Applying (1) with x replaced by xy we see that (4) qx()yq2 =-22()xy qx()+ qy() forall xy,.! G (5) Next we claim that 2 ! Problemen qx()yx = 4qx()yxforall ,.yG (6) Indeed, qx()yx22==qx()yq2222()xy-+qx()22qy() =-24qy()xq()xq+ 2 ()y (4)(5) (4)(3) =-22((qyxq)(yq)(+-24xq)) ()xq+=24()yq()xy .
    [Show full text]