Curve Shortening Flow
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Section 1.7B the Four-Vertex Theorem
Section 1.7B The Four-Vertex Theorem Matthew Hendrix Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition The integer I is called the rotation index of the curve α where I is an integer multiple of 2π; that is, Z 1 k(s) dt = θ(`) − θ(0) = 2πI 0 Definition 2 Let α : [0; `] ! R be a plane closed curve given by α(s) = (x(s); y(s)). Since s is the arc length, the tangent vector t(s) = (x0(s); y 0(s)) has unit length. The tangent indicatrix is 2 0 0 t : [0; `] ! R , which is given by t(s) = (x (s); y (s)); this is a differentiable curve, the trace of which is contained in the circle of radius 1. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 Let α : [0; `] ! R be a plane closed curve given by α(s) = (x(s); y(s)). Since s is the arc length, the tangent vector t(s) = (x0(s); y 0(s)) has unit length. The tangent indicatrix is 2 0 0 t : [0; `] ! R , which is given by t(s) = (x (s); y (s)); this is a differentiable curve, the trace of which is contained in the circle of radius 1. Definition The integer I is called the rotation index of the curve α where I is an integer multiple of 2π; that is, Z 1 k(s) dt = θ(`) − θ(0) = 2πI 0 Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 A vertex of a regular plane curve α :[a; b] ! R is a point t 2 [a; b] where k0(t) = 0. -
Symmetrization of Convex Plane Curves
Symmetrization of convex plane curves P. Giblin and S. Janeczko Abstract Several point symmetrizations of a convex curve Γ are introduced and one, the affinely invariant ‘cen- tral symmetric transform’ (CST) with respect to a given basepoint inside Γ, is investigated in detail. Examples for Γ include triangles, rounded triangles, ellipses, curves defined by support functions and piecewise smooth curves. Of particular interest is the region of basepoints for which the CST is convex (this region can be empty but its complement in the interior of Γ is never empty). The (local) boundary of this region can have cusps and in principle it can be determined from a geometrical construction for the tangent direction to the CST. MR Classification 52A10, 53A04 1 Introduction There are several ways in which a convex plane curve Γ can be transformed into one which is, in some sense, symmetric. Here are three possible constructions. PT Assume that Γ is at least C1. For each pair of parallel tangents of Γ construct two parallel lines with the same separation and the origin (or any other fixed point) equidistant from them. The resulting envelope, the parallel tangent transform, will be symmetric about the chosen fixed point. All choices of fixed point give the same transform up to translation. This construction is affinely invariant. A curve of constant width, where the separation of parallel tangent pairs is constant, will always transform to a circle. Another example is given in Figure 1(a). This construction can also be interpreted in the following way: Reflect Γ in the chosen fixed point 1 to give Γ∗ say. -
One-Dimensional Projective Structures, Convex Curves and the Ovals of Benguria & Loss
published version: Comm. Math. Phys. 336 (2015), 933-952 One-dimensional Projective Structures, Convex Curves and the Ovals of Benguria & Loss JACOB BERNSTEIN AND THOMAS METTLER ABSTRACT. Benguria and Loss have conjectured that, amongst all smooth closed curves in R2 of length 2, the lowest possible eigenvalue of the op- erator L 2 is 1. They observed that this value was achieved on a D C two-parameter family, O, of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in O as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all three problems. 1. Introduction In[ 1], Benguria and Loss conjectured that for any, , a smooth closed curve in R2 2 of length 2, the lowest eigenvalue, , of the operator L satisfied D C 1 1. That is, they conjectured that for all such and all functions f H ./, 2 Z Z 2 2 2 2 (1.1) f f ds f ds; jr j C where f is the intrinsic gradient of f , is the geodesic curvature and ds is the lengthr element. This conjecture was motivated by their observation that it was equivalent to a certain one-dimensional Lieb-Thirring inequality with conjectured sharp constant. They further observed that the above inequality is saturated on a two-parameter family of strictly convex curves which contains the round circle and degenerates into a multiplicity-two line segment. The curves in this family look like ovals and so we call them the ovals of Benguria and Loss and denote the family by 1 O. -
Asymptotic Approximation of Convex Curves
Asymptotic Approximation of Convex Curves Monika Ludwig Abstract. L. Fejes T´othgave asymptotic formulae as n → ∞ for the distance between a smooth convex disc and its best approximating inscribed or circumscribed polygons with at most n vertices, where the distance is in the sense of the symmetric difference metric. In this paper these formulae are extended by specifying the second terms of the asymptotic expansions. Tools are from affine differential geometry. 1 Introduction 2 i Let C be a closed convex curve in the Euclidean plane IE and let Pn(C) be the set of all convex polygons with at most n vertices that are inscribed in C. We i measure the distance of C and Pn ∈ Pn(C) by the symmetric difference metric δS and study the asymptotic behaviour of S i S i δ (C, Pn) = inf{δ (C, Pn): Pn ∈ Pn(C)} S c as n → ∞. In case of circumscribed polygons the analogous notion is δ (C, Pn). For a C ∈ C2 with positive curvature function κ(t), the following asymptotic formulae were given by L. Fejes T´oth [2], [3] Z l !3 S i 1 1/3 1 δ (C, Pn) ∼ κ (t)dt as n → ∞ (1) 12 0 n2 and Z l !3 S c 1 1/3 1 δ (C, Pn) ∼ κ (t)dt as n → ∞, (2) 24 0 n2 where t is the arc length and l the length of C. Complete proofs of these results are due to D. E. McClure and R. A. Vitale [9]. In this article we extend these asymptotic formulae by deriving the second terms in the asymptotic expansions. -
Naming Infinity: a True Story of Religious Mysticism And
Naming Infinity Naming Infinity A True Story of Religious Mysticism and Mathematical Creativity Loren Graham and Jean-Michel Kantor The Belknap Press of Harvard University Press Cambridge, Massachusetts London, En gland 2009 Copyright © 2009 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Library of Congress Cataloging-in-Publication Data Graham, Loren R. Naming infinity : a true story of religious mysticism and mathematical creativity / Loren Graham and Jean-Michel Kantor. â p. cm. Includes bibliographical references and index. ISBN 978-0-674-03293-4 (alk. paper) 1. Mathematics—Russia (Federation)—Religious aspects. 2. Mysticism—Russia (Federation) 3. Mathematics—Russia (Federation)—Philosophy. 4. Mathematics—France—Religious aspects. 5. Mathematics—France—Philosophy. 6. Set theory. I. Kantor, Jean-Michel. II. Title. QA27.R8G73 2009 510.947′0904—dc22â 2008041334 CONTENTS Introduction 1 1. Storming a Monastery 7 2. A Crisis in Mathematics 19 3. The French Trio: Borel, Lebesgue, Baire 33 4. The Russian Trio: Egorov, Luzin, Florensky 66 5. Russian Mathematics and Mysticism 91 6. The Legendary Lusitania 101 7. Fates of the Russian Trio 125 8. Lusitania and After 162 9. The Human in Mathematics, Then and Now 188 Appendix: Luzin’s Personal Archives 205 Notes 212 Acknowledgments 228 Index 231 ILLUSTRATIONS Framed photos of Dmitri Egorov and Pavel Florensky. Photographed by Loren Graham in the basement of the Church of St. Tatiana the Martyr, 2004. 4 Monastery of St. Pantaleimon, Mt. Athos, Greece. 8 Larger and larger circles with segment approaching straight line, as suggested by Nicholas of Cusa. 25 Cantor ternary set. -
Arxiv:1805.06667V4 [Math.NA] 26 Jun 2019 Kowloon, Hong Kong E-Mail: [email protected] 2 B
Version of June 27, 2019 A convergent evolving finite element algorithm for mean curvature flow of closed surfaces Bal´azsKov´acs · Buyang Li · Christian Lubich This paper is dedicated to Gerhard Dziuk on the occasion of his 70th birthday and to Gerhard Huisken on the occasion of his 60th birthday. Abstract A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk's method, and linearly implicit backward difference formulae for time integration. The proposed method dif- fers from Dziuk's approach in that it discretizes Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This nu- merical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency esti- mates to yield optimal-order H1-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix{vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results. Keywords mean curvature flow · geometric evolution equations · evolving surface finite elements · linearly implicit backward difference formula · stability · convergence analysis Mathematics Subject Classification (2000) 35R01 · 65M60 · 65M15 · 65M12 B. -
Hamilton's Ricci Flow
The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland. -
Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
UC San Diego UC San Diego Electronic Theses and Dissertations Title Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere Permalink https://escholarship.org/uc/item/7gt6c9nw Author Louie, Janelle Publication Date 2014 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Janelle Louie Committee in charge: Professor Bennett Chow, Chair Professor Kenneth Intriligator Professor Elizabeth Jenkins Professor Lei Ni Professor Jeffrey Rabin 2014 Copyright Janelle Louie, 2014 All rights reserved. The dissertation of Janelle Louie is approved, and it is ac- ceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2014 iii TABLE OF CONTENTS Signature Page.................................. iii Table of Contents................................. iv List of Figures..................................v Acknowledgements................................ vi Vita........................................ vii Abstract of the Dissertation........................... viii Chapter 1 Introduction............................1 1.1 Previous work and background...............2 1.2 Results and outline of the thesis..............4 Chapter 2 Preliminaries...........................6 2.1 -
Isoptics of a Closed Strictly Convex Curve. - II Rendiconti Del Seminario Matematico Della Università Di Padova, Tome 96 (1996), P
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA W. CIESLAK´ A. MIERNOWSKI W. MOZGAWA Isoptics of a closed strictly convex curve. - II Rendiconti del Seminario Matematico della Università di Padova, tome 96 (1996), p. 37-49 <http://www.numdam.org/item?id=RSMUP_1996__96__37_0> © Rendiconti del Seminario Matematico della Università di Padova, 1996, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Isoptics of a Closed Strictly Convex Curve. - II. W. CIE015BLAK (*) - A. MIERNOWSKI (**) - W. MOZGAWA(**) 1. - Introduction. ’ This article is concerned with some geometric properties of isoptics which complete and deepen the results obtained in our earlier paper [3]. We therefore begin by recalling the basic notions and necessary results concerning isoptics. An a-isoptic Ca of a plane, closed, convex curve C consists of those points in the plane from which the curve is seen under the fixed angle .7r - a. We shall denote by C the set of all plane, closed, strictly convex curves. Choose an element C and a coordinate system with the ori- gin 0 in the interior of C. Let p(t), t E [ 0, 2~c], denote the support func- tion of the curve C. -
Stewart I. Visions of Infinity.. the Great Mathematical Problems
VISIONS OF INFINITY Also by Ian Stewart Concepts of Modern Mathematics Game, Set, and Math The Problems of Mathematics Does God Play Dice? Another Fine Math You’ve Got Me Into Fearful Symmetry (with Martin Golubitsky) Nature’s Numbers From Here to Infinity The Magical Maze Life’s Other Secret Flatterland What Shape Is a Snowflake? The Annotated Flatland Math Hysteria The Mayor of Uglyville’s Dilemma Letters to a Young Mathematician Why Beauty Is Truth How to Cut a Cake Taming the Infinite/The Story of Mathematics Professor Stewart’s Cabinet of Mathematical Curiosities Professor Stewart’s Hoard of Mathematical Treasures Cows in the Maze Mathematics of Life In Pursuit of the Unknown with Terry Pratchett and Jack Cohen The Science of Discworld The Science of Discworld II: The Globe The Science of Discworld III: Darwin’s Watch with Jack Cohen The Collapse of Chaos Figments of Reality Evolving the Alien/What Does a Martian Look Like? Wheelers (science fiction) Heaven (science fiction) VISIONS OF INFINITY The Great Mathematical Problems IAN STEWART A Member of the Perseus Books Group New York Copyright © 2013 by Joat Enterprises Published by Basic Books, A Member of the Perseus Books Group All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 250 West 57th Street, New York, NY 10107. Books published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. -
A Local Regularity Theorem for Mean Curvature Flow
Annals of Mathematics, 161 (2005), 1487–1519 A local regularity theorem for mean curvature flow By Brian White* Abstract This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to 1. Introduction Let Mt with 0 < t < T be a smooth one-parameter family of embed- ded manifolds, not necessarily compact, moving by mean curvature in RN . This paper proves uniform curvature bounds in regions of spacetime where the Gaussian density ratios are close to 1. For instance (see 3.4): § Theorem. There are numbers ε = ε(N) > 0 and C = C(N) < with ∞ the following property. If is a smooth, proper mean curvature flow in an M open subset U of the spacetime RN R and if the Gaussian density ratios of × are bounded above by 1 + ε, then at each spacetime point X = (x, t) of , M M the norm of the second fundamental form of at X is bounded by M C δ(X, U) where δ(X, U) is the infimum of X Y among all points Y = (y, s) U c # − # ∈ with s t. ≤ (The terminology will be explained in 2.) § Another paper [W5] extends the bounds to arbitrary mean curvature flows of integral varifolds. However, that extension seems to require Brakke’s Local Regularity Theorem [B, 6.11], the proof of which is very difficult. The results of this paper are much easier to prove, but nevertheless suffice in many interesting situations. In particular: *The research presented here was partially funded by NSF grants DMS-9803403, DMS- 0104049, DMS-0406209 and by a Guggenheim Foundation Fellowship. -
Biography of N. N. Luzin
BIOGRAPHY OF N. N. LUZIN http://theor.jinr.ru/~kuzemsky/Luzinbio.html BIOGRAPHY OF N. N. LUZIN (1883-1950) born December 09, 1883, Irkutsk, Russia. died January 28, 1950, Moscow, Russia. Biographic Data of N. N. Luzin: Nikolai Nikolaevich Luzin (also spelled Lusin; Russian: НиколайНиколаевич Лузин) was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the co-founder of "Luzitania" (together with professor Dimitrii Egorov), a close group of young Moscow mathematicians of the first half of the 1920s. This group consisted of the higly talented and enthusiastic members which form later the core of the famous Moscow school of mathematics. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics. Luzin started studying mathematics in 1901 at Moscow University, where his advisor was professor Dimitrii Egorov (1869-1931). Professor Dimitrii Fedorovihch Egorov was a great scientist and talented teacher; in addition he was a person of very high moral principles. He was a Russian and Soviet mathematician known for significant contributions to the areas of differential geometry and mathematical analysis. Egorov was devoted and openly practicized member of Russian Orthodox Church and active parish worker. This activity was the reason of his conflicts with Soviet authorities after 1917 and finally led him to arrest and exile to Kazan (1930) where he died from heavy cancer. From 1910 to 1914 Luzin studied at Gottingen, where he was influenced by Edmund Landau. He then returned to Moscow and received his Ph.D.