Extremal Solutions to Some Art Gallery and Terminal-Pairability Problems
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Proofs from the BOOK.Pdf
Martin Aigner Günter M. Ziegler Proofs from THE BOOK Fourth Edition Martin Aigner Günter M. Ziegler Proofs from THE BOOK Fourth Edition Including Illustrations by Karl H. Hofmann 123 Prof. Dr. Martin Aigner Prof.GünterM.Ziegler FB Mathematik und Informatik Institut für Mathematik, MA 6-2 Freie Universität Berlin Technische Universität Berlin Arnimallee 3 Straße des 17. Juni 136 14195 Berlin 10623 Berlin Deutschland Deutschland [email protected] [email protected] ISBN 978-3-642-00855-9 e-ISBN 978-3-642-00856-6 DOI 10.1007/978-3-642-00856-6 Springer Heidelberg Dordrecht London New York c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Paul Erdosliked˝ to talk aboutThe Book, in which God maintainsthe perfect proofsfor mathematical theorems, following the dictum of G. -
Stony Brook University
SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Combinatorics and Complexity in Geometric Visibility Problems A Dissertation Presented by Justin G. Iwerks to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics and Statistics (Operations Research) Stony Brook University August 2012 Stony Brook University The Graduate School Justin G. Iwerks We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Joseph S. B. Mitchell - Dissertation Advisor Professor, Department of Applied Mathematics and Statistics Esther M. Arkin - Chairperson of Defense Professor, Department of Applied Mathematics and Statistics Steven Skiena Distinguished Teaching Professor, Department of Computer Science Jie Gao - Outside Member Associate Professor, Department of Computer Science Charles Taber Interim Dean of the Graduate School ii Abstract of the Dissertation Combinatorics and Complexity in Geometric Visibility Problems by Justin G. Iwerks Doctor of Philosophy in Applied Mathematics and Statistics (Operations Research) Stony Brook University 2012 Geometric visibility is fundamental to computational geometry and its ap- plications in areas such as robotics, sensor networks, CAD, and motion plan- ning. We explore combinatorial and computational complexity problems aris- ing in a collection of settings that depend on various notions of visibility. We first consider a generalized version of the classical art gallery problem in which the input specifies the number of reflex vertices r and convex vertices c of the simple polygon (n = r + c). -
Exploring Topics of the Art Gallery Problem
The College of Wooster Open Works Senior Independent Study Theses 2019 Exploring Topics of the Art Gallery Problem Megan Vuich The College of Wooster, [email protected] Follow this and additional works at: https://openworks.wooster.edu/independentstudy Recommended Citation Vuich, Megan, "Exploring Topics of the Art Gallery Problem" (2019). Senior Independent Study Theses. Paper 8534. This Senior Independent Study Thesis Exemplar is brought to you by Open Works, a service of The College of Wooster Libraries. It has been accepted for inclusion in Senior Independent Study Theses by an authorized administrator of Open Works. For more information, please contact [email protected]. © Copyright 2019 Megan Vuich Exploring Topics of the Art Gallery Problem Independent Study Thesis Presented in Partial Fulfillment of the Requirements for the Degree Bachelor of Arts in the Department of Mathematics and Computer Science at The College of Wooster by Megan Vuich The College of Wooster 2019 Advised by: Dr. Robert Kelvey Abstract Created in the 1970’s, the Art Gallery Problem seeks to answer the question of how many security guards are necessary to fully survey the floor plan of any building. These floor plans are modeled by polygons, with guards represented by points inside these shapes. Shortly after the creation of the problem, it was theorized that for guards whose positions were limited to the polygon’s j n k vertices, 3 guards are sufficient to watch any type of polygon, where n is the number of the polygon’s vertices. Two proofs accompanied this theorem, drawing from concepts of computational geometry and graph theory. -
Paul Erdős Mathematical Genius, Human (In That Order)
Paul Erdős Mathematical Genius, Human (In That Order) By Joshua Hill MATH 341-01 4 June 2004 "A Mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent that theirs, it is because the are made with ideas... The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours of the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." --G.H. Hardy "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is." -- Paul Erdős "One of the first people I met in Princeton was Paul Erdős. He was 26 years old at the time, and had his Ph.D. for several years, and had been bouncing from one postdoctoral fellowship to another... Though I was slightly younger, I considered myself wiser in the ways of the world, and I lectured Erdős "This fellowship business is all well and good, but it can't go on for much longer -- jobs are hard to get -- you had better get on the ball and start looking for a real honest job." ... Forty years after my sermon, Erdős hasn't found it necessary to look for an "honest" job yet." -- Paul Halmos Introduction Paul Erdős (said "Air-daish") was a brilliant and prolific mathematician, who was central to the advancement of several major branches of mathematics. -
Proofs from the BOOK Third Edition Springer-Verlag Berlin Heidelberg Gmbh Martin Aigner Gunter M
Martin Aigner Gunter M. Ziegler Proofs from THE BOOK Third Edition Springer-Verlag Berlin Heidelberg GmbH Martin Aigner Gunter M. Ziegler Proofs from THE BOOK Third Edition With 250 Figures Including Illustrations by Karl H. Hofmann Springer Martin Aigner Gunter M. Ziegler Freie Universitat Berlin Technische Universitat Berlin Institut flir Mathematik II (WE2) Institut flir Mathematik, MA 6-2 Arnimallee 3 StraBe des 17. Juni 136 14195 Berlin, Germany 10623 Berlin, Germany email: [email protected] email: [email protected] Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. Mathematics Subject Classification (2000): 00-01 (General) ISBN 978-3-662-05414-7 ISBN 978-3-662-05412-3 (eBook) DOl 10 .1007/978-3 -662-05412-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Viola tions are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1998,2001,2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004. -
The Art Gallery Theorem
Outline The players The theorem The proof from THE BOOK Variations The Art Gallery Theorem Vic Reiner, Univ. of Minnesota Monthly Math Hour at UW May 18, 2014 Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations 1 The players 2 The theorem 3 The proof from THE BOOK 4 Variations Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Victor Klee, formerly of UW Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Klee’s question posed to V. Chvátal Given the floor plan of a weirdly shaped art gallery having N straight sides, how many guards will we need to post, in the worst case, so that every bit of wall is visible to a guard? Can one do it with N=3 guards? Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Klee’s question posed to V. Chvátal Given the floor plan of a weirdly shaped art gallery having N straight sides, how many guards will we need to post, in the worst case, so that every bit of wall is visible to a guard? Can one do it with N=3 guards? Vic Reiner, Univ. of Minnesota The Art Gallery Theorem Outline The players The theorem The proof from THE BOOK Variations Vasek Chvátal: Yes, I can prove that! Vic Reiner, Univ. -
An Introductory Study on Art Gallery Theorems and Problems
AN INTRODUCTORY STUDY ON ART GALLERY THEOREMS AND PROBLEMS JEFFRY CHHIBBER M Dept of Mathematics, Noorul Islam Centre of Higher Education, Kanyakumari, Tamil Nadu, India. E-mail: [email protected] Abstract - In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Visibility graphs may also be used to calculate the placement of radio antennas, or as a tool used within architecture and urban planningthrough visibility graph analysis. This is a brief survey on the visibility graphs application in Art Gallery Problems and Theorems. Keywords- Art galley theorems, orthogonal polygon, triangulation, Visibility graphs I. INTRODUCTION point y outside of P if the segment xy is nowhere interior to P; xy may intersect ∂P, the boundary of P. In a visibility graph, each node in the graph Star polygon: A polygon visible from a single interior represents a point location, and each edge represents point. Diagonal: A segment inside a polygon whose a visible connectionbetween them. That is, if the line endpoints are vertices, and which otherwise does not segment connecting two locations does not pass touch ∂P. Floodlight: A light that illuminates from the through any obstacle, an edge is drawn between them apex of a cone with aperture α. Vertex floodlight: in the graph. Lozano-Perez & Wesley (1979) attribute One whose apex is at a vertex (at most one per the visibility graph method for Euclidean shortest vertex). paths to research in 1969 by Nils Nilsson on motion planning for Shakey the robot, and also cite a 1973 The problem: description of this method by Russian mathematicians What is the art gallery problem? M. -
3.1 Prime Numbers
Chapter 3 I Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocks in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime if its only positive integer multiplicative factors are 1 and n. Furthermore, every integer can be expressed as a product of primes, and this expression is unique up to the order of the primes in the product. This important insight into the multiplicative structure of the integers has become known as the fundamental theorem of arithmetic . Beneath the simplicity of the prime numbers lies a sophisticated world of insights and results that has intrigued mathematicians for centuries. By the third century b.c.e. , Greek mathematicians had defined prime numbers, as one might expect from their familiarity with the division algorithm. In Book IX of Elements [73], Euclid gives a proof of the infinitude of primes—one of the most elegant proofs in all of mathematics. Just as important as this understanding of prime numbers are the many unsolved questions about primes. For example, the Riemann hypothesis is one of the most famous open questions in all of mathematics. This claim provides an analytic formula for the number of primes less than or equal to any given natural number. A proof of the Riemann hypothesis also has financial rewards. The Clay Mathematics Institute has chosen six open questions (including the Riemann hypothesis)—a complete solution of any one would earn a $1 million prize. Working toward defining a prime number, we recall an important theorem and definition from section 2.2. -
Formalization of Some Central Theorems in Combinatorics of Finite
Kalpa Publications in Computing Volume 1, 2017, Pages 43–57 LPAR-21S: IWIL Workshop and LPAR Short Presentations Formalization of some central theorems in combinatorics of finite sets Abhishek Kr Singh School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai [email protected] Abstract We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth’s decomposition theorem, Mirsky’s theorem, Hall’s marriage theorem and the Erdős-Szekeres theorem. Dilworth’s decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky’s theorem is a dual of Dilworth’s decomposition theorem, which states that in any finite poset, the size of a smallest antichain cover and a largest chain are the same. We use Dilworth’s theorem in the proofs of Hall’s Marriage theorem and the Erdős-Szekeres theorem. The combinatorial objects involved in these theorems are sets and sequences. All the proofs are formalized in the Coq proof assistant. We develop a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets. 1 Introduction Formalization of any mathematical theory is a difficult task because the length of a formal proof blows up significantly. In combinatorics the task becomes even more difficult due to the lack of structure in the theory. Some statements often admit more than one proof using completely different ideas. Thus, exploring dependencies among important results may help in identifying an effective order amongst them. -
1 Books 2 Papers
1 Books • J. Matoušek, Lectures on Discrete Geometry • N. Alon and J. Spencer, The Probabilistic Method, 3rd edition • M. Aigner and G. Ziegler, Proofs from the BOOK • B. Bollobás, The art of mathematics (Coffee time in Memphis) • J. Matousek, Thirty-three miniatures: mathematical and algorithmic ap- plications of linear algebra • S. Jukna, Extremal combinatorics 2 Papers • P. Erdős and S. Fajtlowicz, On a conjecture of Hajós, Combinatorica 1 (1981), 141-143. and C. Thomassen, Some remarks on Hajós conjecture, J. Combin. Theory Ser. B 93 (2005), 95-105. • F. Chung, R. Graham, and R. Wilson, Quasi-random graphs, Combina- torica 9 (1989), 345-362. • J. Shearer, A note on the independence number of triange-free graphs, Discrete Math 46 (1983), 83-87. • A. Schrijver, A short proof of Minc’s conjecture, J. Combinatorial Theory Ser. A 25 (1978), 80-83. (see also Alon and Spencer book, page 22) • P. Erdős, J. Pach, J. Pyber, Isomorphic subgraphs in a graph, in: Com- binatorics (Eger, 1987), Colloq. Math. Soc. János Bolyai, 52, North- Holland, Amsterdam, 1988, 553–556. • M. Goemans and D. Williamson, Improved algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42 (1995), 1115-1145. • P. Erdős and S. Shelah, On a problem of Moser and Hanson, Graph the- ory and applications, Lecture Notes in Math., Vol. 303, Springer, Berlin (1972), 75-79. • S. Brandt, On the structure of graphs with bounded clique number, Com- binatorica 23 (2003), 693-696. 1 • Hamiltonicity and pancyclicity, G. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc., 2 (1952), 69-81. -
Guarding and Searching Polyhedra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Electronic Thesis and Dissertation Archive - Università di Pisa Universita` di Pisa Dipartimento di Informatica Dottorato di Ricerca in Informatica Ph.D. Thesis: XXIV Guarding and Searching Polyhedra Giovanni Viglietta Supervisor Referee Prof. Linda Pagli Prof. Peter Widmayer Referee Prof. Masafumi Yamashita Chair Prof. Fabrizio Broglia November 11, 2012 Abstract Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that br=2c + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that b11e=72c edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was be=6c. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of b7r=12c + 1. -
Monskyn Lause Neliölle
CORE Metadata, citation and similar papers at core.ac.uk Provided by Helsingin yliopiston digitaalinen arkisto Monskyn lause neliölle Linda Lumio 17. marraskuuta 2016 HELSINGIN YLIOPISTO — HELSINGFORS UNIVERSITET — UNIVERSITY OF HELSINKI Tiedekunta/Osasto — Fakultet/Sektion — Faculty Laitos — Institution — Department Matemaattis-luonnontieteellinen Matematiikan ja tilastotieteen laitos Tekijä — Författare — Author Linda Lumio Työn nimi — Arbetets titel — Title Monskyn lause nelille Oppiaine — Läroämne — Subject Matematiikka Työn laji — Arbetets art — Level Aika — Datum — Month and year Sivumäärä — Sidoantal — Number of pages Pro gradu -tutkielma Marraskuu 2016 36 s. Tiivistelmä — Referat — Abstract Tässä työssä tutkitaan neliön tasaosittamiseen liittyvää Monskyn lausetta sekä esitellään sen todistamisessa tarvittava matemaattinen koneisto. Monskyn lause on matemaattinen lause, joka yhdistää kaksi toisistaan näennäisesti erillistä matematiikan osa-aluetta. Lauseen mukaan neliötä ei voida osittaa parittomaan määrään kolmioita, joilla on keskenään sama pinta-ala. Päämää- ränä on esitellä Monskyn lauseen todistamiseen tarvittava koneisto sekä itse lause ja tämän todistus. Monskyn lauseen merkittävyys piilee siinä, että sen todistus rakentuu kahdesta (tai kolmesta) osasta, jotka yhdistävät kaksi näennäisesti erillistä matematiikan osa-aluetta, topologian ja al- gebran. Todistuksen topologinen osuus tiivistyy niin kutsuttuun Spernerin lemmaan, josta on työssä esitetty useampi versio. Todistuksen algebrallinen osuus puolestaan sisältää valuaatiot