Outer Billiards on Kites (AM-171)

Outer Billiards on Kites (AM-171)

Author: Richard Evan Schwartz

Publisher: Princeton University Press

Published: 2009-10-05

Total Pages: 320

ISBN-13: 1400831970

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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950s, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.


Book Synopsis Outer Billiards on Kites (AM-171) by : Richard Evan Schwartz

Download or read book Outer Billiards on Kites (AM-171) written by Richard Evan Schwartz and published by Princeton University Press. This book was released on 2009-10-05 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950s, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.

Man, Play, and Games

Man, Play, and Games

Author: Roger Caillois

Publisher: University of Illinois Press

Published: 2001

Total Pages: 228

ISBN-13: 9780252070334

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According to Roger Caillois, play is an occasion of pure waste. In spite of this - or because of it - play constitutes an essential element of human social and spiritual development. In this study, the author defines play as a free and voluntary activity that occurs in a pure space, isolated and protected from the rest of life.


Book Synopsis Man, Play, and Games by : Roger Caillois

Download or read book Man, Play, and Games written by Roger Caillois and published by University of Illinois Press. This book was released on 2001 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: According to Roger Caillois, play is an occasion of pure waste. In spite of this - or because of it - play constitutes an essential element of human social and spiritual development. In this study, the author defines play as a free and voluntary activity that occurs in a pure space, isolated and protected from the rest of life.

Geometry and Billiards

Geometry and Billiards

Author: Serge Tabachnikov

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 192

ISBN-13: 0821839195

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Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differential geometry, classical mechanics, and geometrical optics. Topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual (or outer) billiards. The book is based on an advanced undergraduate topics course. Minimum prerequisites are the standard material covered in the first two years of college mathematics (the entire calculus sequence, linear algebra). However, readers should show some mathematical maturity and rely on their mathematical common sense. A unique feature of the book is the coverage of many diverse topics related to billiards, for example, evolutes and involutes of plane curves, the four-vertex theorem, a mathematical theory of rainbows, distribution of first digits in various sequences, Morse theory, the Poincare recurrence theorem, Hilbert's fourth problem, Poncelet porism, and many others. There are approximately 100 illustrations. The book is suitable for advanced undergraduates, graduate students, and researchers interested in ergodic theory and geometry. This volume has been copublished with the Mathematics Advanced Study Semesters program at Penn State.


Book Synopsis Geometry and Billiards by : Serge Tabachnikov

Download or read book Geometry and Billiards written by Serge Tabachnikov and published by American Mathematical Soc.. This book was released on 2005 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differential geometry, classical mechanics, and geometrical optics. Topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual (or outer) billiards. The book is based on an advanced undergraduate topics course. Minimum prerequisites are the standard material covered in the first two years of college mathematics (the entire calculus sequence, linear algebra). However, readers should show some mathematical maturity and rely on their mathematical common sense. A unique feature of the book is the coverage of many diverse topics related to billiards, for example, evolutes and involutes of plane curves, the four-vertex theorem, a mathematical theory of rainbows, distribution of first digits in various sequences, Morse theory, the Poincare recurrence theorem, Hilbert's fourth problem, Poncelet porism, and many others. There are approximately 100 illustrations. The book is suitable for advanced undergraduates, graduate students, and researchers interested in ergodic theory and geometry. This volume has been copublished with the Mathematics Advanced Study Semesters program at Penn State.

Letters and Social Aims

Letters and Social Aims

Author: Ralph Waldo Emerson

Publisher:

Published: 1888

Total Pages: 558

ISBN-13:

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Book Synopsis Letters and Social Aims by : Ralph Waldo Emerson

Download or read book Letters and Social Aims written by Ralph Waldo Emerson and published by . This book was released on 1888 with total page 558 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Silicon Web

The Silicon Web

Author: Michael G. Raymer

Publisher: CRC Press

Published: 2009-06-23

Total Pages: 600

ISBN-13: 1439803129

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The technology behind computers, fiber optics, and networks did not originate in the minds of engineers attempting to build an Internet. The Internet is a culmination of intellectual work by thousands of minds spanning hundreds of years. We have built concept upon concept and technology upon technology to arrive at where we are today, in a world co


Book Synopsis The Silicon Web by : Michael G. Raymer

Download or read book The Silicon Web written by Michael G. Raymer and published by CRC Press. This book was released on 2009-06-23 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: The technology behind computers, fiber optics, and networks did not originate in the minds of engineers attempting to build an Internet. The Internet is a culmination of intellectual work by thousands of minds spanning hundreds of years. We have built concept upon concept and technology upon technology to arrive at where we are today, in a world co

Discovering Geometry

Discovering Geometry

Author: Michael Serra

Publisher:

Published: 2003-03-01

Total Pages: 34

ISBN-13: 9781559535885

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Book Synopsis Discovering Geometry by : Michael Serra

Download or read book Discovering Geometry written by Michael Serra and published by . This book was released on 2003-03-01 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Work; A Story of Experience

Work; A Story of Experience

Author: Louisa May Alcott

Publisher: BoD – Books on Demand

Published: 2023-01-16

Total Pages: 482

ISBN-13: 3368335553

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Reproduction of the original.


Book Synopsis Work; A Story of Experience by : Louisa May Alcott

Download or read book Work; A Story of Experience written by Louisa May Alcott and published by BoD – Books on Demand. This book was released on 2023-01-16 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reproduction of the original.

Dictionary of the British English Spelling System

Dictionary of the British English Spelling System

Author: Greg Brooks

Publisher: Open Book Publishers

Published: 2015-03-30

Total Pages: 524

ISBN-13: 1783741074

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This book will tell all you need to know about British English spelling. It's a reference work intended for anyone interested in the English language, especially those who teach it, whatever the age or mother tongue of their students. It will be particularly useful to those wishing to produce well-designed materials for teaching initial literacy via phonics, for teaching English as a foreign or second language, and for teacher training. English spelling is notoriously complicated and difficult to learn; it is correctly described as much less regular and predictable than any other alphabetic orthography. However, there is more regularity in the English spelling system than is generally appreciated. This book provides, for the first time, a thorough account of the whole complex system. It does so by describing how phonemes relate to graphemes and vice versa. It enables searches for particular words, so that one can easily find, not the meanings or pronunciations of words, but the other words with which those with unusual phoneme-grapheme/grapheme-phoneme correspondences keep company. Other unique features of this book include teacher-friendly lists of correspondences and various regularities not described by previous authorities, for example the strong tendency for the letter-name vowel phonemes (the names of the letters ) to be spelt with those single letters in non-final syllables.


Book Synopsis Dictionary of the British English Spelling System by : Greg Brooks

Download or read book Dictionary of the British English Spelling System written by Greg Brooks and published by Open Book Publishers. This book was released on 2015-03-30 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book will tell all you need to know about British English spelling. It's a reference work intended for anyone interested in the English language, especially those who teach it, whatever the age or mother tongue of their students. It will be particularly useful to those wishing to produce well-designed materials for teaching initial literacy via phonics, for teaching English as a foreign or second language, and for teacher training. English spelling is notoriously complicated and difficult to learn; it is correctly described as much less regular and predictable than any other alphabetic orthography. However, there is more regularity in the English spelling system than is generally appreciated. This book provides, for the first time, a thorough account of the whole complex system. It does so by describing how phonemes relate to graphemes and vice versa. It enables searches for particular words, so that one can easily find, not the meanings or pronunciations of words, but the other words with which those with unusual phoneme-grapheme/grapheme-phoneme correspondences keep company. Other unique features of this book include teacher-friendly lists of correspondences and various regularities not described by previous authorities, for example the strong tendency for the letter-name vowel phonemes (the names of the letters ) to be spelt with those single letters in non-final syllables.

Daly's Billiard Book

Daly's Billiard Book

Author: Maurice Daly

Publisher:

Published: 1914

Total Pages: 320

ISBN-13:

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Book Synopsis Daly's Billiard Book by : Maurice Daly

Download or read book Daly's Billiard Book written by Maurice Daly and published by . This book was released on 1914 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Spherical CR Geometry and Dehn Surgery (AM-165)

Spherical CR Geometry and Dehn Surgery (AM-165)

Author: Richard Evan Schwartz

Publisher: Princeton University Press

Published: 2007-01-29

Total Pages: 199

ISBN-13: 1400837197

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This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids quotations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.


Book Synopsis Spherical CR Geometry and Dehn Surgery (AM-165) by : Richard Evan Schwartz

Download or read book Spherical CR Geometry and Dehn Surgery (AM-165) written by Richard Evan Schwartz and published by Princeton University Press. This book was released on 2007-01-29 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids quotations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.