Exterior Billiards

Author : Alexander Plakhov
Publisher : Springer Science & Business Media
Page : 296 pages
File Size : 50,7 Mb
Release : 2012-09-11
Category : Mathematics
ISBN : 9781461444800

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Exterior Billiards by Alexander Plakhov Pdf

A billiard is a dynamical system in which a point particle alternates between free motion and specular reflections from the boundary of a domain. Exterior Billiards presents billiards in the complement of domains and their applications in aerodynamics and geometrical optics. This book distinguishes itself from existing literature by presenting billiard dynamics outside bounded domains, including scattering, resistance, invisibility and retro-reflection. It begins with an overview of the mathematical notations used throughout the book and a brief review of the main results. Chapters 2 and 3 are focused on problems of minimal resistance and Newton’s problem in media with positive temperature. In chapters 4 and 5, scattering of billiards by nonconvex and rough domains is characterized and some related special problems of optimal mass transportation are studied. Applications in aerodynamics are addressed next and problems of invisibility and retro-reflection within the framework of geometric optics conclude the text. The book will appeal to mathematicians working in dynamical systems and calculus of variations. Specialists working in the areas of applications discussed will also find it useful.

Author : Marcel Berger
Publisher : Springer Science & Business Media
Page : 831 pages
File Size : 55,5 Mb
Release : 2010-07-23
Category : Mathematics
ISBN : 9783540709978

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Geometry Revealed by Marcel Berger Pdf

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Author : Leopold Flatto
Publisher : American Mathematical Soc.
Page : 259 pages
File Size : 48,5 Mb
Release : 2024-06-10
Category : Mathematics
ISBN : 9780821886267

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Poncelet's Theorem by Leopold Flatto Pdf

Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics. The author's original research in queuing theory and dynamical systems figures prominently in the book. This book stresses the modern approach to the subject and contains much material not previously available in book form. It also discusses the relation between Poncelet's theorem and some aspects of queueing theory and mathematical billiards. The proof of Poncelet's theorem presented in this book relates it to the theory of elliptic curves and exploits the fact that such curves are endowed with a group structure. The book also treats the real and degenerate cases of Poncelet's theorem. These cases are interesting in themselves, and their proofs require some other considerations. The real case is handled by employing notions from dynamical systems. The material in this book should be understandable to anyone who has taken the standard courses in undergraduate mathematics. To achieve this, the author has included in the book preliminary chapters dealing with projective geometry, Riemann surfaces, elliptic functions, and elliptic curves. The book also contains numerous figures illustrating various geometric concepts.

Author : Richard Evan Schwartz
Publisher : Princeton University Press
Page : 320 pages
File Size : 54,7 Mb
Release : 2009-10-05
Category : Mathematics
ISBN : 9781400831975

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Outer Billiards on Kites (AM-171) by Richard Evan Schwartz Pdf

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.

Author : Karl-Fredrik Berggren
Publisher : World Scientific
Page : 286 pages
File Size : 47,9 Mb
Release : 2001
Category : Science
ISBN : 9789810247119

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Quantum Chaos Y2K by Karl-Fredrik Berggren Pdf

Quantum chaos is becoming a very wide field that ranges from experiments to theoretical physics and purely mathematical issues. In view of this grand span, Nobel Symposium 116 focused on experiments and theory, and attempted to encourage interplay between them. There was emphasis on the interdisciplinary character of the subject, involving a broad range of subjects in physics, including condensed matter physics, nuclear physics, atomic physics and elementary particle physics. The physics involved in quantum chaos has much in common with acoustics, microwaves, optics, etc., and therefore the symposium also covered aspects of wave chaos in this broader sense. The program was structured according to the following areas: manifestations of classical chaos in quantum systems; transport phenomena; quantal spectra in terms of periodic orbits; semiclassical and random matrix approaches; quantum chaos in interacting systems; chaos and tunneling; wave-dynamic chaos. This important book constitutes the proceedings of the symposium.

Author : Jürgen Moser
Publisher : Birkhäuser
Page : 134 pages
File Size : 42,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034880572

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Selected Chapters in the Calculus of Variations by Jürgen Moser Pdf

0.1 Introduction These lecture notes describe a new development in the calculus of variations which is called Aubry-Mather-Theory. The starting point for the theoretical physicist Aubry was a model for the descrip tion of the motion of electrons in a two-dimensional crystal. Aubry investigated a related discrete variational problem and the corresponding minimal solutions. On the other hand, Mather started with a specific class of area-preserving annulus mappings, the so-called monotone twist maps. These maps appear in mechanics as Poincare maps. Such maps were studied by Birkhoff during the 1920s in several papers. In 1982, Mather succeeded to make essential progress in this field and to prove the existence of a class of closed invariant subsets which are now called Mather sets. His existence theorem is based again on a variational principle. Although these two investigations have different motivations, they are closely re lated and have the same mathematical foundation. We will not follow those ap proaches but will make a connection to classical results of Jacobi, Legendre, Weier strass and others from the 19th century. Therefore in Chapter I, we will put together the results of the classical theory which are the most important for us. The notion of extremal fields will be most relevant. In Chapter II we will investigate variational problems on the 2-dimensional torus. We will look at the corresponding global minimals as well as at the relation be tween minimals and extremal fields. In this way, we will be led to Mather sets.

Author : Marion Wynne Brunzie
Publisher : Unknown
Page : 160 pages
File Size : 45,6 Mb
Release : 1994
Category : Electronic
ISBN : UCAL:X54163

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Generalized Dual Billiards by Marion Wynne Brunzie Pdf

Author : Alexander Plakhov,Tatiana Tchemisova,Adelaide Freitas
Publisher : Springer
Page : 207 pages
File Size : 54,6 Mb
Release : 2015-06-10
Category : Computers
ISBN : 9783319203522

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Optimization in the Natural Sciences by Alexander Plakhov,Tatiana Tchemisova,Adelaide Freitas Pdf

This book constitutes the refereed proceedings of the 30th Euro Mini-Conference, EmC-ONS 2014, held in Aveiro, Portugal, in February 2014. The 13 revised full papers presented were carefully reviewed and selected from 70 submissions. The papers are organized in topical sections on dynamical systems; optimization and applications; modeling and statistical techniques for data analysis.

Author : Jon M. Boyd,Michael J. Scheffler
Publisher : ASTM International
Page : 335 pages
File Size : 48,8 Mb
Release : 1999
Category : Dampness in buildings
ISBN : 9780803126077

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Water Problems in Building Exterior Walls by Jon M. Boyd,Michael J. Scheffler Pdf

Author : Richard Evan Schwartz
Publisher : American Mathematical Soc.
Page : 212 pages
File Size : 52,8 Mb
Release : 2014-07-03
Category : Mathematics
ISBN : 9781470415228

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The Octagonal PETs by Richard Evan Schwartz Pdf

A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.

Author : Bogdan Grechuk
Publisher : Springer Nature
Page : 437 pages
File Size : 55,6 Mb
Release : 2021-09-21
Category : Mathematics
ISBN : 9783030806279

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Landscape of 21st Century Mathematics by Bogdan Grechuk Pdf

Landscape of 21st Century Mathematics offers a detailed cross section of contemporary mathematics. Important results of the 21st century are motivated and formulated, providing an overview of recent progress in the discipline. The theorems presented in this book have been selected among recent achievements whose statements can be fully appreciated without extensive background. Grouped by subject, the selected theorems represent all major areas of mathematics: number theory, combinatorics, analysis, algebra, geometry and topology, probability and statistics, algorithms and complexity, and logic and set theory. The presentation is self-contained with context, background and necessary definitions provided for each theorem, all without sacrificing mathematical rigour. Where feasible, brief indications of the main ideas of a proof are given. Rigorous yet accessible, this book presents an array of breathtaking recent advances in mathematics. It is written for everyone with a background in mathematics, from inquisitive university students to mathematicians curious about recent achievements in areas beyond their own.

Author : Arne Jensen
Publisher : World Scientific
Page : 743 pages
File Size : 49,6 Mb
Release : 2014
Category : Science
ISBN : 9789814449243

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XVIIth International Congress on Mathematical Physics by Arne Jensen Pdf

This is an in-depth study of not just about Tan Kah-kee, but also the making of a legend through his deeds, self-sacrifices, fortitude and foresight. This revised edition sheds new light on his political agonies in Mao's China over campaigns against capitalists and intellectuals.

Author : Anonim
Publisher : Unknown
Page : 286 pages
File Size : 49,9 Mb
Release : 1994
Category : Mathematics
ISBN : PSU:000052403781

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Report PM by Anonim Pdf

Author : D. B. Fuks,Serge Tabachnikov
Publisher : American Mathematical Soc.
Page : 463 pages
File Size : 42,6 Mb
Release : 2007
Category : Mathematics
ISBN : 9780821843161

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Mathematical Omnibus by D. B. Fuks,Serge Tabachnikov Pdf

The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an accomplished artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.

Author : Richard Evan Schwartz
Publisher : Princeton University Press
Page : 280 pages
File Size : 51,6 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9780691188997

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The Plaid Model by Richard Evan Schwartz Pdf

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz’s Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites. Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.