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Chaos & Fractals the Mathematics Behind
This volume contains the proceedings of a highly successful AMS Short Course on Chaos and Fractals, held during the AMS Centennial Celebration in Providence, Rhode Island in August 1988. Chaos and fractals have been the subject of great interest in recent years and have proven to be useful in a variety of areas of mathematics and the sciences. The purpose of the short course was to provide a solid introduction to the mathematics underlying the notions of chaos and fractals. The papers in this book range over such topics as dynamical systems theory, Julia sets, the Mandelbrot set, attractors, the Smale horseshoe, calculus on fractals, and applications to data compression. The authors represented here are some of the top experts in this field. Aimed at beginning graduate students, college and university mathematics instructors, and non-mathematics researchers, this book provides readable expositions of several exciting topics of contemporary research.
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First published January 1, 1989
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April 12, 2018Truth be told, I got this book for the illustrations. I remember one lecture in which the lecturer explained that we study fractals because they're beautiful (a more than adequate reason, for a mathematician). Then he said "But if there are any administrators here, they're useful, as well."
But if all I wanted was the illustrations, it wouldn't be necessary to cram the whole book, lovely and durable as it is, in overcrowded library shelves. So I decided to read it, despite doubts about whether my math was up to it (I barely got into integral calculus--and there's a LOT of integral calculus herein). If worst came to worst, I reckoned, I could just skip the equations--though it'd leave a lot of gaps, since there are about 4-5 equations/page on average.
Turned out it was harder than that, even. The text is often more than a little opaque, as well. It's not just that there are words I'd never heard before (I have a pretty extensive vocabulary, but even I had to look up words like 'ergodic'). It's also that there are terms that LOOK like vernacular terms--but are evidently used in a technical sense. There are references to various people's Theorems, which it might be possible to look up in the bibliographies, but which are NOT found in Bartlett's.
All that is manageable. But unless the reader is au courant with set theory, differential and integral calculus, etc, the best bet would be to seek out and print up a visual guide to unfamiliar mathematical operators. I did Google them at one point, but I didn't have the facilities to make a copy. I did figure out, finally, that what looked like a capital eta seems to mean 'is a member of' in set theory. But what am I to make of an upside-down capital a? Also, there are unelaborated acronyms throughout.
From the start I felt that I'd be better advised to find a source for the computer game Life, since it seems to underlie a lot of the theory. Unfortunately, I've been looking for one for years, without success.
I did work out a lot of it--but it took a long time--and quite a bit remains unclear. If this is a course, I would say it was an advanced level course. It's too full of the 'from this it obviously follows that...' lines. It may be obvious to some; it's not to me.
All that aside, it's also out of date. I don't think many people program in BASIC anymore, for one thing.
Still, I don't regret reading it. And others may find it useful, as well--though probably not as an introduction. If there's room, I'll try to add a list of the first 38 books in the series. If not, at least I should say that there IS a list of them in the front of the book. [Finally finished this-VS]
Oh, and I should say there's a subtitle to this volume: "The Mathematics behind The Computer Graphics".
CONTENTS:
I SERIES LIST: AMS SHORT COURSE LECTURE NOTES: INTRODUCTORY SURVEY LECTURES: A SUBSERIES OF PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
VOLUME I: NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA (ed by E Reissner, Brown U, Aug, 1947)
VOLUME II: ELECTROMAGNETIC THEORY: (ed by A H Taub, MIT, July 1948)
VOLUME III: ELASTICITY: (ed by R V Churchill, U of Massachusetts, June 1949)
VOLUME IV: FLUID DYNAMICS: (ed by M H Martin, U of Maryland, June 1951)
VOLUME V: WAVE MOTION AND VIBRATION THEORY: (ed by A E Heins, C(arnegie)IT, June 1952)
VOLUME VI: NUMERICAL ANALYSIS: (ed by J H Curtiss, Santa Monica City College, Aug 1953)
VOLUME VII: APPLIED PROBABILITY: (ed by L A MacColl, Polytechnic Institute of Brooklyn, Apr 1955)
VOLUME VIII: CALCULUS OF VARIATION AND ITS APPLICATIONS: (ed by L M Graves, U of Chicago, Apr, 1956)
VOLUME IX: ORBIT THEORY: (ed by G Birkhoff & R E Langer, (Columbia U), Apr, 1958)
VOLUME X: COMBINATORIAL ANALYSIS: (ed by R Bellman & M Hall, Jr, NYU, Apr, 1957 [?sic]
VOLUME XI: NUCLEAR REACTOR THEORY: (ed by G Birkhoff & EP Wigner, NYC, Apr, 1959)
VOLUME XII: STRUCTURE OF LANGUAGE & ITS MATHEMATICAL ASPECTS: (ed by R Bellman, NYC, Apr 1960)
VOLUME XIII: HYDRODYNAMIC INSTABILITY: (ed by R Bellman, G Birkhoff, & CC Lin [note: this approximately the right time, if not the right place, for this to be the CC Lin who worked with my father in the late '60s-early '70s], NYC, Apr, 1960)
VOLUME XIV: MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES: (ed by R Bellman, NYC, 1961)
VOLUME XV: EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, & MATHEMATICS: (ed by NC Metropolis et al, Atlantic City & Chicago, Apr, 1962)
VOLUME XVI: STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS & ENGINEERING: (ed by R Bellman, NYC, Apr 1961)
VOLUME XVII: APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS: (ed by R Finn, NYC, Apr 1964)
VOLUME XVIII: MAGNETO-FLUID & PLASMA DYNAMICS: (ed by H Grad, NYC, Apr 1965)
VOLUME XIX: MATHEMATICAL ASPECTS OF COMPUTER SCIENCE: (ed by JT Schwarz, NYC, Apr 1966)
VOLUME XX: THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH & EDUCATION: (ed by JP LaSalle, U of MT, Aug 1973 [?sic--whence the gap?])
VOLUME XXI: MATHEMATICAL ASPECTS OF PRODUCTION & DISTRIBUTION OF ENERGY: (ed by PD Lax, San Antonio TX, Jan 1976)
VOLUME XXII: NUMERICAL ANALYSIS: (ed by GH Golub & J Oliger, Atlanta, GA, Jan 1978)
VOLUME XXIII: MODERN STATISTICS: METHODS & APPLICATIONS: (by RV Hogg, San Antonio, TX, Jan 1980)
VOLUME XXIV: GAME THEORY AND ITS APPLICATIONS: (ed by WF Lucas, Biloxi, MS, Jan 1979)
VOLUME XXV: OPERATIONS RESEARCH: MATHEMATICS & MODELS: (ed by SI Gass, Duluth, MN, Aug 1979)
VOLUME XXVI: THE MATHEMATICS OF NETWORKS: (ed by SA Burr, Pittsburgh, PA, Aug 1981)
VOLUME XXVII: COMPUTED TOMOGRAPHY: (ed by LA Shepp, Cincinnati, OH, Jan, 1982)
VOLUME XXVIII: STATISTICAL DATA ANALYSIS: (ed by R Gnanadesikan, T0ronto, ON, Canada, Aug, 1982)
VOLUME XXIX: APPLIED CRYPTOGRAPHY, CRYPTOLOGY PROTOCOLS, & COMPUTER SECURITY MODELS: (ed by RA DeMillo et al, SF CA, Jan 1981)
VOLUME XXX: POPULATION BIOLOGY: (ed by Simon A Levin, Albany, NY, Aug 1983)
VOLUME XXXI: COMPUTER COMMUNICATIONS: (ed by B Gopinath, Denver, CO, Jan 1983)
VOLUME XXXII: ENVIRONMENTAL & NATURAL RESOURCE MATHEMATICS: (ed by RW McKelvey, Eugene, OR, Aug, 1984)
VOLUME XXXIII: FAIR ALLOCATION: (ed by H Peyton Young, Anaheim, CA, Jan 1985)
VOLUME XXXIV: MATHEMATICS OF INFORMATION PROCESSING: (ed by Michael Anshel & William Gewirtz, Louisville, KY, Jan 1984)
VOLUME XXXV: ACTUARIAL MATHEMATICS: (ed by Harry J Panjer, Laramie WY, Aug 1985)
VOLUME XXXVI: APPROXIMATION THEORY: (ed by Carl de Boor, New Orleans, LA, Jan 1986)
VOLUME XXXVII: MOMENTS IN MATHEMATICS: (ed by Henry J Landau, San Antonio, TX, Jan 1987)
VOLUME XXXVIII: COMPUTATIONAL COMPLEXITY THEORY: (ed by Juris Hartmanis, Atlanta, GA, Jan 1988)
Which brings us to the present volume, which is #XXXIX. There are almost certainly more volumes. Given that this one was published 30 years ago, and that the volumes were published an average of once a year, there are probably about 69 by now--but the list in this volume doesn't even list one more in press.
II PREFACE: (signed by the editors) A description of the symposium itself.
III DYNAMICS OF SIMPLE MAPS (by Robert L Devaney): This is the sort of thing I mean. Terms like 'simple', 'continuous', and 'filled' are not defined as they are to be interpreted herein. There's a DESPERATE need for a glossary--but though there's an Index and bibliographies, there's no glossary),
IV NONLINEAR OSCILLATIONS AND THE SMALL HORSESHOE MAP (by Philip Holmes)
V FRACTAL BASIN BOUNDARIES AND CHAOTIC ATTRACTORS (by Kathleen T Alligood and James A Yorke) Add logical syllogisms--and add 'theorem' vs 'lemma' to the glossary).
VI JULIA SETS (by Linda Keen)
VII THE MANDELBROT SET (by Bodil Banner): One confusing thing is that the capital m sometimes indicates the Mandelbrot set, and sometimes not--and this is almost never noted.
VIII AN INTRODUCTION TO FRACTALS (by Jenny Harrison)
IX LECTURE NOTES ON ITERATED FUNCTION SYSTEMS (by Michael F Barnsley) Despite the fact that the phrase 'iterated function systems' is spelled out only in the title, this is easily the most readable chapter in the book. I have a tendency to slog through things from start to finish--but for those who don't, I'd recommend reading this part first--it'll explain a lot of stuff that the other chapters just assume people know. Which might be useful even to subject experts...if it's been a while since they studied this subject in detail.
INDEX
But if all I wanted was the illustrations, it wouldn't be necessary to cram the whole book, lovely and durable as it is, in overcrowded library shelves. So I decided to read it, despite doubts about whether my math was up to it (I barely got into integral calculus--and there's a LOT of integral calculus herein). If worst came to worst, I reckoned, I could just skip the equations--though it'd leave a lot of gaps, since there are about 4-5 equations/page on average.
Turned out it was harder than that, even. The text is often more than a little opaque, as well. It's not just that there are words I'd never heard before (I have a pretty extensive vocabulary, but even I had to look up words like 'ergodic'). It's also that there are terms that LOOK like vernacular terms--but are evidently used in a technical sense. There are references to various people's Theorems, which it might be possible to look up in the bibliographies, but which are NOT found in Bartlett's.
All that is manageable. But unless the reader is au courant with set theory, differential and integral calculus, etc, the best bet would be to seek out and print up a visual guide to unfamiliar mathematical operators. I did Google them at one point, but I didn't have the facilities to make a copy. I did figure out, finally, that what looked like a capital eta seems to mean 'is a member of' in set theory. But what am I to make of an upside-down capital a? Also, there are unelaborated acronyms throughout.
From the start I felt that I'd be better advised to find a source for the computer game Life, since it seems to underlie a lot of the theory. Unfortunately, I've been looking for one for years, without success.
I did work out a lot of it--but it took a long time--and quite a bit remains unclear. If this is a course, I would say it was an advanced level course. It's too full of the 'from this it obviously follows that...' lines. It may be obvious to some; it's not to me.
All that aside, it's also out of date. I don't think many people program in BASIC anymore, for one thing.
Still, I don't regret reading it. And others may find it useful, as well--though probably not as an introduction. If there's room, I'll try to add a list of the first 38 books in the series. If not, at least I should say that there IS a list of them in the front of the book. [Finally finished this-VS]
Oh, and I should say there's a subtitle to this volume: "The Mathematics behind The Computer Graphics".
CONTENTS:
I SERIES LIST: AMS SHORT COURSE LECTURE NOTES: INTRODUCTORY SURVEY LECTURES: A SUBSERIES OF PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
VOLUME I: NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA (ed by E Reissner, Brown U, Aug, 1947)
VOLUME II: ELECTROMAGNETIC THEORY: (ed by A H Taub, MIT, July 1948)
VOLUME III: ELASTICITY: (ed by R V Churchill, U of Massachusetts, June 1949)
VOLUME IV: FLUID DYNAMICS: (ed by M H Martin, U of Maryland, June 1951)
VOLUME V: WAVE MOTION AND VIBRATION THEORY: (ed by A E Heins, C(arnegie)IT, June 1952)
VOLUME VI: NUMERICAL ANALYSIS: (ed by J H Curtiss, Santa Monica City College, Aug 1953)
VOLUME VII: APPLIED PROBABILITY: (ed by L A MacColl, Polytechnic Institute of Brooklyn, Apr 1955)
VOLUME VIII: CALCULUS OF VARIATION AND ITS APPLICATIONS: (ed by L M Graves, U of Chicago, Apr, 1956)
VOLUME IX: ORBIT THEORY: (ed by G Birkhoff & R E Langer, (Columbia U), Apr, 1958)
VOLUME X: COMBINATORIAL ANALYSIS: (ed by R Bellman & M Hall, Jr, NYU, Apr, 1957 [?sic]
VOLUME XI: NUCLEAR REACTOR THEORY: (ed by G Birkhoff & EP Wigner, NYC, Apr, 1959)
VOLUME XII: STRUCTURE OF LANGUAGE & ITS MATHEMATICAL ASPECTS: (ed by R Bellman, NYC, Apr 1960)
VOLUME XIII: HYDRODYNAMIC INSTABILITY: (ed by R Bellman, G Birkhoff, & CC Lin [note: this approximately the right time, if not the right place, for this to be the CC Lin who worked with my father in the late '60s-early '70s], NYC, Apr, 1960)
VOLUME XIV: MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES: (ed by R Bellman, NYC, 1961)
VOLUME XV: EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, & MATHEMATICS: (ed by NC Metropolis et al, Atlantic City & Chicago, Apr, 1962)
VOLUME XVI: STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS & ENGINEERING: (ed by R Bellman, NYC, Apr 1961)
VOLUME XVII: APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS: (ed by R Finn, NYC, Apr 1964)
VOLUME XVIII: MAGNETO-FLUID & PLASMA DYNAMICS: (ed by H Grad, NYC, Apr 1965)
VOLUME XIX: MATHEMATICAL ASPECTS OF COMPUTER SCIENCE: (ed by JT Schwarz, NYC, Apr 1966)
VOLUME XX: THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH & EDUCATION: (ed by JP LaSalle, U of MT, Aug 1973 [?sic--whence the gap?])
VOLUME XXI: MATHEMATICAL ASPECTS OF PRODUCTION & DISTRIBUTION OF ENERGY: (ed by PD Lax, San Antonio TX, Jan 1976)
VOLUME XXII: NUMERICAL ANALYSIS: (ed by GH Golub & J Oliger, Atlanta, GA, Jan 1978)
VOLUME XXIII: MODERN STATISTICS: METHODS & APPLICATIONS: (by RV Hogg, San Antonio, TX, Jan 1980)
VOLUME XXIV: GAME THEORY AND ITS APPLICATIONS: (ed by WF Lucas, Biloxi, MS, Jan 1979)
VOLUME XXV: OPERATIONS RESEARCH: MATHEMATICS & MODELS: (ed by SI Gass, Duluth, MN, Aug 1979)
VOLUME XXVI: THE MATHEMATICS OF NETWORKS: (ed by SA Burr, Pittsburgh, PA, Aug 1981)
VOLUME XXVII: COMPUTED TOMOGRAPHY: (ed by LA Shepp, Cincinnati, OH, Jan, 1982)
VOLUME XXVIII: STATISTICAL DATA ANALYSIS: (ed by R Gnanadesikan, T0ronto, ON, Canada, Aug, 1982)
VOLUME XXIX: APPLIED CRYPTOGRAPHY, CRYPTOLOGY PROTOCOLS, & COMPUTER SECURITY MODELS: (ed by RA DeMillo et al, SF CA, Jan 1981)
VOLUME XXX: POPULATION BIOLOGY: (ed by Simon A Levin, Albany, NY, Aug 1983)
VOLUME XXXI: COMPUTER COMMUNICATIONS: (ed by B Gopinath, Denver, CO, Jan 1983)
VOLUME XXXII: ENVIRONMENTAL & NATURAL RESOURCE MATHEMATICS: (ed by RW McKelvey, Eugene, OR, Aug, 1984)
VOLUME XXXIII: FAIR ALLOCATION: (ed by H Peyton Young, Anaheim, CA, Jan 1985)
VOLUME XXXIV: MATHEMATICS OF INFORMATION PROCESSING: (ed by Michael Anshel & William Gewirtz, Louisville, KY, Jan 1984)
VOLUME XXXV: ACTUARIAL MATHEMATICS: (ed by Harry J Panjer, Laramie WY, Aug 1985)
VOLUME XXXVI: APPROXIMATION THEORY: (ed by Carl de Boor, New Orleans, LA, Jan 1986)
VOLUME XXXVII: MOMENTS IN MATHEMATICS: (ed by Henry J Landau, San Antonio, TX, Jan 1987)
VOLUME XXXVIII: COMPUTATIONAL COMPLEXITY THEORY: (ed by Juris Hartmanis, Atlanta, GA, Jan 1988)
Which brings us to the present volume, which is #XXXIX. There are almost certainly more volumes. Given that this one was published 30 years ago, and that the volumes were published an average of once a year, there are probably about 69 by now--but the list in this volume doesn't even list one more in press.
II PREFACE: (signed by the editors) A description of the symposium itself.
III DYNAMICS OF SIMPLE MAPS (by Robert L Devaney): This is the sort of thing I mean. Terms like 'simple', 'continuous', and 'filled' are not defined as they are to be interpreted herein. There's a DESPERATE need for a glossary--but though there's an Index and bibliographies, there's no glossary),
IV NONLINEAR OSCILLATIONS AND THE SMALL HORSESHOE MAP (by Philip Holmes)
V FRACTAL BASIN BOUNDARIES AND CHAOTIC ATTRACTORS (by Kathleen T Alligood and James A Yorke) Add logical syllogisms--and add 'theorem' vs 'lemma' to the glossary).
VI JULIA SETS (by Linda Keen)
VII THE MANDELBROT SET (by Bodil Banner): One confusing thing is that the capital m sometimes indicates the Mandelbrot set, and sometimes not--and this is almost never noted.
VIII AN INTRODUCTION TO FRACTALS (by Jenny Harrison)
IX LECTURE NOTES ON ITERATED FUNCTION SYSTEMS (by Michael F Barnsley) Despite the fact that the phrase 'iterated function systems' is spelled out only in the title, this is easily the most readable chapter in the book. I have a tendency to slog through things from start to finish--but for those who don't, I'd recommend reading this part first--it'll explain a lot of stuff that the other chapters just assume people know. Which might be useful even to subject experts...if it's been a while since they studied this subject in detail.
INDEX
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