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A Gateway to Modern Mathematics Adventures in Iterations II ( Volume 2 ) by Shailesh A Shirali Ramanujan Mathematical Society Little Mathematical Treasures INMO IMO Math Olympiad PDF

257 Pages·2019·58.03 MB·English
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Preview A Gateway to Modern Mathematics Adventures in Iterations II ( Volume 2 ) by Shailesh A Shirali Ramanujan Mathematical Society Little Mathematical Treasures INMO IMO Math Olympiad

A Gateway to MODERN MATHEMATICS Adventures in Iteration II Universities Press Ww SHAILESH A SHIRALI the RMS series Little The books inthis series school students who are mathematics a ite ol curriculum offers. Iterations isan exciting topic of study and should interest. both the amateur as well as. the professional. Many of the iterations in elementary mathematics o ope for extended gateway to important ates Shailesh A Shirl served a Principal of Rishi Valley School for over a decade, and has been teaching mathemati Jom the President of India in 003, He has a deep interest in 12590, 2 problem journal for ol and college students. He tly heads a Community th Center, located in Rishi which conducts Editorial Policy In times past, mathematics was studied and researched by a select few—an elite class who valued the subject for its intellectual elegance and perhaps also for its philosophic connotations. But today itis studied and practised by an extremely large number of individuals, for mathematics has acquired 2 pivotal role inthe sustenance ofa technologically advanced society. Broadly speaking, there are two distinct groups of students to whom expository material in mathematics may be addressed: (a) students who have a deep interest in mathematics and wish to pursue a career involving research and teaching of mathematics; (b) students who are good at mathematics but wish to pursue a career in some other discipline. Clearly such students require suitable material (in addition to contact with inspiring teachers and practising mathematicians) to nurture their talent in the subject. To this list we may add professionals in industry and government and teachers in mathematics and other disciplines who require a good understanding of certain areas of mathematics for their professional work. ‘The proposed series is addressed to mathematically mature readers and to bright students in their last two years of school education. Iti envisaged that the books will contain expository material not generally included in standart school or college texts. New developments in mathematics will be presente attractively using mathematical tools familiar at the high school and undergraduate levels. There will be problem sets scattered through the texts which will serve to draw the reader into a closer hands-on study of the subject. Readers will be invited to grapple with the subject, and so experience the creative joy of discovery and contact with beauty. A thing of beauty is a joy forever.... So it is with mathematics, ‘The discoveries of Archimedes, Apollonius and Diophantus have been sources of Joy from very ancient times. It is our hope that these books will serve our readers ina similar manner. Proposals for publication of a book in the series is invited with a deat of the proposed book. The decision of the Editorial Board will be final, LITTLE MATHEMATICAL TREASURES VOLUME 2 A Gateway to MODERN MATHEMATICS Adventures in Iteration II SHAILESH A SHIRALI Rishi Valley School Andhra Pradesh w Universities Press Contents Preface xi Review ait 1. Insights from calculus 1 1.1 Error estimates 2 1.2 Attraction and repulsion 3 1.3 What if the slope is 1 or 1? 4 1.4 Applications 5 1.5 Comments on cobwebbing 2 1.6 Convergence to 2-cycles 1B 1.7 Contraction mappings, 3B 18 An iteration for x 16 1.9 Exercises 18 2. Solution of equations 19 2.1 ‘The Newton—Raphson algorithm 20 2.2 Examples 2 2.3 Comments and extensions 4 24 Halley's method 6 2.5 The method of false position 29 2.6 A route to square roots 31 2.7 A route to cube roots 34 2.8 Exercises 36 3. Atower of exponents 37 3.1. An infinite exponential 37 3.2. Number crunching I 37 3.3 The number e 40 4 3.4 Explanations “a 35 Number crunching It 3.6 Maximising x'/* 3.7 The occurrence of 2-cycles 38 The outcome when's = el/@ 3.9 The outcome when 0.<u<e~* 3.10 The outcome when u = e~* 3.11. Number crunching IL 3.12 Exercises ‘The drifters 4.1. The iteration x (22 —1)/2e 42 Explanations 43 The iteration x —+ ex(1-2) 44 Explanations 45 Period doubling forthe iteration x—+ ex(1 ~2) 46 Exercises . Memorable problems from the Olympiads 5.1 Problem IMO 1986/3 5.2. Problem IMO 1993/6 53 Ex 54 Two USAMO problems ‘555 An almost-ran IMO problem 56 Exercises 6. Miscellaneous problems 6.1 Four problems 62 Solutions 63 Ramanujan’s problem {64 A problem from the CMJ 65 Another GCD iteration 66 The Mersenne iteration 6.7 Exercises Contos & 7. Sarkovskii’s theorem, us 7.1 Proof of the theorem of Li and Yorke 18 8, Estimating the speed 13 81 Theiteration x + x— 2 ps 82 Theiteraions—+2—2 17 83 The iterated sine 128 84 Theiteraion s+ 2-4 1/x 132 85 The Tower of Exponents 133 86 Exercises 136 9. Fermat's two-squares theorem 137 9.1 Zagier’s proof 137 9.2 A constructive proof 141 9.3 Proof ofthe algorithm 12 94 Exercises 144 10. Graphics through iteration 146 10.1 The Cantor set 146 102 The Sierpinski gasket 149 103 The Chaos Game 150 1044 erated function systems (IFS) 152 105 Turtle walk 155 106 Exercises 163 11, Fractional linear maps over C 164 11.1 Functions defined on the extended complex plane 164 11.2 Linear mappings 166 112.1 The case a= 1 166 11.22 The case a1 166 11.23 The case when ais a root of unity 169 11.3 Mobius transformations 170 114 Iterations under Mobius transformations 13 IS Exercises 182 R 13. ‘Quadratic maps over C 12.1 Sample orbits of the map z+ 2? +e 122 Exercises ‘The Mandelbrot set 131 Julia—Fatou sets 132 Iulia set for the logistic map 133 The Mandelbrot set 134 Shape ofthe Mandelbrot set 135 Exercises 136 MATHEMATICA programs Appendix A Appendix B Appendix. Appendix D Index Preface ‘This book isa sequel to the author's A Gateway to Modern Mathematics: Adventures Iteration, Volume I (which we shall refer to as Adventures 1). The idea of iteration ‘was introduced in that work, together with various associated notions (fixed points, ‘orbits, cycles, limit points, convergence, solution of equations, cobwebbing, and so ‘on), and a large number of examples were studied from the world of arithmetic, algebra and geometry. The present work continues the study of iteration, but at a higher level. However, itis largely self-contained, and can be read without reference to Adventures I ‘Students who are preparing for the Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) will find this book useful. It is suitable for self-study by students in the age range 15-21 years (this includes students of mathematics atthe college evel, who will enjoy seeing “in action"” some of the facts learnt in their early courses in calculus and analysis), and for lay readers, ‘who will enjoy learning about a topic which is of great current interest. It can also ‘be used by teachers in a school mathematics club. Preview of book [At the start of the book, in the unnumbered ‘‘Review"” chapter, we present a very brief review of the material covered in Adventures I Following this, in Chapter 1, we examine the insights on iteration provided by differential calculus. Then, in Chapter 2, we study various approaches 10 the numerical solution of equations (c.g., the Newton—Raphson method). These approaches, as also the iterations studied in Chapters 3 and 4, offer very good ‘examples for illustrating the applications of calculus. In Chapters 5 and 6, we tackle ‘an assorted list of problems. Some of these have their origins in the mathematical ‘olympiads, but we also discuss in detail a well-known problem of Ramanujan's that ‘was once used in the Putnam Lowell competition, and an open problem related 10 “Mersenne primes. In Chapter 7, we give brief account of two fascinating discoveries ‘made in recent years—the theorems of Li and Yorke, and of Sarkovskii. In Chapter 8, ‘we study the speeds of convergence in iterative processes. Here some knowledge is required of the use of L'Hospital’s rule for computing limits. In Chapter 9, we consider the well-known two-squares theorem of Fermat concerning primes of the form 4n-+ 1; we prove it using an iterative technique, and in the process arrive at a constructive proof, In Chapter 10, we study how recursion makes for a simple and clegant construction of certain tree-like structures which have an amazingly “real” appearance. Finally, in Chapter 11, 12 and 13, we step into the complex plane, and discuss Julia sets of fractional linear maps and quadratic maps, and some associated notions, including the famous Mandelbrot set. Price ‘There are plenty of problems scattered throughout the book, with answers or solution to all except a few given atthe end of the book. Appendices ‘Appendices A and B have brief reviews of analysis and calculus (notions of limit, continuity, slope, derivative, etc.) and the algebra of complex numbers (notions ‘of magnitude and argument, de Moivre's theorem, etc), respectively; Appendix C ives a list of books and internet sites for further reading: and Appendix D gives solutions to many of the problems in the book. Prerequisites ‘The book presupposes some degree of mathematical maturity on the part of the reader—an elusive quality that means, roughly, the ability to rhink mathematically. It's the earmest and committed belief ofthis author that working through this book will itself greatly enhance te learner's mathematical maturity. Acknowledgements Grateful acknowledgements of thanks are due, for the assistance that I received from numerous individuals: to Professor Phoolan Prasad, Dr. Jayant Kirtane and ‘Mr. Siddhartha Menon, for going through early versions of the book and offering helpful and critical comments; to Mr. and Mrs, Mohan Subramaniam, for their help, generously given, in procuring the IIEX software; to Mr. K Seshadri, for technical help withthe computer systems; to Professor E Sampathkumar, for his enthusiastic supportin publishing this book; tomy friends inthe “Olympiad cel,” C I Pranesachar ("CRP"), J Venkatachala “BJV") and C'S Yogananda ""CSY" ‘with whom I have had numerous iterated adventures; and to Padmapriya, my wife, for her iterated support.

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Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.