Number-crunching : taming unruly computational problems from mathematical physics to science fiction

Paul J. Nahin

How do technicians repair broken communications cables at the bottom of the ocean without actually seeing them? What's the likelihood of plucking a needle out of a haystack the size of the Earth? And is it possible to use computers to create a universal library of everything ever written or every photo ever taken? These are just some of the intriguing questions that best-selling popular math writer Paul Nahin tackles in "Number-Crunching". Through brilliant math ideas and entertaining stories, Nahin demonstrates how odd and unusual math problems can be solved by bringing together basic physics ideas and today's powerful computers. Some of the outcomes discussed are so counterintuitive they will leave readers astonished. Nahin looks at how the art of number-crunching has changed since the advent of computers, and how high-speed technology helps to solve fascinating conundrums such as the three-body, Monte Carlo, leapfrog, and gambler's ruin problems. Along the way, Nahin traverses topics that include algebra, trigonometry, geometry, calculus, number theory, differential equations, Fourier series, electronics, and computers in science fiction. He gives historical background for the problems presented, offers many examples and numerous challenges, supplies MATLAB codes for all the theories discussed, and includes detailed and complete solutions. Exploring the intimate relationship between mathematics, physics, and the tremendous power of modern computers, "Number-Crunching" will appeal to anyone interested in understanding how these three important fields join forces to solve today's thorniest puzzles.

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  • Introduction x Chapter 1: FEYNMAN MEETS FERMAT 1 1.1 The Physicist as Mathematician 1 1.2 Fermat's Last Theorem 2 1.3 "Proof" by Probability 3 1.4 Feynman's Double Integral 6 1.5 Things to come 10 1.6 Challenge Problems 11 1.7 Notes and References 13 Chapter 2: Just for Fun: Two Quick Number-Crunching Problems 16 2.1 Number-Crunching in the Past 16 2.2 A Modern Number-Cruncher 20 2.3 Challenge Problem 25 2.4 Notes and References 25 Chapter 3: Computers and Mathematical Physics 27 3.1 When Theory Isn't Available 27 3.2 The Monte Carlo Technique 28 3.3 The Hot Plate Problem 34 3.4 Solving the Hot Plate Problem with Analysis 38 3.5 Solving the Hot Plate Problem by Iteration 44 3.6 Solving the Hot Plate Problem with the Monte Carlo Technique 50 3.7 ENIAC and MANIAC-I: the Electronic Computer Arrives 55 3.8 The Fermi-Pasta-Ulam Computer Experiment 58 3.9 Challenge Problems 73 3.10 Notes and References 74 Chapter 4: The Astonishing Problem of the Hanging Masses 82 4.1 Springs and Harmonic Motion 82 4.2 A Curious Oscillator 87 4.3 Phase-Plane Portraits 96 4.4 Another (Even More?) Curious Oscillator 99 4.5 Hanging Masses 104 4.6 Two Hanging Masses and the Laplace Transform 108 4.7 Hanging Masses and MATLAB 113 4.8 Challenge Problems 124 4.9 Notes and References 124 Chapter 5: The Three-Body Problem and Computers 131 5.1 Newton's Theory of Gravity 131 5.2 Newton's Two-Body Solution 139 5.3 Euler's Restricted Three-Body Problem 147 5.4 Binary Stars 155 5.5 Euler's Problem in Rotating Coordinates 166 5.6 Poincare and the King Oscar II Competition 177 5.7 Computers and the Pythagorean Three-Body Problem 184 5.8 Two Very Weird Three-Body Orbits 195 5.9 Challenge Problems 205 5.10 Notes and References 207 Chapter 6: Electrical Circuit Analysis and Computers 218 6.1 Electronics Captures a Teenage Mind 218 6.2 My First Project 220 6.3 "Building" Circuits on a Computer 230 6.4 Frequency Response by Computer Analysis 234 6.5 Differential Amplifiers and Electronic Circuit Magic 249 6.6 More Circuit Magic: The Inductor Problem 260 6.7 Closing the Loop: Sinusoidal and Relaxation Oscillators by Computer 272 6.8 Challenge Problems 278 6.9 Notes and References 281 Chapter 7: The Leapfrog Problem 288 7.1 The Origin of the Leapfrog Problem 288 7.2 Simulating the Leapfrog Problem 290 7.3 Challenge Problems 296 7.4 Notes and References 296 Chapter 8: Science Fiction: When Computers Become Like Us 297 8.1 The Literature of the Imagination 297 8.2 Science Fiction "Spoofs" 300 8.3 What If Newton Had Owned a Calculator? 305 8.4 A Final Tale: the Artificially Intelligent Computer 314 8.5 Notes and References 324 Chapter 9: A Cautionary Epilogue 328 9.1 The Limits of Computation 328 9.2 The Halting Problem 330 9.3 Notes and References 333 Appendix (FPU Computer Experiment MATLAB Code) 335 Solutions to the Challenge Problems 337 Acknowledgments 371 Index 373 Also by Paul J. Nahin 377

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書名 Number-crunching : taming unruly computational problems from mathematical physics to science fiction
著作者等 Nahin, Paul J.
出版元 Princeton University Press
刊行年月 c2011
ページ数 xxvi, 376 p.
大きさ 24 cm
ISBN 9780691144252
NCID BB06918339
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言語 英語
出版国 アメリカ合衆国