Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

by Paul J. Nahin

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Overview

More stimulating mathematics puzzles from bestselling author Paul 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.

Product Details

ISBN-13: 9780691144252
Publisher: Princeton University Press
Publication date: 08/28/2011
Pages: 408
Product dimensions: 6.40(w) x 9.30(h) x 1.10(d)

About the Author

Paul J. Nahin is the author of many best-selling popular math books, including Mrs. Perkins's Electric Quilt, Digital Dice, Chases and Escapes, Dr. Euler's Fabulous Formula, When Least Is Best, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.

Table of Contents

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 Poincaré 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

What People are Saying About This

From the Publisher

"While there is a plethora of computational physics books, only this one brings the sheer joy and fascination of the subject to the general reader. The problems in Number-Crunching are nicely selected, the prose is clear and humorous, and the solutions range from the interesting to the gloriously counterintuitive. With the ubiquity of powerful personal computers and the easy availability of scientific software, this is a very timely book."—Lawrence Weinstein, coauthor of Guesstimation

"Nahin's work is always highly interesting. His book tackles a large number of problems that can be handled by an adroit combination of pencil-and-paper work and, where the analytics get too difficult, numerical computation. This book is great and I really enjoyed it."—Charles Adler, St. Mary's College

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