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.
|Publisher:||Princeton University Press|
|Product dimensions:||6.40(w) x 9.30(h) x 1.10(d)|
About the Author
Table of Contents
Chapter 1: FEYNMAN MEETS FERMAT 11.1 The Physicist as Mathematician 11.2 Fermat's Last Theorem 21.3 "Proof" by Probability 31.4 Feynman's Double Integral 61.5 Things to come 101.6 Challenge Problems 111.7 Notes and References 13
Chapter 2: Just for Fun: Two Quick Number-Crunching Problems 162.1 Number-Crunching in the Past 162.2 A Modern Number-Cruncher 202.3 Challenge Problem 252.4 Notes and References 25
Chapter 3: Computers and Mathematical Physics 273.1 When Theory Isn't Available 273.2 The Monte Carlo Technique 283.3 The Hot Plate Problem 343.4 Solving the Hot Plate Problem with Analysis 383.5 Solving the Hot Plate Problem by Iteration 443.6 Solving the Hot Plate Problem with the Monte Carlo Technique 503.7 ENIAC and MANIAC-I: the Electronic Computer Arrives 553.8 The Fermi-Pasta-Ulam Computer Experiment 583.9 Challenge Problems 733.10 Notes and References 74
Chapter 4: The Astonishing Problem of the Hanging Masses 824.1 Springs and Harmonic Motion 824.2 A Curious Oscillator 874.3 Phase-Plane Portraits 964.4 Another (Even More?) Curious Oscillator 994.5 Hanging Masses 1044.6 Two Hanging Masses and the Laplace Transform 1084.7 Hanging Masses and MATLAB 1134.8 Challenge Problems 1244.9 Notes and References 124
Chapter 5: The Three-Body Problem and Computers 1315.1 Newton's Theory of Gravity 1315.2 Newton's Two-Body Solution 1395.3 Euler's Restricted Three-Body Problem 1475.4 Binary Stars 1555.5 Euler's Problem in Rotating Coordinates 1665.6 Poincaré and the King Oscar II Competition 1775.7 Computers and the Pythagorean Three-Body Problem 1845.8 Two Very Weird Three-Body Orbits 1955.9 Challenge Problems 2055.10 Notes and References 207
Chapter 6: Electrical Circuit Analysis and Computers 2186.1 Electronics Captures a Teenage Mind 2186.2 My First Project 2206.3 "Building" Circuits on a Computer 2306.4 Frequency Response by Computer Analysis 2346.5 Differential Amplifiers and Electronic Circuit Magic 2496.6 More Circuit Magic: The Inductor Problem 2606.7 Closing the Loop: Sinusoidal and Relaxation Oscillators by Computer 2726.8 Challenge Problems 2786.9 Notes and References 281
Chapter 7: The Leapfrog Problem 2887.1 The Origin of the Leapfrog Problem 2887.2 Simulating the Leapfrog Problem 2907.3 Challenge Problems 2967.4 Notes and References 296
Chapter 8: Science Fiction: When Computers Become Like Us 2978.1 The Literature of the Imagination 2978.2 Science Fiction "Spoofs" 3008.3 What If Newton Had Owned a Calculator? 3058.4 A Final Tale: the Artificially Intelligent Computer 3148.5 Notes and References 324
Chapter 9: A Cautionary Epilogue 3289.1 The Limits of Computation 3289.2 The Halting Problem 3309.3 Notes and References 333
Appendix(FPU Computer Experiment MATLAB Code) 335Solutions to the Challenge Problems 337
Acknowledgments 371Index 373Also by Paul J. Nahin 377
What People are Saying About This
"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