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Number-Crunching

Book Description

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.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Contents
  6. Introduction
  7. 1 Feynman Meets Fermat
    1. 1.1 The Physicist as Mathematician
    2. 1.2 Fermat’s Last Theorem
    3. 1.3 “Proof” by Probability
    4. 1.4 Feynman’s Double Integral
    5. 1.5 Things to come
    6. 1.6 Challenge Problems
    7. 1.7 Notes and References
  8. 2 Just for Fun: Two Quick Number-Crunching Problems
    1. 2.1 Number-Crunching in the Past
    2. 2.2 A Modern Number-Cruncher
    3. 2.3 Challenge Problem
    4. 2.4 Notes and References
  9. 3 Computers and Mathematical Physics
    1. 3.1 When Theory Isn’t Available
    2. 3.2 The Monte Carlo Technique
    3. 3.3 The Hot Plate Problem
    4. 3.4 Solving the Hot Plate Problem with Analysis
    5. 3.5 Solving the Hot Plate Problem by Iteration
    6. 3.6 Solving the Hot Plate Problem with the Monte Carlo Technique
    7. 3.7 Eniac and Maniac-I: the Electronic Computer Arrives
    8. 3.8 The Fermi-Pasta-Ulam Computer Experiment
    9. 3.9 Challenge Problems
    10. 3.10 Notes and References
  10. 4 The Astonishing Problem of the Hanging Masses
    1. 4.1 Springs and Harmonic Motion
    2. 4.2 A Curious Oscillator
    3. 4.3 Phase-Plane Portraits
    4. 4.4 Another (Even More?) Curious Oscillator
    5. 4.5 Hanging Masses
    6. 4.6 Two Hanging Masses and the Laplace Transform
    7. 4.7 Hanging Masses and MATLAB
    8. 4.8 Challenge Problems
    9. 4.9 Notes and References
  11. 5 The Three-Body Problem and Computers
    1. 5.1 Newton’s Theory of Gravity
    2. 5.2 Newton’s Two-Body Solution
    3. 5.3 Euler’s Restricted Three-Body Problem
    4. 5.4 Binary Stars
    5. 5.5 Euler’s Problem in Rotating Coordinates
    6. 5.6 Poincaré and the King Oscar II Competition
    7. 5.7 Computers and the Pythagorean Three-Body Problem
    8. 5.8 Two Very Weird Three-Body Orbits
    9. 5.9 Challenge Problems
    10. 5.10 Notes and References
  12. 6 Electrical Circuit Analysis and Computers
    1. 6.1 Electronics Captures a Teenage Mind
    2. 6.2 My First Project
    3. 6.3 “Building” Circuits on a Computer
    4. 6.4 Frequency Response by Computer Analysis
    5. 6.5 Differential Amplifiers and Electronic Circuit Magic
    6. 6.6 More Circuit Magic: The Inductor Problem
    7. 6.7 Closing the Loop: Sinusoidal and Relaxation Oscillators by Computer
    8. 6.8 Challenge Problems
    9. 6.9 Notes and References
  13. 7 The Leapfrog Problem
    1. 7.1 The Origin of the Leapfrog Problem
    2. 7.2 Simulating the Leapfrog Problem
    3. 7.3 Challenge Problems
    4. 7.4 Notes and References
  14. 8 Science Fiction: When Computers Become Like Us
    1. 8.1 The Literature of the Imagination
    2. 8.2 Science Fiction “Spoofs”
    3. 8.3 What If Newton Had Owned a Calculator?
    4. 8.4 A Final Tale: the Artificially Intelligent Computer
    5. 8.5 Notes and References
  15. 9 A Cautionary Epilogue
    1. 9.1 The Limits of Computation
    2. 9.2 The Halting Problem
    3. 9.3 Notes and References
  16. Appendix (FPU Computer Experiment MATLAB Code)
  17. Solutions to the Challenge Problems
  18. Acknowledgments
  19. Index
  20. Also by Paul J. Nahin