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Computer Algebra Recipes a Gourmet's Guide to Mathematical Models of Science

by Richard H. Enns

  • ISBN: 9780387951485
  • ISBN10: 0387951482

Computer Algebra Recipes a Gourmet's Guide to Mathematical Models of Science

by Richard H. Enns

  • List Price: $154.00
  • Binding: Hardcover
  • Publisher: Springer Verlag
  • Publish date: 06/01/2001
  • ISBN: 9780387951485
  • ISBN10: 0387951482
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Description: Preface.- A Computer Algebra Systems.- B The Spiral Staircase to Learning.- C How to Climb the Spiral Staircase.- I The Appetizers.- 1 The Pictures of Science.- 1.1 Introduction.- 1.2 Data and Function Plots.- 1.2.1 Correcting for Inflation.- 1.2.2 The Plummeting Badminton Bird.- 1.2.3 Minimizing the Travel Time.- 1.3 Log-Log (Power Law) Plots.- 1.3.1 Chimpanzee Brain Size.- 1.3.2 Scaling Arguments and Gulliver''s Travels.- 1.4 Contour and Gradient Plots.- 1.4.1 The Secret Message.- 1.4.2 Designing a Ski Hill.- 1.5 Animated Plots.- 1.5.1 Waves Are Dynamic.- 1.5.2 The Sands of Time.- 2 Deriving Model Equations.- 2.1 Introduction.- 2.2 Linear Correlation.- 2.2.1 What Is Linear Correlation'.- 2.2.2 The Corn Palace.- 2.3 Least Squares Derivations.- 2.3.1 Regression Analysis.- 2.3.2 Will You Be Better Off Than Your Parents'.- 2.3.3 What Was the Heart Rate of a Brachiosaurus'.- 2.3.4 Senate Renewal.- 2.3.5 Bikini Sales and the Logistic Curve.- 2.3.6 Following the Dow Jones Index.- 2.3.7 Variation of "g" with Latitude.- 2.3.8 Finding Romeo a Juhet.- 2.4 Multiple Regression Equations.- 2.4.1 Real Estate Appraisals.- 2.4.2 And the Winner Is'.- 3 Algebraic Models.- 3.1 Introduction.- 3.2 Algebraic Examples.- 3.2.1 Bombs Versus Schools.- 3.2.2 Kirchhoff Rules the Electrical World.- 3.2.3 The Window Washer''s Secret.- 3.2.4 The Science Student''s Summer Job Interview.- 3.2.5 Envelope of Safety.- 3.2.6 Rainbow County.- 3.3 Integral Examples.- 3.3.1 The Great Pyramid of Cheops.- 3.3.2 Noah''s Ark.- 3.4 Vector Examples.- 3.4.1 Vectoria''s Mathematical Heritage.- 3.4.2 Ain''t She Sweet.- 3.4.3 Born Curl Free.- 3.4.4 Of Flux and Circulation and Coordinates Too.- 4 Monte Carlo Methods.- 4.1 Introduction.- 4.2 Random Walks.- 4.2.1 The Concept.- 4.2.2 The Soccer Fan''s Drunken Walk.- 4.2.3 Blowin'' in the Wind.- 4.2.4 Flight of Penelope Jitter Bug.- 4.2.5 That Meandering Perfume Molecule.- 4.3 Monte Carlo Integration.- 4.3.1 Standard Numerical Integration Algorithms.- 4.3.2 Monte Carlo Integration.- 4.3.3 Wait and Buy Later!.- 4.3.4 Wait and Buy Later! The Sequel.- 4.3.5 Estimating '.- 4.3.6 Chariot of Fire and Destruction.- 4.4 Probability Distributions.- 4.4.1 Of Nuts and Bolts and Hospital Beds Too.- 4.4.2 The Ice Wines of Rainbow County.- 4.5 Monte Carlo Statistical Distributions.- 4.5.1 Estimating e.- 4.5.2 Vapor Deposition.- II The Entrees.- 5 Phase-Plane Portraits.- 5.1 Introduction.- 5.2 Phase-Plane Portraits.- 5.2.1 Stationary or Singular Points.- 5.3 Linear ODE Models.- 5.3.1 Tenure Policy at Erehwon University.- 5.3.2 Vectoria Investigates the RLC Circuit.- 5.4 Nonlinear ODE Models.- 5.4.1 Classification of Stationary Points.- 5.4.2 Rabbits and Foxes.- 5.4.3 The Mona Lisa of Nonlinear Science.- 5.4.4 Mike Creates a Higher-Order Singular Point.- 5.4.5 The Gnus and Sung of Erehwon.- 5.5 Nonautonomous ODEs.- 5.5.1 Can an Unstable Spring Find Stability'.- 5.5.2 The Period Doubling Route to Chaos.- 6 Linear ODE Models.- 6.1 Introduction.- 6.2 Solving Linear ODEs with Maple.- 6.3 First-Order ODE Models.- 6.3.1 There Goes Louie''s Alibi.- 6.3.2 The Water Skier.- 6.4 Second-Order ODE Models.- 6.4.1 Shrinking the Safety Envelope.- 6.4.2 Halley''s Comet.- 6.4.3 Frank N Stein Is Not Heartless.- 6.4.4 Vectoria Feels the Force and Hits the Bottle.- 6.5 Bessel and Legendre ODE Models.- 6.5.1 Introduction to Special Functions.- 6.5.2 The Vibrating Bungee Cord.- 6.5.3 Wheel of misFortune.- 6.5.4 TheWeedeater.- 7 Nonlinear ODE Models.- 7.1 Introduction.- 7.2 First-Order Models.- 7.2.1 The Nonlinear Diode.- 7.2.2 The Bad Bird Equation.- 7.2.3 The Struggle for Existence.- 7.3 Second-Order Models.- 7.3.1 Pirates of the Caribbean.- 7.3.2 Oh What Sounds We Hear!.- 7.3.3 Those Lennard-Jones Vibrational Blues.- 7.3.4 Golf Is Such an "Uplifting" Experience.- 7.3.5 This Would Be a Great Amusement Park Ride.- 7.4 Limit Cycles.- 7.4.1 The Bizarre World of the Tunnel Diode Oscillator.- 7.4.2 Follow That Rabbit.- 8 Difference Equation Models.- 8.1 Introduction.- 8.2 Linear Difference Equation Models.- 8.3 First-Order Linear Models.- 8.3.1 Those Dratted Gnats.- 8.3.2 Gone Fishing.- 8.4 Second-Order Linear Models.- 8.4.1 Fibonacci''s Adam and Eve Rabbit.- 8.4.2 How Red Is Your Blood'.- 8.4.3 Fermi-Pasta-Ulam Is Not a Spaghetti Western.- 8.5 Nonlinear Difference Equation Models.- 8.6 First-Order Nonhnear Models.- 8.6.1 Competition for Available Resources.- 8.6.2 The Logistic Map and Cobweb Diagrams.- 8.7 Second-Order Nonhnear Models.- 8.7.1 The Bouncing Ball Art Gallery.- 8.7.2 Onset of Chaos: A Model for the Outbreak of War.- 8.8 Numerically Solving ODEs.- 8.8.1 Finite Difference Approximations to Derivatives.- 8.8.2 Rabbits and Foxes: The Sequel.- 8.8.3 Glycolytic Oscillator.- 9 Some Analytic Approaches.- 9.1 Introduction.- 9.2 Checking Solutions.- 9.2.1 The Palace of the Governors.- 9.2.2 Play It, Sam.- 9.2.3 The Tliree-Piece String.- 9.3 Calculus of Variations.- 9.3.1 Dress Design, The Erehwonese Way.- 9.3.2 Queen Dido''s Problem.- 9.3.3 The Human Fly Plans His Escape Route.- 9.4 Fourier Series.- 9.4.1 Hi C Is Not Always a Drink.- 9.4.2 Play It, Sam: A New Perspective.- 9.4.3 Vectoria Sums a Series.- 10 Fractal Patterns.- 10.1 Introduction.- 10.2 Difference Equation Patterns.- 10.2.1 Wallpaper for the Mind.- 10.2.2 Sierpinski''s Fractal Gasket.- 10.2.3 Barnsley''s Fern.- 10.2.4 Douadv''s Rabbit and Other Fauna and Flora.- 10.2.5 The Rings of Saturn.- 10.3 ODE Patterns.- 10.3.1 The Butterfly Attractor.- 10.3.2 Rssler''s Strange Attractor.- 10.4 Cellular Automata Patterns.- 10.4.1 A Navaho Rug Design.- 10.4.2 The One Out of Eight Rule.- III The Desserts.- 11 Diagnostic Tools for Nonlinear Dynamics.- 11.1 Introduction.- 11.2 The Poincar Section.- 11.2.1 The Concept.- 11.2.2 A Rattler Signals Chaos.- 11.3 The Power Spectrum.- 11.3.1 The Concept.- 11.3.2 The Rattler Returns.- 11.4 The Bifurcation Diagram.- 11.4.1 The Concept.- 11.4.2 Pitchforks and Other Bifurcations.- 11.5 The Lyapunov Exponent.- 11.5.1 The Concept.- 11.5.2 Mr Lyapunov Agrees.- 11.6 Reconstructing an Attractor.- 11.6.1 The Concept.- 11.6.2 Chaos Versus Noise.- 12 Linear PDE Models.- 12.1 Introduction.- 12.1.1 The Linear PDEs of Mathematical Physics.- 12.1.2 Separation of Variables.- 12.2 Diffusion and Laplace''s Equation Models.- 12.2.1 Freeing Excalibur.- 12.2.2 Aussie Barbecue.- 12.2.3 Erehwon Institute of Technology.- 12.2.4 Hugo and the Atomic Bomb.- 12.2.5 Hugo Prepares for his Job Interview.- 12.3 Wave Equation Models.- 12.3.1 Vectoria Encounters Simon Legree.- 12.3.2 Homer''s Jiggle Test.- 12.3.3 Vectoria''s Second Problem.- 12.4 Semi-Infinite and Infinite Domains.- 12.4.1 Vectoria''s Third Problem.- 12.4.2 Assignment Complete!.- 12.4.3 Radioactive Contamination.- 12.4.4 "Play It Sara" Revisited.- 13 Nonlinear PDE Models: Soliton Solutions.- 13.1 Introduction.- 13.2 Solitary Waves.- 13.3 The Graphical Hunt for Solitons.- 13.3.1 Of Kinks and Antikinks.- 13.3.2 In Search of Bright Solitons.- 13.3.3 Can Three Solitary Waves Live Together'.- 13.4 Analytic Soliton Solutions.- 13.4.1 Follow That Wave!.- 13.4.2 Looking for a Kinky Solution.- 14 Simulating PDE Models.- 14.1 Introduction.- 14.2 Diffusion and Wave Equation Models.- 14.2.1 Freeing Excalibur the Numerical Way.- 14.2.2 Vectoria Secret.- 14.2.3 Enjoy the Klein-Gordon Vibes.- 14.3 Soliton Colhsions.- 14.3.1 To Be or Not to Be a Soliton.- 14.3.2 Are Diamonds a Kink''s Best Friend'.- Epilogue.
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