Five more golden rules : knots, codes, chaos, and other great theories of 20th century mathematics

"Casti is one of the great science writers." -San Francisco Examiner "Casti's gift is to be able to let the nonmathematical reader share in his understanding of the beauty of a good theory." -Christian Science Monitor Following up the acclaimed Five Golden Rules, another quintet of gleaming math discoveries With Five More Golden Rules, readers are treated to another fascinating set of theoretical gems from acclaimed popular science author John Casti. Injecting all-new ingredients into his trademark recipe of real-world examples, historical anecdotes, and straightforward explanations, Casti once again brings math to thrilling life. All who enjoyed the unique pleasures of the original will love this follow-up survey highlighting the creme de la creme of math in the last century. Explores how knot theory informs the classic tale of Alexander the Great and the Gordian Knot \* Considers how the Shannon Coding Theory applies to decoding the human genome John L. Casti, PhD (Santa Fe, NM), a resident member of the Santa Fe Institute, is a professor at the Technical University of Vienna and the author of Would-Be Worlds (Wiley) and Cambridge Quintet. Chapter 1 The Alexander Polynomial: Knot Theory Knot History, Mathematics · Have Knot, Will Unravel · What Color Is Your Knot? · Unwinding DNA · Alexander's Great Invariant · Getting Physical · All Tangled Up . Appendix: Knots and Energy Chapter 2 The Hopf Bifurcation Theorem: Dynamical System Theory What Is a Dynamical System? · In the Long Run · Stability. At the Center of Things · Forks in the Flow · Gradient Dynamical Sys- tems and Elementary Catastrophes · A Strangeness in the Attraction · Up, Up, Down and Away · The Lyapunov Exponents · Chaos among the Planets · The Correlation Dimension · Gold Bugs and Ef- ficient Markets · Domains of Attraction · A Mountain Is Not a Cone and a Cloud Is Not a Sphere · Computing the Fractal Dimen- sion · The Fractal Dimension of a River Basin . Bach and Fractal Music · Fractals and Domains of Attraction Chapter 3 The Kalman Filter: Control Theory Looking for Life · Can You Get There from Here? . The Problem of Reachability · The Problem of Observation · Complete Observabil- ity · Duality · A Nonlinear Interlude · Sharks and Minnows · Linear Stability Analysis . What's Best? . The Pontryagin Mini- mum Principle · Feedback Control and Dynamic Programming · The Minimum Principle versus Dynamic Programming · Getting the Best of It · State Estimation · The Kalman Filter · Duality-Yet Again Chapter 4 The Hahn-Banach Theorem: Functional Analysis The Big Picture . Problems at Infinity · The Tale of Two Coffee Houses · Spaces and Functionals · Minimum Distances and Maxi- mum Profits · The Other Side of a Space · But Does It Exist? · Big- Time Operators · Quantum Mechanics and Functional Analysis · The Importance of Being Nonlinear Chapter 5 The Shannon Coding Theorem: Information Theory Communication, Information, and Life . Making Codes · Huffing and Puffing and Squeezing the Message Down · Symbols, Signals, and Noise . If It Ain't Broken, Fix It Anyway . The Information of the Genes · Randomness, Information, and Computation · Zipf Up That Lip. Appendix: Derivation of the Power Law Form of Zipf's Law References Index How Did The Classic Tale Of Alexander The Great Influence The Principal Problem Facing Knot Theorists Today? What Effect Did The Exalted Kalman Filter Have On Space Travel? How Did The Shannon Coding Theory Make Mapping Human Dna Possible? In This Follow-up To Five Golden Rules, Popular Science Author John Casti Treats Us To Another Quintet Of Theories To Answer These And Other Questions. Casti Reveals The Origins Of Some Of The Most Monumental Mathematical Discoveries Of The Last Century - And Sheds Light On How These Revolutionary Ideas Have Reshaped Our Lives.--jacket. Alexander Polynomial: Knot Theory -- Hopf Bifurcation Theorem: Dynamical System Theory -- Kalman Filter: Control Theory -- Hahn-banach Theorem: Functional Analysis -- Shannon Coding Theorem: Information Theory. John L. Casti. Includes Bibliographical References (p. [255]-262) And Index. "How did the classic tale of Alexander the Great influence the principal problem facing knot theorists today? What effect did the exalted Kalman filter have on space travel? How did the Shannon Coding Theory make mapping human DNA possible? In this follow-up to Five Golden Rules, popular science author John Casti treats us to another quintet of theories to answer these and other questions.". "Casti reveals the origins of some of the most monumental mathematical discoveries of the last century - and sheds light on how these revolutionary ideas have reshaped our lives."--BOOK JACKET. A follow-up to the critically acclaimed and strong-selling Five Golden Rules, this book introduces five more of the greatest mathematical theorems discovered in the 20"th" century.