Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics Book

Chapter 1	The Alexander Polynomial: Knot Theory	1
	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	35
	What Is a Dynamical System?
	In the Long Run
	Stability
	At the Center of Things
	Forks in the Flow
	Gradient Dynamical Systems 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 Efficient Markets
	Domains of Attraction
	A Mountain Is Not a Cone and a Cloud Is Not a Sphere
	Computing the Fractal Dimension
	The Fractal Dimension of a River Basin
	Bach and Fractal Music
	Fractals and Domains of Attraction
Chapter 3	The Kalman Filter: Control Theory	101
	Looking for Life
	Can You Get There from Here?
	The Problem of Reachability
	The Problem of Observation
	Complete Observability
	Duality
	A Nonlinear Interlude
	Sharks and Minnows
	Linear Stability Analysis
	What's Best?
	The Pontryagin Minimum 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	155
	The Big Picture
	Problems at Infinity
	The Tale of Two Coffee Houses
	Spaces and Functionals
	Minimum Distances and Maximum 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	207
	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	255
	Index	263