This book is a follow-up to John Casti's Five Golden Rules. In this installment, the author talks about the development of five (more) mathematical theories and their applications.
Chapter 1 The Alexander Polynomial: Knot Theory
In the first chapter, we are introduced to the theory of knots: when (and how) a knot can be unraveled and the idea of knot invariants. Some of the invariants include knot colorings, linking numbers, twisting numbers, writhing numbers, and the Alexander polynomial. Casti manages to relate the theory (at least in part) to the works of Albrecht Durer, to minimal energy configurations, and to the raveling and unraveling of DNA. This last topic yields one of the most interesting observations I've seen in a while:
...the amino acids making up the proteins in living organisms as well as the nucleic acids forming the cellular DNA all come in both left- and right-handed forms. These two forms, although chemically identical in the sense of being formed from exactly the same atoms, have entirely different chemical actions as a result of their "twisting" in opposite directions in space. Interestingly, in observations of galactic clouds in space, as well as in experiments in earthly chemical laboratories, it seems that both forms arise naturally in more or less equal proportions. Yet all life forms on Earth use exclusively left-handed amino acids to form proteins and right-handed nucleic acids to form the genetic material. As a consequence of this puzzling fact, you would starve to death on a world where the steaks were all made out of right-handed proteins because your body chemistry would be unable to break these proteins down to extract their energy.
Chapter 2 The Hopf Bifurcation Theorem: Dynamical System Theory
In this chapter, we learn of the theory of dynamical systems, and the idea of stability and attractors. Casti provides many examples of such systems and uses them to roll out many of the major results from the theory: the Linear Stability Theorem, the Hartman-Grobman Theorem (on linear approximation of a dynamical system near the origin), the Center Manifold Theorem (on equilibrium solutions), and the Hopf Bifurcation Theorem (on the stability of a system's equilibrium). This leads to discussions on randomness and deterministic dynamical systems, the motion of planets, fractals, and the music of Bach.
Chapter 3 The Kalman filter: Control Theory
This chapter starts with the idea of trying to determine if, given a set of rules governing the motion of an object (control inputs), we can reach a given position. This problem of reachability is solved for linear dynamical systems. Casti then turns to the problem of observation: observation of things (like concentration of a drug in a patient's blood stream) which cannot be measured directly. The problem of complete observability is solved. The solutions to the problems of reachability and observability look remarkably similar, which is no accident: the problems turn out to be duals of each other. Next, Bernoulli's brachistochrone problem is used to introduce the calculus of variations and the theory of optimal control. In studying optimal control processes, we are lead to the Pontryagin Minimum Principle and we find that the solutions to the optimal control problems have the form of an "open-loop" control law. On the other hand, in studying the stability of a control system, we are led to the idea of feedback control and dynamic programming. It turns out that the calculus of variations and dynamic programming are duals of each other. The Kalman filter is then introduced as a way of observing a system in which each observation is corrupted by some background noise. This piece of mathematics leads to the following application:
The Kalman filter is used in just about every inertial navigation system in existence. For example, Hexad, the gyroscopic system on the Boeing 777 aircraft uses a Kalman filter to estimate the errors in each of the six gyros with respect to the others.
Chapter 4 The Hahn-Banach Theorem: Functional Analysis
This chapter provides a nice history of the development of some of the early topics of functional analysis. Casti once again exploits the idea of duality, relating functions to functionals. The study of functionals leads us to the standard results: the Riesz Representation Theorem and the Hahn-Banach Theorem. The author then turns to the idea of operators and linear transformations, giving the Contraction Mapping Theorem and the Spectral Theorem. These ideas are used to describe John von Neumann's attempt to apply functional analysis to quantum mechanics. The remainder of the chapter is reserved for what happens with nonlinear operators.
Chapter 5 The Shannon Coding Theorem: Information Theory
This final chapter gives some of the history of information theory, which leads to the study of making codes. McMillan's Theorem on uniquely decipherable codes and Kraft's Theorem on instantaneous codes are mentioned, and a nice description of Huffman's instantaneous coding scheme is given. The topic of optimal codes (codes with minimal average code-word length) culminates in Shannon's Noiseless Coding Theorem. Casti then turns to the matter of signals and error-correcting codes, which leads to a description of Hamming's error-correcting codes. We return to the topic of DNA, which is itself a coding scheme--a code which is both instantaneous and optimal. The chapter ends with an interesting discussion on the relation between word rank and frequency of use (Zipf's Law).
Casti's presentation of these topics is very readable. Many of the topics start as simple ideas, which he uses to lead us to some very rich mathematics. He sprinkles the topics with occasional brief quotes and stories from historical characters. And he provides a nice set of references for each of the five themes, in which he not only lists, but also gives a short description of, each book or article.
With regards to the nonmathematical reader, this book walks a fine line: giving enough of the general view of each theory and its applications to keep the reader interested, while mathematically justifying the steps (even if this reader does end up skipping over the equations and formulas). Mathematicians, however, should enjoy seeing how these abstract theories play themselves out in the "real world". Perhaps a word of caution is in order, though: the mathematics (at least in the second, third, and fourth chapters) is heavily skewed toward analysis. So, if you really don't care for integration and differential equations, then you may want to restrict yourself to the first and last chapters. But any professional mathematician should appreciate the beauty of these ideas.
Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at email@example.com.