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More Golden Rules P: Knots, Codes, Chaos and Other Great Theories of 20th-century Mathematics Paperback – 9 Feb 2001


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Product details

  • Paperback: 272 pages
  • Publisher: John Wiley & Sons; New Ed edition (9 Feb 2001)
  • Language: English
  • ISBN-10: 0471395285
  • ISBN-13: 978-0471395287
  • Product Dimensions: 15.7 x 1.5 x 23.4 cm
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 1,822,096 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Amazon Review

Bring more joy to your favourite maths-head with Five More Golden Rules from science writer and national treasure John L. Casti. Though a quick glance through the book will cause an intense fight-or-flight response in the numbers-phobic, Casti's writing is lovely and lucid as ever, explaining not just equations and theorems but their significance in our lives. Having discovered in Five Golden Rules that he couldn't restrict himself to just five important 20th-century mathematical theories, this follow-up explores the intricacies of knot theory, functional analysis, control theory, chaotic systems and information theory. Each of the five lively chapters introduces its subject with a seemingly-unrelated anecdote that is (of course) informed by the theory in question. Then it's headlong into the wonderful details of postulation and demonstration that makes maths so much fun. Unlike a textbook, Five More Golden Rules meanders and breaks away from its proofs to discover relations between the symbols and the real world from the stock market to the coastline of Norway. Besides giving the reader a break, this makes the abstract, almost ethereal concepts concrete and provides a definite advantage to the interested student. Perhaps textbook publishers should take note of this technique; until they do, we'll have to curl up with Casti's Five More Golden Rules if we want to have fun with our higher maths.--Rob Lightner --This text refers to an out of print or unavailable edition of this title.

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"...any reader with a nodding acquaintance with calculus should find the book interesting and informative..." (Mathematika)

“…any reader with a nodding acquaintance with calculus should find the book interesting and informative…”(Mathematika)

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First Sentence
According to legend, one of the world's first knot theorists was the simple Greek peasant, Gordius, who was named king of Phrygia when he arrived in a public square in an oxen cart around 1200 B.C., thus fulfilling an oracle's pronouncement that the future king would arrive in a wagon. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Most Helpful Customer Reviews

Format: Hardcover
This book is not for the faint hearted but if you like mathematics at high level (i.e. degree level and beyond) then you may well appreciate this book. If you enjoyed dynamical systems theory, control systems theory and/or manifold theory at university, or at least thought you did (or thought you should have done), then this book may give you the insights that the drier, purely mathematical (text book) approaches failed to do. If you are a professor of mathematics, you might hate the book on the grounds that the mainly non-mathematical treatment lacks rigour. But, hey, it's a popular science book!
Amazingly, Casti covers five fascinating theories with only the barest minimum of absolutely essential but non-trivial(!) mathematics. The cleverness of this book, for me at least, is that it provides deep insights to sometimes highly abstract mathematical notions and connects one to the usual mathematical equations for each area without the need for preliminary mathematics (but this requires the reader to be comfortable with diving in at the deep end). Casti's powers of explanation using plain English replace the need for much of the mathematics that one typically finds in books about these five theorems. But it's not easy going. It's a book that demands your undivided attention and concentration. A good time to read it would be on a long aeroplane journey.
In summary, most people, in fairness, won't be equipped to read this book and therefore won't like it (one star). If the title piques your interest and you're not in the first category, you might really like it, as I did, hence 4 stars. Maths purists might be cooler about it: maybe 2 - 3 stars.
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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 7 reviews
26 of 26 people found the following review helpful
nice sequel to Five golden rules but heavy on the mathematics 24 Jan 2008
By Michael R. Chernick - Published on Amazon.com
Format: Hardcover
In Five Golden Rules John Casti wrote a wonderful book about important theorems in mathematics that were discovered in the 20th Century. The style and description was such that a layperson could understand, enjoy and appreciate the results. All the theorems were discovered before 1950 and they all dealt with topics in applied mathematics and particularly game theory and operations research.
Perhaps he found the list of five golden rules too restrictive and thus comes the sequel "Five More Golden Rules". Again, it would be hard to argue the choices. Casti goes into the details of the theorems and the theory related to them much like he did in the first book. However, in this book, he has chosen topics from very abstract areas of mathematics. I have a masters degree in mathematics and a Ph.D. in statistics and yet I had no familiarity with knot theory. So I learned a lot from chapter 1 but found it to be difficult reading, more like a mathematics textbook than a popular book for the scientist and layman.

This feeling continued as I read the other four chapters even though I was treading on territory that was very familiar to me (e.g. the Kalman filter of control theory). It was reassuring to me to see that this impression was also shared by the three customers that had already written reviews on the book.

I recommend this book wholeheartedly for mathematicians and other with strong math training. The Hahn-Banach theorem was the most important theorem that I learned about when I took my functional analysis course at the University of Maryland some 26 years ago. But I have not had much use for it since and I completely forgot what it said. Casti provides me with a nice reminder and shows how this result is a generalization of very practical results that relate to quantum mechanics and other results in physics.

The latter part of the 20th Century saw a great deal of activity in nonlinear dynamics. This is connected by Casti to the Hopf Bifurcation theorem. That chapter deals with many topics that grasped the attention of applied mathematicians, including chaos and catastrophy theory, strange attractors and the beautiful geometry of fractals. This material is not for a layperson. On the other hand, the introduction to the chapter, covering what a dynamical system is, provides a wonderful analogy to a treasure hunt in Central Park that can be appreciated by everyone.

The Kalman filter provides an example of how linearization of real dynamic systems allows one to write a prediction equation for the state at the next time point recursively as a function of the current state and the new measurements. This recursive formulation leads to the same solution that Wiener had found much earlier, but because of the recursion, it is much more suitable for real time computer applications. This was essential to controlling space vehicles and is the important result that made the trip to the moon possible. Casti covers the theory of Kalman filtering very well but emphasizes many of the interesting abstract concepts rather than the more concrete aspects of the solution.

The finally chapter on the Shannon Coding Theorem takes us into the realm of information theory. Casti provides the key references. Electronic communication in the 20th century has benefitted from the efficient coding of information that makes transmissions faster easier and error free. This is very important work with unforeseen applications. Casti points to applications in genetics.

Another interesting feature of the book is the connection made between the knot theory associated with Alexander's polynomials and DNA sequencing, a subject to be further explored in the 21st Century.
32 of 33 people found the following review helpful
Beauty that is not made accessible to the layperson 3 Dec 2000
By Daniel Lalonde - Published on Amazon.com
Format: Hardcover
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«The linear dynamical system (**) is completely reachable if and only if the block matrix C contains n linearly independant vectors, that is, rank C = N»
If you don't feel completely at ease with this sentence, do not read this book. Every page contains mathematical propositions of such level, and such level of mathematical fluency is required in order to fully appreciate the content of John Casti's book. The content is interesting but the reading is made rendered somewhat tedious by this high density of maths. I have a degree in engineering, and I often fast forwarded trough the equations in an effort to not lose sight of the big picture Casti want to show the reader.
At the end you will be smarter, but it will not have been a relaxed reading. If you are looking for food for toughts, I would recommand, among others, «Paradigms Lost : Tackling the Unanswered Mysteries of Modern Science», by the same author.
19 of 21 people found the following review helpful
Not bad--very scientific, not for the lay person 19 Jun 2000
By A Customer - Published on Amazon.com
Format: Hardcover
This book was a good look at some of the newer, yet important laws that govern physisc, and science altogether. I found it well-written, fascinating and a fun book to read. I would not recommend it to a person without a strong scientific background, as the author makes many assumptions about the readers knowledge.
Dreadful 25 July 2010
By Mark Twian - Published on Amazon.com
Format: Paperback
As a popular science/maths work - horrible. In the middle of the chapter on the Kalman filter he starts talking about Lie algebras without actually explaining what they are, or their relevance. The whole topic is just dropped in from above and then disappears from the discussion more or less without further comment. I understand what they are, but if I were coming to this as a layperson I'd be completely in "WFT??" mode.

Further examples of this abound in this chapter - econometrics is chucked in for no particular reason and then just dropped and ignored. A good example of a child on a swing is not fully explained. What is the variable 'l'?. I would guess the hinge point where the child's hands are on the ropes of the swing but that is never articulated. Langragian dynamics appears but it's just chucked in as a couple of formula without any identification of what it is or how he's going to use it. Technical phrases like "equation of state" are just used without any explanation.

To top it all off the chapter starts by asserting that the Kalman filter is one of the most important results of the 20th C but doesn't actually bother to state what it is until the last page, and then does a bad job of it.

I'm not sure I'd even accept this as a series of lecture notes. (It certainly reads as though it was sourced from a set though - a bad set.)
More Difficult than Five Golden Rules 13 Aug 2000
By Keith Halbasch - Published on Amazon.com
Format: Hardcover
The prospective reader is warned that this book requires considerably more mathematical background than Five Golden Rules, a fact that is not made clear on the book jacket nor in the nonexistent forward or table of contents. In addition, Casti seems to have found it necessary to leave many more points incompletely explained than in Five Golden Rules; though perhaps this was unavoidable given the more difficult subject matter. I am puzzled about what audience Casti thought would read the book.
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