31 of 31 people found the following review helpful:
3.0 out of 5 stars
Beauty that is not made accessible to the layperson, December 2, 2000
This review is from: Five More Golden Rules: Knots, Codes, Chaos and Other Great Theories of 20th-Century Mathematics (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.
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26 of 26 people found the following review helpful:
5.0 out of 5 stars
nice sequel to Five golden rules but heavy on the mathematics, January 24, 2008
This review is from: Five More Golden Rules: Knots, Codes, Chaos and Other Great Theories of 20th-Century Mathematics (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.
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19 of 21 people found the following review helpful:
4.0 out of 5 stars
Not bad--very scientific, not for the lay person, June 19, 2000
By A Customer
This review is from: Five More Golden Rules: Knots, Codes, Chaos and Other Great Theories of 20th-Century Mathematics (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.
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