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Title: The Essential John Nash by John F. Nash, Sylvia Nasar, Harold Kuhn, Harold W. Kuhn ISBN: 0-691-09527-2 Publisher: Princeton University Press Pub. Date: 01 December, 2001 Format: Hardcover Volumes: 1 List Price(USD): $31.95 |
Average Customer Rating: 4.45 (11 reviews)
Rating: 5
Summary: An excellent compilation
Comment: Having written about the life of the mathematician John Nash in the excellent biography "A Beautiful Mind", Sylvia Nasar teams up with the mathematician Harold W. Kuhn to produce a book that introduces the mathematical contributions of Nash, something that was done only from a "popular" point of view in Nasar's biography. For those who have the background, this book is a fine overview of just what won Nash acclaim in the mathematical community, and won him a Nobel Prize in economics.
It is always easy to dismiss ideas as trivial after they have been discovered and have been put into print. This is apparently what John von Neumann did after discussing with Nash his ideas on noncooperative games, dismissing his ideas as a mere "fixed point theorem". At the time of course, the only game-theoretic ideas that had any influence were those of von Neumann and his collaborator, the Princeton economist Oskar Morgenstern. The rejection of ideas by those whose who hold different ones is not uncommon in science and mathematics, and, from von Neumann's point of view at the time, he did not have the advantage that we do of examining the impact that Nash's ideas would have on economics and many other fields of endeavor. Therefore, von Neumann was somewhat justified, although not by a large measure, in dismissing what Nash was proposing. Nash's thesis was relatively short compared to the size on the average of Phd theses, but it has been applied to many areas, a lot of these listed in this book, and others that are not, such as QoS provisioning in telecommunication and packet networks. The thesis is very readable, and employs a few ideas from algebraic topology, such as the Brouwer fixed point theorem.
The paper on real algebraic manifolds though is more formidable, and will require a solid background in differential geometry and algebraic geometry. However, from a modern point of view the paper is very readable, and is far from the sheaf and scheme-theoretic points of view that now dominate algebraic geometry. It is interesting that Nash was able to prove what he did with the concepts he used. The result could be characterized loosely as a representation theory employing algebraic analytic functions. These functions are defined on a closed analytic manifold and serve as well-behaved imbedding functions for the manifold, which is itself analytic and closed. These manifolds have been called 'Nash manifolds' in the literature, and have been studied extensively by a number of mathematicians.
I first heard about John Nash by taking a course in algebraic topology and characteristic classes in graduate school. The instructor was discussing the imbedding problem for Riemannian manifolds, and mentioned that Nash was responsible for one of the major results in this area. His contribution is included in this book, and is the longest chapter therein. Here again, the language and flow of Nash's proof is very understandable. This is another example of the difference in the way mathematicians wrote back then versus the way they do now. Nash and other mathematicians of his time were more 'wordy' in their presentations, and this makes the reading of their works much more palatable. This is to be contrasted with the concisness and economy of thought expressed in modern papers on mathematics. These papers frequently employ a considerable amount of technical machinery, and thus the underlying conceptual foundations are masked. Nash explains what he is going to do before he does it, and this serves to motivate the constructions that he employs. His presentation is so good that one can read it and not have to ask anyone for assistance in the understanding of it. This is the way all mathematical papers should be written, so as to alleviate any dependence on an 'oral tradition' in mathematical developments.
Nash's proof illuminates nicely just what happens to the derivatives of a function when the smoothing operation is applied. The smoothing operator consists of essentially of extending a function to Euclidean n-space, applying a convolution operator to the extended function, and then restricting the result to the given manifold. Nash gives an intuitive picture of this smoothing operator as a frequency filter, passing without attenuation all frequencies below a certain parameter, omitting all frequencies above twice this parameter, and acting as a variable attenuator between these two, resulting in infinitely smooth function of frequency.
The next stage of the proof of the imbedding theorem is more tedious, and consists of using the smoothing operator and what Nash calls 'feed-back' to construct a 'perturbation device' in order to study the rate of change of the metric induced by the imbedding. Nash's description of the perturbation process is excellent, again for its clarity in motivating what he is going to do. The feed-back mechanism allows him to get a handle of the error term in the infinitesimal perturbation, isolating the smoother parts first, and handling the more difficult parts later. Nash reduces the perturbation process to a collection of integral equations, and then proves the existence of solutions to these equations. A covariant symmetric tensor results from these endeavors, which is CK-smooth for k greater than or equal to 3, and which represents the change in the metric induced by the imbedding of the manifold. The imbedding problem is then solved for compact manifolds by proving that only infinitesimal changes in the metric are needed. The non-compact case is treated by reducing it to the compact case. The price paid for this strategy is a weakening of the bound on the required dimension of the Eucliden imbedding space.
The last chapter concerns Nash's contribution to nonlinear partial differential equations. I did not read this chapter, so I will omit its review.
Rating: 5
Summary: A Most Welcome Mathematical Banquet
Comment: I can't begin to express how deeply satisfying it was to peruse these papers by John Nash. You almost felt you were right there at his side, as he penned them.
There is even something in the book for non-mathematical types: Sylvia Nasar's Introduction and the autobiographical essay (Chapter Two). But for me the greatest interest resided in the remaining chapters: 4-11.
Of these, I particularly enjoyed reading the original presentation of Nash's Thesis on 'Non-Cooperative Games' (Chapter 6), and was fascinated not only with the air-tight logic of his proofs, but the use of hand written-in symbols.
Of course, Chapter 7 is just the re-hashing of Ch. 6, but in proper type-set form, rather than Nash's original script. But - give me the former any day! Reading the original form and format almost made me feel like Nash's Thesis aupervisor, including the same excitement of a new discovery!
Chapter 8 'Two person Cooperative Games' nicely extends the mathematical basis to cover this species of interaction.(And in many ways, people will find the cooperative game model easier to understand than the non-cooperative).
Chapter 9 is important because it delves into the issue of parallel control, and logical functions such as used in high speed digital computers. This chapter was of much interest to me since particular aspects of parallel control figured in my own model of consciousness - recently presented in Chapter Five of my book, 'The Atheist's Handbook to Modern Materialism'. Astute readers who read both books will quickly see the analog between the Schematic of Logical Unit Function (p. 122) and my own Figure 5-13 ('Development of Neural Assemblies', p. 156).
I enjoyed Chapter 10, 'Real Algebraic Manifolds' because of my ongoing interest in Algebraic Topology, and especially homology and homotopy theory. In his chapter, Nash presents a cornucopia of methods for representation, which I am still playing with for different manifolds.
Chapter 11, 'The Imbedding Problem for Riemannian Manifolds', is a delight for anyone familiar with Einstein's General Relativity, or even differential geometry. When you read through this chapter, you also will understand why Nash is still very interested (and involved) in research to do with general relativity and cosmology. Particularly fun for me was his section on 'Smoothing of Tensors' (p. 163) and 'Derivative Size Concept for Tensors' (p. 164).
Chapter 12, 'Continuity of Solutions of Parabolic and Elliptic Equations' is like 'dessert' for anyone who is intensely interested (as I am) in modular functions, which themselves are related intimately to elliptic equations.
In short, I think this book has something for both mathematicians and non-math types alike. Obviously, the former are likely to get more out of it, so the question the latter group must ask is whether the purchase is worth satiating their curiosity about Nash.
I know how I would answer, even if I couldn't tell a derivative from a differential. However, this book can be read on all kinds of levels, and that's the beauty of it.
Rating: 5
Summary: excellent
Comment: Personally, I found this book to be very interestring. The proofs and ideas are presented in clear and non-rigomorphic fashion. One is able to read the works of Nash in the way he himself presented them, and hopefully appropriate some mental strategies used by this genius. There is much that goes on behind the scene of creation of proofs. I think mathematicians of today would greatly benefit from availability of larger number of books which would contain the mathematical works in the way they were originally presented. This is certainly a major step in that direction.
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Title: A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash by Sylvia Nasar ISBN: 0743224574 Publisher: Simon & Schuster Pub. Date: 27 November, 2001 List Price(USD): $16.00 |
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Title: Prisoner's Dilemma/John Von Neumann, Game Theory and the Puzzle of the Bomb by William Poundstone ISBN: 038541580X Publisher: Anchor Pub. Date: 01 February, 1993 List Price(USD): $15.95 |
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Title: Game Theory: A Nontechnical Introduction by Morton D. Davis ISBN: 0486296725 Publisher: Dover Publications Pub. Date: 01 July, 1997 List Price(USD): $9.95 |
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Title: Essays on Game Theory by John F. Nash, John F., Jr Nash ISBN: 1858984262 Publisher: Edward Elgar Publishing Pub. Date: 01 January, 1997 List Price(USD): $50.00 |
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Title:A Brilliant Madness ASIN: B000067S10 Publisher: Warner Home Video Pub. Date: 25 June, 2002 List Price(USD): $19.98 Comparison N/A, buy it from Amazon for $17.98 |
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