List price: $14.95 (that's 30% off!)
Used price: $6.90
List price: $19.95 (that's 30% off!)
Used price: $10.87
Buy one from zShops for: $13.92
Buy one from zShops for: $10.00
List price: $31.95 (that's 30% off!)
Used price: $19.90
Buy one from zShops for: $21.23
Like so many excellent mathematicians, Nash also did some work in recreational mathematics, and he independently invented the game now known as Hex. Played on a board of hexagonal sections, the object is to create a continuous chain of your color from one side to the other. A short chapter explains the basis of the game, although it does not do justice to the complexity .
Nash's work in game theory is outstanding, and the reason why he won the Nobel prize in economics. The bulk of the book is a recreation of his seminal work in this area, with his Ph. D. thesis being presented twice. The first is a photocopy of the work and the second is the thesis in text form. In reading the material, it is easy to see why it has applications in so many areas.
Nash was also interested in computing and he wrote an imaginative paper on parallel computing, which is included in the book. Given the state of computing at the time it was written, it shows imagination and fundamental understanding of the basics of computing.
The last two papers in the book deal with manifolds. The first concerns real algebraic manifolds and the second examines abstract Riemannian manifolds. Once again, you can see aspects of genius in the papers and avenues for further exploration.
It is a mathematical tragedy that John Nash was almost totally unable to work for so many years. In fact, when it was announced that he had won the Nobel prize, many were surprised to hear that he was still alive. In reading these papers from the early years of his career, it is clear to see that had he not became ill, he would have had a shot at being labeled the best mathematician of the century. Long after memories of the movie have faded away, Nash's work will still be applied to the problems of the world.
Published in Journal of Recreational Mathematics, reprinted with permission.
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.
Where Nash is weak is in her descriptions of mathematical formulae. She does not appear to have any real understanding of the mathematics and I would have thought a plain English explanation of his work would have strengthened the biography. I got a little frustrated that she did not tackle this task. Yet it is perhaps a measure of Nash's genius that the ideas are so complex they cannot be easily reduced to a paragraph. Still she could have tried harded in this area. Nasar tends to get around this problem, by getting another expert to describe the brilliance of the idea, rather than the mathemtical idea itself.
Based on my own experiences with people with schizophrenia, Nash's recovery is remarkable and this is the section is probably the most interesting, perhaps because it is so startling. Even after reading the biogrpahy, I still find it hard to believe that someone could recover given the severity of the illness, so it gives some hope to people who suffer this disability and those close to them.
An absorbing biography and close to a great one.
Sylvia Nasar deserves credit for her extensive and thorough account of Nash's life. She also must be highly commended for producing a biography of a mathematican/economist that could be read and enjoyed by just about anyone- but I'm sure she will be the first to admit that Nash's twisted mind made this possible.
A Beautiful Mind is good on so many levels. It provides wonderful insight into the whole process of becoming a research faculty. It is also a great informal history of 20th century mathematical research. Although there is a some discussion of mathematical theory in the book, it is written for the general reader and should not be problematic for anyone who has an interest in math.
On top of that it is a great biography of a person with a difficult personality and it is a sensitive treatment of schizophrenia. All in all a delightful read if you don't get easily depressed by the tragic illness that changed this man's life.
This paperback edition published in 2001 contains an Epilogue that provides an update on events since the original 1998 edition appeared. As such it is preferable because of the additional information it contains.
Used price: $14.00
Collectible price: $21.94
Used price: $16.95
Collectible price: $17.77
Used price: $9.94
Collectible price: $12.71
are still offerring it for sale even after you
sent me my money back????????