The essential John Nash. Edited by Harold W. Kuhn and Sylvia Nasar. (Q2760412)

The book consists of 12 chapters. The first three, as well as the preface (by Harold W. Kuhn) and introduction (by Sylvia Nasar) give some information about J. F. Nash. The remaining nine contain the most important papers by J. F. Nash and chapter 2 is the autobiography by J. F. Nash and chapter 3, written by J. Milnor, is the description of Hex -- the game invented independently by Piet Hein in Denmark and by J. F. Nash in USA.NEWLINENEWLINE Chapter 4 contains the paper ``The Bargaining problem''. The author states some axioms and explains what he means by the solution of this problem and how to obtain it.NEWLINENEWLINE The three subsequent chapters are devoted to the three different proofs of the existence of Nash equilibria in non-cooperative games. The first one ``Equilibrium points in \(n\)-person games'' is an article originally published in Proc. Natl. Acad. Sci. USA 36, 48--49 (1950; Zbl 0036.01104) containing the proof based on Kakutani's fixed point theorem.NEWLINENEWLINE The next two chapters have the same title ``Non-cooperative Games''. The first one is the facsimile of his Ph. D. Thesis and there are introduced the concept of non-cooperative games as well as mathematical methods of analysis of such games. Here J. F. Nash gives the proof of the existence of equilibrium points using only Brouwer's theorem. The second one is the shortened version of the above mentioned thesis, originally published in Ann. Math. (2) 54, 286--295 (1951; Zbl 0045.08202). In the chapter 8 the article ``Two-Person Cooperative Games'' from Econometrica 21, 128--140 (1953; Zbl 0050.14102) is reprinted. Here the author extends his earlier treatment of ``The Bargaining Problem'' to a wider class of situations in which threats can play a role. The threat concept is develoved in this paper. Two methods of the presentation of the game are given here. The first one reduces it to a non-cooperative game. The second one is an axiomatic method. The author states seven axioms which enable him to determine the solution uniquely.NEWLINENEWLINE The next article originally published as John F. Nash ``Parallel Control'' RAND/RM -- 1361 Santa Monica, Cal.: RAND, 8--27--54 was devoted to the problem of computations when several operations can be performed simultaneously.NEWLINENEWLINE Chapter 10 contains the paper ``Real Algebraic Manifolds'' originally published in Ann. Math. (2) 56, 405--421 (1952; Zbl 0048.38501). The author investigates the real algebraic manifold i.e. a closed analytic manifold \({\mathfrak M}\) with a ring \({\mathfrak R}\) of functions on it such that: a) Each function of \({\mathfrak R}\) is a single valued real function analytic at all points of \({\mathfrak M}\). b) There is a basic set of \({\mathfrak M}\) composed of functions in \({\mathfrak R}\). c) If a set of functions in \({\mathfrak R}\) contains more functions than the number of dimensions of \({\mathfrak M}\) then the functions of this set must satisfy some non-trivial polynomial dependence relation. d) Finally \({\mathfrak R}\) must be maximal within the class of rings satisfying the above conditions. According to the definition of the basic set the manifold \({\mathfrak M}\) can be imbedded in some Euclidean space using imbedding functions, that are in its ring \({\mathfrak R}\) of algebraic analytic functions. Such an imbedding is called a representation of the algebraic manifold \((\mathfrak{M,R})\). The main results are the following theorems: 1) A differentiable imbedding \({\mathfrak D}\) in \(E^n\) of a closed differentiable manifold may be approximated by an algebraic representation of the manifold. 2) A closed differentiable manifold always has a proper algebraic representation in the Euclidean space of one more than twice its number of dimensions. There are also given further properties of algebraic manifolds.NEWLINENEWLINE The next chapter is devoted to the paper ``The Imbedding Problem for Riemannian Manifolds'', originally published in Ann. Math. (2) 63, 20--63 (1956; Zbl 0070.38603). It consists of four parts. In the first one -- A) the smoothing operator of the special type is constructed. In part B) a perturbation process is developed and applied to construct a small finite perturbation of an imbedding such that the perturbed imbedding induces a metric that differs by a specified small amount from the metric induced by the original imbedding. In part C) the main result is proved, namely: Every compact Riemannian \(n\)-manifold is realizable as a submanifold of Euclidean \((n/2)(3n+11)\)-space. In part D) this fact is applied to the proof of the theorem: Any Riemannian \(n\)-manifold with \(C^k\) positive metric, where \(3\leq k \leq\infty\), has a \(C^k\) isometric imbedding in \(((3/2)n^3+ 7n^2+(11/2)n)\)-space; in fact, in any small portion of this space.NEWLINENEWLINE The last paper ``Continuity of Solutions of Parabolic and Elliptic Equations'' was originally published in Am. J. Math. 80, 931--954 (1958; Zbl 0096.06902). There Nash considered linear parabolic equations of the form \(\nabla\cdot (C(x,t)\cdot \nabla T)=T_t\), where \(C_{ij}\) form a symmetric real matrix \(C(x,t)\) for each point \(x\) and time \(t\). There is assumed that there are universal bounds \(c_2\geq c_1>0\) on the eigenvalues of \(C\) so that any eigenvalue \(t_n\) satisfies \(c_1\leq t_n\leq c_2\). The continuity estimate for a solution \(T(x,t)\) of the above equation satisfying \(| T|\leq B\) and defined for \(t\geq t_0\) is NEWLINE\[NEWLINE\bigl| T(x_1,t_1) -T(x_2, t_2)\bigr |\leq BA\Bigl\{\bigl[ | x_1-x_2|/(t_1-t_0)^{(1/2)}\bigr ]^\alpha +\bigl[ (t_2-t_1)/(t_1-t_0) \bigr]^{((1 /2) \alpha/(1+a))} \Bigr\},NEWLINE\]NEWLINE where \(t_2\geq t_1>t_0\). Here \(A\) and \(\alpha\) are a priori constants which depend only on \(c_1\) and \(c_2\) and the space dimension \(n\). As a corollary the author obtains a continuity estimate for the solution of elliptic equations. There are also given other results concerning continuity at the boundary in the Dirichlet problem and a Harnack inequality.