Notice Ulam, Feynman, and von Neumann at Los Alamos in http://www-history.mcs.st-and.ac.uk/BigPictures/Ulam_Feynman_von_Neumann.jpeg http://www-history.mcs.st-

Notice Ulam, Feynman, and von Neumann at Los Alamos in http://www-history.mcs.st-and.ac.uk/BigPictures/Ulam_Feynman_von_Neumann.jpeg http://www-history.mcs.st-and.ac.uk/BigPictures/Ulam_Feynman... For the Banach match box problem, that is, flip a fair coin x + y times and find the probability of getting heads exactly x times. So, it's a binomial probability problem. A Banach space is a complete, normed linear space. There is a nice chapter on Banach space in Walter Rudin, Real and Complex Analysis. There are some nice applications of the Hahn-Banach theorem in David G. Luenberger, Optimization by Vector Space Methods. In Patrick Billingsley, Convergence of Probability Measures. is a nice presentation of Ulam's result in measure theory Le Cam called tightness: Roughly, IIRC, for any probability measure P and any a > 0 no matter how small, there exists a sphere S of finite radius so that P(S) > 1 - a. Intuitively the probability mass can't just keep avoiding all spheres; eventually some sphere, if large enough, must cover nearly all the mass. There are some cute technical details. Once in a paper I used Ulam's result to show that a goofy distribution-free statistical hypothesis test was not trivial. And I've seen other applications. The hypothesis test was to improve on our work in artificial intelligence for zero day monitoring for problems in server farms and networks. So, Ulam's tightness has played a role in at least one piece of work intended to be practical! IIRC, Ulam was long head of Los Alamos. Once I heard his lecture on the role in evolution of having two sexes. There was a Time-Life book on math with a few pages on Ulam. IIRC, Ulam did by hand or mechanical calculator some of the early Monte-Carlo evaluations of critical mass. At http://www.brainyquote.com/quotes/quotes/s/stanislawu312043.html http://www.brainyquote.com/quotes/quotes/s/stanislawu312043.... is "It is still an unending source of surprise for me how a few scribbles on a blackboard or on a piece of paper can change the course of human affairs." Stanislaw Ulam