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Page 13
... limits in the ergodic theorem itself ... We wanted to show that on every manifold ( a space representing the possible states of a dynamical system ) —the kind used in statistical mechanics - such ergodic be- havior is the rule ... It ...
... limits in the ergodic theorem itself ... We wanted to show that on every manifold ( a space representing the possible states of a dynamical system ) —the kind used in statistical mechanics - such ergodic be- havior is the rule ... It ...
Page 30
... limit theorem , and he showed me how to derive the same result by an elegant argument with ordinary square roots . Stan did his best work in fields where no one dared to tread , where he would be sure of having the first shot , free ...
... limit theorem , and he showed me how to derive the same result by an elegant argument with ordinary square roots . Stan did his best work in fields where no one dared to tread , where he would be sure of having the first shot , free ...
Page 45
... limit the variety of such patterns . Each of us was defending a different meaning of " complex . " I already knew of his work ( or play ) with computer- generated growth patterns ( Ulam 1962 ) but hadn't realized fully the range of ...
... limit the variety of such patterns . Each of us was defending a different meaning of " complex . " I already knew of his work ( or play ) with computer- generated growth patterns ( Ulam 1962 ) but hadn't realized fully the range of ...
Page 53
... limit- ing laws governing the long - term dynam- ics of nonlinear systems that are analo- gous to the major limiting theorems in classical probability theory . The attempt to construct appropriate probability mea- sures for such systems ...
... limit- ing laws governing the long - term dynam- ics of nonlinear systems that are analo- gous to the major limiting theorems in classical probability theory . The attempt to construct appropriate probability mea- sures for such systems ...
Page 57
... limit as the number of faces becomes infinitely large . Despite this awkwardness the general result is quite simple : If the probability of a head in one toss is p , 0 ≤ p ≤ 1 , and the probability of a tail is 1 Pq , then the ...
... limit as the number of faces becomes infinitely large . Despite this awkwardness the general result is quite simple : If the probability of a head in one toss is p , 0 ≤ p ≤ 1 , and the probability of a tail is 1 Pq , then the ...
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Popular passages
Page 299 - Talibus orabat dictis, arasque tenebat, cum sic orsa loqui vates : ' Sate sanguine divom, 125 Tros Anchisiade, facilis descensus Averno ; noctes atque dies patet atri janua Ditis ; sed revocare gradum superasque evadere ad auras, hoc opus, hie labor est.
Page 231 - But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.
Page 210 - The low-velocity equation of state for a lattice gas can be written as p - ^p (l - |v2), where p is the pressure, p is the density, and v is the flow speed.
Page 316 - Southeastern Conference on Combinatorics. Graph Theory, and Computing (Florida Atlantic University, Boca Raton, Florida, April 2-6, 1979), 3-18.
Page 121 - He is a member of the National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences, the American Physical Society and the American Association for the Advancement of Science.
Page 231 - A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws...
Page 224 - Instead, we conclude by remarking that really efficient high-speed computing devices may, in the field of non-linear partial differential equations as well as in many other fields, which are now difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress.
Page 272 - Let us say here that the results of our computations show features which were, from the beginning, surprising to us. Instead of a gradual, continuous flow of energy from the first mode to the higher modes, all of the problems show an entirely different behavior.
Page 58 - That was dramatized by Bertrand, who considered the problem of finding the probability that a chord of a circle chosen "at random" be longer than the side of an equilateral triangle inscribed in the circle.