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Page 13
... defined the notion of " transitivity . " Oxtoby and I worked on the completion to the existence of limits in the ergodic theorem itself ... We wanted to show that on every manifold ( a space representing the possible states of a ...
... defined the notion of " transitivity . " Oxtoby and I worked on the completion to the existence of limits in the ergodic theorem itself ... We wanted to show that on every manifold ( a space representing the possible states of a ...
Page 30
... defined by specific equations . After a tortuous period of abstraction , the point - set topologists in this cen- tury arrived at the abstract notion of a topological space , which renders in pre- cise terms our intuitive grasp of the ...
... defined by specific equations . After a tortuous period of abstraction , the point - set topologists in this cen- tury arrived at the abstract notion of a topological space , which renders in pre- cise terms our intuitive grasp of the ...
Page 38
... define a probability measure on strange attractors that could be used to calculate the average properties of complex nonlinear systems . In the article that follows , Paul Stein takes us back a step in history to the computer studies ...
... define a probability measure on strange attractors that could be used to calculate the average properties of complex nonlinear systems . In the article that follows , Paul Stein takes us back a step in history to the computer studies ...
Page 45
... define a measure of biological complexity , or organization , in information- theory terms , and we were immediately at loggerheads . He wanted to insist that very simple instructions could produce very complex patterns and I that such ...
... define a measure of biological complexity , or organization , in information- theory terms , and we were immediately at loggerheads . He wanted to insist that very simple instructions could produce very complex patterns and I that such ...
Page 47
... defined above ) between two randomly selected individuals from the same generation was , on the average , about 4N . It was shown subsequently , by Kahane and Marr , that the average distance is precisely 4 ( N − 1 ) . Intrigued by all ...
... defined above ) between two randomly selected individuals from the same generation was , on the average , about 4N . It was shown subsequently , by Kahane and Marr , that the average distance is precisely 4 ( N − 1 ) . Intrigued by all ...
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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.