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By Cherny A.S.

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Now let us assume that the cells evolve according to the following rule. In the next step a cell will be black if it or either of its two closest neighbors were black in the previous step. According to this rule in the second step the system grows as follows, 47 August 4, 2008 48 10:37 9in x 6in B-634 ch07 Randomnicity Fig. 13. Cellular automaton — 2nd step. in the third step it grows as follows, Fig. 14. Cellular automaton — 3rd step. and so on. Apparently, we get a simple growing pattern uniformly filled with black.

But this is not fair. ” “It may seem unfair,” continues Aristotle, “but the alternative is even worse. ” asks Alexander with interest. “Let us assume that there is no chance and that everything is predetermined. Let us further assume that your predetermined destiny is to conquer the world. Since everything is cast in stone, there will be no change in your destiny, no matter what your actions will be. ” “It makes sense. . ” “Right,” replies Alexander knowing that his teacher is setting up a trap.

0 1 −1 1/2 −1/2 2 −2 3 −3 etc. Therefore, all rational numbers (positive, negative and zero) have a seat. The set containing the seat number is the set of the counting numbers. It follows that the set of rational numbers is also countable. August 4, 2008 10:37 9in x 6in B-634 Fuller than Full ch04 31 So, fine! The whole numbers, the even numbers, the odd numbers, the integers and the rational numbers are all sets of the same size and they are countable. But, how about the rest of the numbers in the number line?

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Stochastic Differential Equations by Cherny A.S.


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