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And usually the only way to be sure of this is to have a more or less complete analysis of all possible behavior that the system can exhibit.
less computational sophistication than is implied by the principle, while others will be based on believing that they have more.
Yet while this means that there is in effect less new information in each theorem that is proved, it turns out that in most areas of mathematics these theorems are usually the ones that are considered interesting.
But if one in effect explicitly searches all 8 trillion or so rules that involve less than four colors, it turns out that one can find 4277 three-color rules that work.
If there are n nodes in such a network, then if any blocks are excluded, the shortest one of them must be of length less than n .
The method of computing square roots shown in the main text is less efficient (it computes t digits in t steps), but illustrates more of the mechanisms involved.
But as in iterated maps, the main part of this process that has been analyzed is simply the excavation of progressively less significant digits in the number a[0] .
However, particularly on timescales less than a day, it has in the past decade become clear that, as suggested by Benoit Mandelbrot in the early 1960s, large price fluctuations are significantly more common than a Gaussian distribution would imply.
And in case (c), the fluctuations are determined by the total number of 1's that occur in the digit sequences of all numbers less than n .
And indeed, while there are some traits—such as eye color and blood type in humans—that are more or less discrete, the vast majority of traits seen, say, in the breeding of plants and animals, show quite smooth variation.
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