r is 0 the rst instance points out similarities

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Unformatted text preview: digits could be expressed more compactly. Evidently not, so far as we can tell.) What about the second sequence? That may not be so familiar, but it is the first 14 digits of the decimal expansion of Feigenbaum’s constant, not as compactly described as the ratio of the circumference to the diameter of a circle, but still shorter than listing the whole infinite sequence. We use a specific feature of this sort of randomness: an infinite random sequence must contain all finite sequences of all lengths. Why is this so? It is simpler to see if we look at binary sequences, but the extension to any base is easy. So, suppose {i1 , i2 , i3 , . . . } is a binary sequence that does not contain the sequence 1, 1. That is, for no j ≥ 1 does ij = ij +1 = 1. This immediately gives rise to a description more compact than listing every element of the sequence: whenever ij = 1, then we do not need to list ij +1 because we know ij +1 must be 0. With this property of randomness, we are able to prove the equivalence of the random...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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