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The main peak is at position 1/3, and in the power spectrum this peak contains half of the total. … (c) (Cantor set) In the limit, no single peak contains a nonzero fraction of the power spectrum.
And in practice the n th digit can be found just by computing slightly over n terms of the sum, according to
Round[FractionalPart[ Sum[FractionalPart[PowerMod[2, n - k, k]/k], {k, n}] + Sum[2 n - k /k, {k, n + 1, n + d}]]]
where several values of d can be tried to check that the result does not change.
And often the rather absurd claim is made that all the order we see in the universe must just be a fluctuation—leaving little explanatory power for principles such as the Second Law.
To go further one begins by defining an analog to the Ackermann function of page 906 :
[1][n_] = 2n; [s_][n_] := Nest[ [s - 1], 1, n]
[2][n] is then 2 n , [3] is iterated power, and so on.
They can be thought of as having dimensions 2 - a and smoothed power spectra ω -(1 + 2a) .
Quantitative comparisons of pure power laws implied by the simplest fractals with observations of natural systems have had somewhat mixed success, leading to the introduction of multifractals with more parameters, but Mandelbrot's general idea of the importance of fractals is now well established in both science and mathematics.
As emphasized by Benoit Mandelbrot in the 1970s and 1980s, topography and contour lines (notably coastlines) seem to show apparently random structure on a wide range of scales—with definite power laws being measured in quite a few cases.
MatrixPower[ m[Map[Length, list]], r] . w/Length[w]]
then forming Sum[ ξ [Abs[r]] Cos[2 π r ω ], {r, -n/2, n/2}] and taking the limit n ∞ .
And given t elements operating in parallel one can consider the class NC studied by Nicholas Pippenger in 1978 of computations that can be done in a number of steps that is at most some power of Log[t] .
And by the 1980s natural selection had become firmly enshrined as a force of practically unbounded power, assumed—though without specific evidence—to be capable of solving almost any problem and producing almost any degree of complexity.