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Since the late 1970s, it has been common to assume that the response of a cell can be modelled by derivatives of Gaussians such as those shown below, or perhaps by Gabor functions given by products of trigonometric functions and Gaussians.
Zeta function For real s the Riemann zeta function Zeta[s] is given by Sum[1/n s , {n, ∞ }] or Product[1/(1 - Prime[n] s ), {n, ∞ }] .
But the fact that the form of flow should depend only on Reynolds number means that in the pictures in the main text for example it is not necessary to specify absolute sizes or speeds: one need only know the product U L that appears in the Reynolds number.
The overall fraction of a length n input that consists of repeats of length at least b is greater than 1 - 2 b /n and is essentially 1 - Sum[(1 - 2 -b ) i Product[1 + (1 - 2 -b ) j - (1 - 2 -b - 1 ) j , {j, i - b + 1, i - 1}], {i, b, n - b}]/(n - 2b + 1)
Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2, Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3]) Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2 Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}] (As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .)
It exhibits a nested structure, and can be obtained as in the pictures below from the evolution of a 2D substitution system, or equivalently from a Kronecker product as in Nest[Flatten2D[Map[# {{1, 1}, {1, -1}} &, #, {2}]] &, {{1}}, s] with Flatten2D[a_] := Apply[Join, Apply[Join, Map[Transpose,a], {2}]] (c) is known as dyadic or Paley order.
The result is a product of: rate of formation of suitable stars; fraction with planetary systems; number of Earth-like planets per system; fraction where life develops; fraction where intelligence develops; fraction where technology develops; time communicating civilizations survive.
As discussed in the note below, this can be viewed as a consequence of the fact that the probability distribution in a random walk depends only on Sum[Outer[Times, e 〚 s 〛 , e 〚 s 〛 ], {s, Length[e]}] and not on products of more of the e 〚 s 〛 .
The first 2 m elements in the sequence can be obtained from (see page 1081 ) (CoefficientList[Product[1 - z 2 s , {s, 0, m - 1}], z] + 1)/2 The first n elements can also be obtained from (see page 1092 ) Mod[CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/ (2(1 + x)), {x, 0, n - 1}], x], 2] The sequence occurs many times in this book; it can for example be derived from a column of values in the rule 150 cellular automaton pattern discussed on page 885 .
But a crucial point noted by Carl-Friedrich Gauss in the 1820s is that the product of such curvatures—the so-called Gaussian curvature—is always independent of how the surface is laid out, and can thus be viewed as intrinsic to the surface itself, and for example determined purely from the metric for the 2D space corresponding to the surface.
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