And in the late 1980s, building on work of mine from 1984 (described on page 276 ), James Crutchfield made a study of such models in which he defined the complexity of a model to be equal to -p Log[p] summed over all connections in the network.

After t steps, the width of the pattern shown here is about Sqrt[Log[2, 3] t] .

After a large number of steps t , the number of distinct positions visited will be proportional to t , at least above 2 dimensions (in 2D, it is proportional to t/Log[t] and in 1D √ t ). Note that the outer boundaries of patterns like those on page 330 formed by n random walks tend to become rougher when t is much larger than Log[n] .

In this pattern, the color of a particular cell can be obtained directly from the digit sequences for t and n by 1 - Sign[BitAnd[-t, n]] or (see page 583 ) With[{d = Ceiling[Log[2, Max[t, n] + 1]]}, If[FreeQ[ IntegerDigits[t, 2, d] - IntegerDigits[n, 2, d], -1], 1, 0]]

Closely related is the total probability for each form of behavior, given for example by Sum[2^-(Ceiling[Log[2, i]]) h[i], {i, ∞ }] .

In the continued fraction for a randomly chosen number, the probability to find a term of size s is Log[2, (1 + 1/s)/(1 + 1/(s + 1))] , so that the probability of getting a 1 is about 41.50%, and the probability of getting a large term falls off like 1/s 2 . … Other less spectacular examples include Exp[ π ]- π and 163/Log[163] . … The pictures below show as a function of n the quantity With[{r = FromContinuedFraction[ContinuedFraction[x, n]]}, -Log[Denominator[r], Abs[x - r]]] which gives a measure of the closeness of successive rational approximations to x .

For a cylinder, there are difficulties with boundary conditions at infinity, but the drag coefficient was nevertheless calculated by William Oseen in 1915 to be 8 π /(R (1/2 + Log[8/R] - EulerGamma)) . … A simple model due to Theodore von Kármán from 1911 predicts a relative spacing of π /Log[1 + √ 2 ] between vortices, and bifurcation theory analyses have provided some justification for some such result.

(Some self-similarity is also present in standard log-periodic antennas.)

And now the repetition period for odd n divides q[n]=2^MultiplicativeOrder[2, n, {1,-1}] - 1 The exponent here always lies between Log[k, n] and (n-1)/2 , with the upper bound being attained only if n is prime.

It is known that Exp[n] and Log[n] for whole numbers n (except 0 and 1 respectively) are transcendental.