.}  Length[k]]]] There are a total of 2 m Fibonacci[m+2] black cells in the pattern obtained up to step 2 m , implying fractal dimension Log[2, 1 + Sqrt[5]] .

The smallest such t is given by MultiplicativeOrder[k, n] , which always divides EulerPhi[n] (see page 1093 ), and has a value between Log[k, n] and n - 1 , with the upper limit being attained only if n is prime.

In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.) … (Typically one needs to generalize formulas that are initially set up with integer numbers of terms; examples include taking Power[x, y] to be Exp[Log[x] y] and x!

[Examples of] reducible systems The color of a cell at step t and position x can be found by starting with initial condition Flatten[With[{w = Max[Ceiling[Log[2, {t, x}]]]}, {2 Reverse[IntegerDigits[t, 2, w]] + 1, 5, 2 IntegerDigits[x, 2, w] + 2}]] then for rule 188 running the cellular automaton with rule {{a : (1 | 3), 1 | 3, _}  a, {_, 2 | 4, a : (2 | 4)}  a, {3, 5 | 10, 2}  6, {1, 5 | 7, 4}  0, {3, 5, 4}  7, {1, 6, 2}  10, {1, 6 | 11, 4}  8, {3, 6 | 8 | 10 | 11, 4}  9, {3, 7 | 9, 2}  11, {1, 8 | 11, 2}  9, {3, 11, 2}  8, {1, 9 | 10, 4}  11, {_, a_ /; a > 4, _}  a, {_, _, _}  0} and for rule 60 running the cellular automaton with rule {{a : (1 | 3), 1 | 3, _}  a, {_, 2 | 4, a : (2 | 4)}  a, {1, 5, 4}  0, {_, 5, _}  5, {_, _, _}  0}

And in general if h is associative the result Nest[h[r, #]&, s, t] of t steps of evolution can be rewritten for example using the repeated squaring method as h[Fold[If[#2  0, h[#1, #1], h[r, h[#1, #1]]] &, r, Rest[IntegerDigits[t, 2]]], s] which requires only about Log[t] rather than t applications of h . … For any associative function h the repeated squaring method allows the result of t steps of evolution to be computed with only about Log[t] applications of h . … But when m ≃ n FFT-related methods allow this to be reduced to about n Log[n] operations.

For when one enters a command such as Log[15] , what actually happens is that the program which implements the Mathematica language interprets this command

Using this procedure one can certainly compute the color of any cell on row n by doing about n Log[n] 3 operations—instead of the n 2 needed if one carried out the cellular automaton evolution explicitly.

Symbolic expressions Expressions like Log[x] and f[x] that give values of functions are familiar from mathematics and from typical computer languages.

If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution Exp[-(Log[x] - μ ) 2 /(2 σ 2 )]/(Sqrt[2 π ] x σ ) For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution Exp[(x - μ )/ β ] Exp[-Exp[(x - μ )/ β ]]/ β related to the Weibull distribution used in reliability analysis.

Formally, there is evidence that the Navier–Stokes equations in 2D might have a ∇ 2 Log[∇ 2 ] viscosity term, rather than a ∇ 2 one.