The argument is based on showing that an algebraic function always exists for which the coefficients in its power series correspond to any given nested sequence when reduced modulo some p . … But then there is a general result that if a particular sequence of power series coefficients can be obtained from an algebraic (but not rational) function modulo a particular p , then it can only be obtained from transcendental functions modulo any other p —or over the integers.

As emphasized by Benoit Mandelbrot in connection with a variety of systems in nature, the same is also true for random walks whose step lengths follow a power-law distribution, but are unbounded.

Note that if n is not between 1 and 4, it must be multiplied or divided by an appropriate power of 4 before starting this procedure.

This quantity can be computed using power tree methods (see below ), though as discussed on page 609 , even more efficient methods are also available.

MatrixPower[m, t] , where init gives the initial number of elements of each color— {1, 0} for case (c) above. … For neighbor-independent rules, the growth for large t must follow an exponential or an integer power less than the number of possible colors.

The key idea is to represent data of any kind by a symbolic expression of the general form head[arg 1 , arg 2 , …] . ( a + b 2 is thus Plus[a, Power[b, 2]] , {a, b, c} is List[a, b, c] and a = b + 1 is Set[a, Plus[b, 1]] .)

Cycles and zeta functions The number of sequences of n cells that can occur repeatedly, corresponding to cycles in the network, is given in terms of the adjacency matrix m by Tr[MatrixPower[m,n]] .

With multiplier m row t corresponds to the power m t .

But given t steps in this sequence as a list of 0's and 1's, the following function will reconstruct the rightmost t digits in the starting value of n : IntegerDigits[First[Fold[{Mod[If[OddQ[#2], 2 First[#1] - 1, 2 First[#1] PowerMod[5, -1, Last[#1]]], Last[#1]], 2 Last[#1]} &, {0, 2}, Reverse[list]]], 2, Length[list]]

Intensity or power spectra are obtained by squaring the quantities shown.