Mathematical properties [of branching model] If an element c of the list b is real, so that there is a stem that goes straight up, then the limiting height of the center of the pattern is obtained by summing a geometric series, and is given by 1/(1 - c) . The overall limiting pattern will be finite so long as Abs[c] < 1 for all elements of b .

Powers of 3/2 The n th value shown in the plot here is Mod[(3/2) n , 1] . … It is then possible to find special values of u (an example is 0.166669170371...) which make the first digit in the fractional part of u (3/2) n always nonzero, so that Mod[u (3/2) n , 1] > 1/6 . In general, it seems that Mod[u (3/2) n , 1] can be kept as large as about 0.30 (e.g. with u = 0.38906669065... ) but no larger.

For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law 2Sqrt[1 - x 2 ] UnitStep[1 - x 2 ]/ π while the distribution of spacings between tends to 1/2( π x)Exp[1/4(- π )x 2 ] The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions. (See also 1/f noise on page 969 .)

If we assume that the density varies slowly with position and time, then we can make series expansions such as f[x + dx, t]  f[x , t] + ∂ x f[x, t] dx + 1/2 ∂ xx f[x, t] dx 2 + … where the coordinates are scaled so that adjacent cells are at positions x - dx , x , x + dx , etc. If we then assume perfect underlying randomness, the density at a particular position must be given in terms of the densities at neighboring positions on the previous step by f[x, t + dt]  p 1 f[x - dx, t] + p 2 f[x, t] + p 3 f[x + dx, t] Density conservation implies that p 1 + p 2 + p 3  1 , while left-right symmetry implies p 1  p 3 . And from this it follows that f[x, t + dt]  c (f[x - dx, t] + f[x + dx, t]) + (1 - 2c)f[x, t] Performing a series expansion then yields f[x, t] + dt ∂ t f[x, t]  f[x, t] + c dx 2 ∂ xx f[x, t] which in turn gives exactly the usual 1D diffusion equation ∂ t f[x, t]  ξ ∂ xx f[x, t] , where ξ is the diffusion coefficient for the system.

Each neuron is assumed to have a value between -1 and 1 corresponding roughly to a firing rate. … For example, with three inputs and one output, w = {{-1, +1, -1}} yields essentially the rule for the rule 178 elementary cellular automaton. … (The VC dimension is n + 1 for such systems.)

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, ∞ }] . … The picture in the main text shows RiemannSiegelZ[t] , defined as Zeta[1/2 +  t] Exp[  RiemannSiegelTheta[t]] , where RiemannSiegelTheta[t_] = Arg[Gamma[1/4 +  t/2]] - t Log[ π ]/2 The first term in an approximation to RiemannSiegelZ[t] is 2 Cos[RiemannSiegelTheta[t]] ; to get results to a given precision requires summing a number of terms that increases like √ t , making routine computation possible up to t ~ 10 10 . It is known that: • The average spacing between zeros decreases like 1/Log[t] . • The amplitude of wiggles grows with t , but more slowly than t 0.16 . • At least the first 10 billion zeros have Re[s]  1/2 .

It is a special case of Hypergeometric2F1 and JacobiP and satisfies a second-order ordinary differential equation in z . The GegenbauerC[n, d/2 - 1, z] form a set of orthogonal functions on a d -dimensional sphere. The GegenbauerC[n, 1/2, z] obtained for d = 3 are LegendreP[n, z] .

The maximum repetition period for rule 90 is 2 (n - 1)/2 - 1 .

The dimensions of the limiting networks are respectively Log[2,3] ≃ 1.58 and Log[3, 7] ≃ 1.77 .

One starts from the substitution system with rules {1  {{3}}, 2  {{13, 1}, {4, 10}}, 3  {{15, 1}, {4, 12}}, 4  {{14, 1}, {2, 9}}, 5  {{13, 1}, {4, 12}}, 6  {{13, 1}, {8, 9}}, 7  {{15, 1}, {4, 10}}, 8  {{14, 1}, {6, 10}}, 9  {{14}, {2}}, 10  {{16}, {7}}, 11  {{13}, {8}}, 12  {{16}, {3}}, 13  {{5, 11}}, 14  {{2, 9}}, 15  {{3, 11}}, 16  {{6, 10}}} This yields the nested pattern below which contains only 51 of the 65,536 possible 2 × 2 blocks of cells with 16 colors.