Usually the representations that are used for r and s will be quite different, and the function h will have no special properties. … This same basic scheme can be used with any associative function h — Max , GCD , And , Dot , Join or whatever—so long as suitable forms for r and s are used. … 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 .

This result follows directly from the generating function formula (1 - 2 x z + x 2 ) -m  Sum[GegenbauerC[n, m, z] x n , {n, 0, ∞ }]

Specifying an operator f (taken in general to have n arguments with k possible values) by giving the rule number u for f[p, q, …] , the rule number for an expression with variables vars can be obtained from With[{m = Length[vars]}, FromDigits[ Block[{f = Reverse[IntegerDigits[u, k, k n ]] 〚 FromDigits[ {##}, k] + 1 〛 &}, Apply[Function[Evaluate[vars], expr], Reverse[Array[IntegerDigits[# - 1, k, m] &, k m ]], {1}]], k]]

The picture below shows the repetition periods as a function of the numerical size of the quantity m/n .

Sarkovskii's theorem For any iterated map based on a continuous function such as a polynomial it was shown in 1962 that if an initial condition exists that gives period 3, then other initial conditions must exist that give any other period.

DNF minimization From a table of values for a Boolean function one can immediately get a DNF representation just by listing cases where the value is 1. … The problem of minimization is then to find the minimal set of hyperplanes that will cover the corners for a particular Boolean function. … Other procedures work slightly more efficiently, but in general the problem of finding the minimal DNF for a Boolean function of n variables is NP-complete (see page 768 ) and is thus expected to grow in difficulty faster than any polynomial in n .

Cellular automaton rules as formulas The value a[t, i] for a cell on step t at position i in any of the cellular automata in this chapter can be obtained from the definition a[t_, i_] := f[a[t - 1, i - 1], a[t - 1, i], a[t - 1, i + 1]] Different rules correspond to different choices of the function f . … The definition of the function f for rule 90 that we gave above is essentially just a look-up table. But it is also possible to define this function in an algebraic way f[p_, q_, r_] := Mod[p + r, 2] Algebraic definitions can also be given for other rules: • Rule 254 (page 24 ): 1 - (1 - p)(1 - q)(1 - r) • Rule 250 (page 25 ): p + r - p r • Rule 30 (page 27 ): Mod[p + q + r + q r, 2] • Rule 110 (page 32 ): Mod[(1 + p) q r + q + r, 2] In these definitions, we represent the values of cells by the numbers 1 or 0.

Ulam systems Having formulated the system around 1960, Stanislaw Ulam and collaborators (see page 877 ) in 1967 simulated 120 steps of the process shown below, with black cells after t steps occurring at positions Map[First, First[Nest[UStep[p[q[r[#1], #2]] &, {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}, #] &, ({#, #} &)[{{{0, 0}, {0, 0}}}], t]]] UStep[f_, os_, {a_, b_}] := {Join[a, #], #} &[f[Flatten[ Outer[{#1 + #2, #1} &, Map[First, b], os, 1], 1], a]] r[c_]:= Map[First, Select[Split[Sort[c], First[#1]  First[#2] &], Length[#]  1 &]] q[c_, a_] := Select[c, Apply[And, Map[Function[u, qq[#1, u, a]], a]] &] p[c_]:= Select[c, Apply[And, Map[Function[u, pp[#1, u]], c]] &] pp[{x_, u_}, {y_, v_}] := Max[Abs[x - y]] > 1 || u  v qq[{x_, u_}, {y_, v_}, a_] := x  y || Max[Abs[x - y]] > 1 || u  y || First[Cases[a, {u, z_}  z]]  y These rules are fairly complicated, and involve more history than ordinary cellular automata.

Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

And it then turns out that in zero spacetime dimensions the complete path integral for the theory can be evaluated exactly—yielding in effect a generating function for the number of possible networks. Parametric differentiation (to yield n -point correlation functions) then gives results for n -sided regions.