Non-power bases One can consider representing numbers by Sum[a[n] f[n], {n, 0, ∞ }] where the f[n] need not be k n .

Distribution of chaotic behavior For iterated maps, unlike for discrete systems such as cellular automata, one can get continuous ranges of rules by varying parameters.

Symmetry [of discrete space] A system like a cellular automaton that consists of a large number of identical cells must in effect be arranged like a crystal, and therefore must exhibit one of the limited number of possible crystal symmetries in any particular dimension, as discussed on page 929 .

One approach to factoring a number n is just to try dividing it by each of the numbers up to √ n . A sequence of much faster methods have however been developed over the past few decades, one simple example that works for most n being the so-called rho method of John Pollard (compare the quadratic residue sequences discussed below): Module[{f = Mod[# 2 + 1, n] &, a = 2, b = 5, c}, While[(c = GCD[n, a - b])  1, {a, b} = {f[a], f[f[b]]}]; c] Most existing methods depend on facts in number theory that are fairly easy to state, though implementing them for maximum efficiency tends to lead to complex programs.

In general any given function will allow many DNF representations; minimal ones can be found as described below. … As in the note on multilevel formulas below, one can also in effect introduce intermediate variables to get recursive formulas of larger depth, somewhat analogous to results from Collect .

Gaming regulations typically require dice to be perfect cubes to within one part in a few thousand; casinos normally retire dice after a few hundred rolls. … One notable consequence were traces of insertion order among the 366 capsules used in the 1970 draft lottery in the U.S.

Typical ones include: • Edge connectivity: the minimum number of connections that must be removed to make the network disconnected. • Diameter: the maximum distance between any two nodes in the network. The pictures below show the largest planar trivalent networks with diameters 1, 2 and 3, and the largest known ones with diameters 4, 5 and 6

If one steps through frequencies with an AM radio scanner, one sometimes hears intelligible speech—from radio or TV broadcasts, or two-way radio communication. But in many frequency bands one hears instead either very regular or seemingly quite random signals. … In general to send many signals together one just needs to associate each with a function f[i, t] orthogonal to all other functions f[j, t] (see page 1072 ).

= {} Given a set of sequences of values represented by a particular network, the set obtained after one step of cellular automaton evolution is given by NetCAStep[{k_, r_, rtab_}, net_] := Flatten[ Map[Table[# /. … This can be done by first creating a "deterministic" network in which at most one arc of each value comes out of each node, then combining equivalent nodes. … To obtain such trimmed networks one can apply the function TrimNet[net_] := With[{m = Apply[Intersection, Map[FixedPoint[ Union[#, Flatten[Map[Last, net 〚 # 〛 , {2}]]]&, #]&, Map[List, Range[Length[net]]]]]}, net 〚 m 〛 /.

Starting in the 1950s, electronic devices were the primary ones used for cryptography. Linear feedback shift registers and perhaps nonlinear ones seem to have been common, though little is publicly known about military cryptographic systems after World War II. … My cellular automaton cryptographic system is one of the very few fundamentally different systems to have been introduced in recent years.