IntegerDigits[m, 2, 8]] For rule 22, for example, this means that if the density at a particular step is p , then the density on the next step should be 3 p (1 - p) 2 , and the densities on subsequent steps should be obtained by iterating this function. … For two steps, one must consider probabilities for all 32 combinations of 5 cells, and for rule 22 the function becomes p (1 - p) 2 (2 + 3p 2 ) , yielding density 0.35012; for three steps it is p (1 - p) 2 (p 4 - 18 p 3 + 41 p 2 - 22 p + 6) yielding density 0.379. … (For rules 90 and 30 the functions obtained after one step are respectively 2 p (1 - p) and p (2 p 2 - 5 p + 3) , both of which turn out to imply correct final densities of 1/2 ).

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 ). Current electronics (with analog elements such as phase-locked loops) make it easy to handle functions Sin[ ω t] , but other functions can yield better data density and perhaps better signal propagation.

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]]

Other smooth functions typically nevertheless yield identical results.

Comments on Mathematica functions CenterList works by first creating a list of n 0's, then replacing the middle 0 by a 1. … Many other evolution functions in these notes use the same mechanism.

A Flat function has the mathematical property of being associative.) … And by using rules such as s[x___, 1, 0, y___]  {s[x, 0, 1, 0, y], Length[s[x]]} one can keep track of the positions at which substitutions are made. ( StringReplace replaces all occurrences of a given substring, not just the first one, so cannot be used directly as an alternative to having a flat function.)

Somewhat related to the curves shown here is the function MoebiusMu[n] , equal to 0 if n has a repeated prime factor and otherwise (-1)^Length[FactorInteger[n]] .

Defining v[u] = -Integrate[f[u], u] the field then has Lagrangian density (( ∂ t u) 2 - ( ∂ x u) 2 )/2 - v[u] and conserves the Hamiltonian (energy function) Integrate[(( ∂ t u) 2 + ( ∂ t u) 2 )/2 + v[u], {x, - ∞ , ∞ }] With the choice for f[u] made here (with a ≥ 0 ), v[u] is bounded from below, and as a result it follows that no singularities ever occur in u[t, x] .

Surprisingly enough, this simple procedure, which can be represented by the function s[list_] := Flatten[ Transpose[Reverse[Partition[list, Length[list]/2]]]] with or without the Reverse , is able to produce orderings which at least in some respects seem quite random.

In traditional mathematics it is normally assumed that once one has an explicit formula involving standard mathematical functions then one can in effect always evaluate this formula immediately.