History [of substitution systems] In their various representations, 1D substitution systems have been invented independently many times for many different purposes. … So-called 0L systems correspond to my neighbor-independent substitution systems; 1L systems correspond to the neighbor-dependent substitution systems on page 85 . … Paths representing sequences from 1D substitution systems can be generated by 2D geometrical substitution systems, as on page 189 .

Junctional calculus Expressions are equivalent when Union[Level[expr, {-1}]] is the same, and this canonical form can be obtained from the axiom system of page 803 by flattening using (a ∘ b) ∘ c  a ∘ (b ∘ c) , sorting using a ∘ b  b ∘ a , and removing repeats using (a ∘ a)  a .

The argument for this is similar to the one on pages 941 and 954 for 1D cellular automata.

In general, if a period m is possible then so must all periods n for which p = {m, n} satisfies OrderedQ[Transpose[If[MemberQ[p/#, 1], Map[Reverse, {p/#, #}], {#, p/#}]] &[2^IntegerExponent[p, 2]]] Extensions of this to other types of systems seem difficult to find, but it is conceivable that when viewed as continuous mappings on a Cantor set (see page 869 ) at least some cellular automata might exhibit similar properties.

Implementation [of conserved quantity test] Whether a k -color cellular automaton with range r conserves total cell value can be determined from Catch[Do[ (If[Apply[Plus, CAStep[rule, #] - #] ≠ 0, Throw[False]] &)[ IntegerDigits[i, k, m]], {m, w}, {i, 0, k m - 1}]; True] where w can be taken to be k 2r , and perhaps smaller.

Known methods for high-precision evaluation of special functions—usually based in the end on series representations—typically require of order n 1/s m[n] operations, where s is often 2 or 3. (Examples of more difficult cases include HypergeometricPFQ[a, b, 1] and StieltjesGamma[k] , where logarithmic series can require an exponential number of terms. … The best-known algorithms for evaluating Zeta[1/2 +  x] (see page 918 ) to fixed precision take roughly √ x operations—or 2 n/2 operations if x is an n -digit integer.

(The case shown corresponds to iteration of the map z  z - (z 3 - 1)/(3z 2 ) corresponding to Newton's method for finding the complex roots of z 3  1 .)

Among k = 2 , r = 1 elementary cellular automata it turns out that this happens precisely for those 30 rules that are additive with respect to at least the first or last position on which they depend (see pages 601 and 1087 ); this includes both rules 90 and 150 and rules 30 and 45. … (Thus for example additive rules such as 90 and 150, as well as one-sided additive rules such as 30 and 45 are always 4-to-1.)

In 1923 Louis de Broglie had suggested that just as light—which in optics was traditionally described in terms of waves—seemed in some respects to act like discrete particles, so conversely particles like electrons might in some respects act like waves. … To find predictions from this theory a so-called perturbation expansion was made, with successive terms representing progressively more interactions, and each having a higher power of the so-called coupling constant α ≃ 1/137 .

The pictures below show what happens if the programs operate by applying elementary cellular automaton rules t times to 2t + 1 inputs.