Operator representations Instead of repeatedly applying an operation to a sequence of digits one can consider forming integers (or other numbers) by performing trees of operations on a single constant.

If only one color of element ever appears this is the complete condition for a solution—and for r = 2 solutions exist if Apply[Times, d] < 0 and are then of length at least Apply[Plus[##]/GCD[##]&, Abs[d]] . … The undecidability of PCP can be seen to follow from the undecidability of the halting problem through the fact that the question of whether a tag system of the kind on page 93 with initial sequence s ever reaches a halting state (where none of its rules apply) is equivalent to the question of whether there is a way to satisfy the PCP constraint TSToPCP[{n_, rule_}, s_] := Map[Flatten[IntegerDigits[#, 2, 2]] &, Module[{f}, f[u_] := Flatten[Map[{1, #} &, 3u]]; Join[Map[{f[Last[#]], RotateLeft[f[First[#]]]} &, rule], {{f[s], {1}}}, Flatten[ Table[{{1, 2}, Append[RotateLeft[f[IntegerDigits[j, 2, i]]], 2]}, {i, 0, n - 1}, {j, 0, 2 i - 1}], 1]]], {2}] Any PCP constraint can also immediately be related to the evolution of a multiway tag system of the kind discussed in the note below. Assuming that the upper string is never shorter than the lower one, the rules for the relevant tag system are given simply by Apply[Append[#2, s___]  Prepend[#1, s] &, p, {1}] In the case of example (e) the existence of a solution of length 24 can then be seen to follow from the fact that MWTSEvolve[rule, {{"B"}}, 22] contains {"B","A"} .

But the table below gives for example the actual algebraic formulas obtained in the case a = 4 after applying FullSimplify —and shows that these increase quite rapidly in complexity. … The series has an accumulation of poles on the circle Abs[a] 2  1 ; the coefficient of x m turns out to have denominator 2^(m - DigitCount[m, 2, 1]) Apply[Times, Table[Cyclotomic[s, a]^Floor[(m - 1)/s], {s, m - 1}]] For other iterated maps general formulas also seem rare.

Applying BitReverseOrder to this matrix yields a matrix which has an essentially nested form, and for size n = 2 s can be obtained from Nest[With[{c = BitReverseOrder[Range[0, Length[#] - 1]/ Length[#]]}, Flatten2D[MapIndexed[#1 {{1, 1}, {1, -1} (-1)^c 〚 Last[#2] 〛 } &, #, {2}]]] &, {{1}}, s] Using this structure, one obtains the so-called fast Fourier transform which operates in n Log[n] steps and is given by With[{n = Length[data]}, Fold[Flatten[Map[With[ {k = Length[#]/2}, {{1, 1}, {1, -1}} .

During the evolution the rule can apply only to the inner part FixedPoint[Replace[#, ℯ [x_]  x] &, expr] of an expression.

With more than two piles it was discovered in 1901 that one player can in general force the other to lose by arranging that after each of their moves Apply[BitXor, h]  0 , where h is the list of heights.

Inverse [cellular automaton] rules Some reversible rules are self-inverse, so that applying the same rule twice yields the identity.

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 〛 /.

Perfect numbers Perfect numbers with the property that Apply[Plus, Divisors[n]]  2n have been studied since at least the time of Pythagoras around 500 BC.

A single step in evolution of a general cellular automaton with state a and rule number num is then given by Map[IntegerDigits[num, k, k^Length[os]] 〚 -1 - # 〛 &, Apply[Plus, MapIndexed[k^(Length[os] - First[#2]) RotateLeft[a, #1] &, os]], {-1}] or equivalently by Map[IntegerDigits[num, k, k^Length[os]] 〚 -# - 1 〛 &, ListCorrelate[Fold[ReplacePart[k #1, 1, #2 + r + 1] &, Array[0 &, Table[2r + 1, {d}]], os], a, r + 1], {d}]