Fourier[data] can be thought of as multiplication by the n × n matrix Array[Exp[2 π  #1 #2/n] &, {n, n}, 0] . 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}} . {Take[#, k], Drop[ #, k](-1)^(Range[0, k - 1]/k)}] &, Partition[##]]] &, BitReverseOrder[data], 2^Range[Log[2, n]]]/ √ n ] (See also page 1080 .)

For a square lattice, it still nevertheless always holds up to n=2 (so that the analogs of moments of inertia satisfy Ι xx  Ι yy , Ι xy  Ι yx  0 ). … For isotropy, only the n = 0 moment can be nonzero. … In continuous systems such as partial differential equations, isotropy requires that coordinates in effect appear only in ∇.

{dr[r_, n_]  d[r, n + First[#2]], dm[r_, z_]  d[r, z /. adrs]})&, Flatten[segs]]] TMCompile[_  z:{_, _, 1}] := f[z, {1, 2}] TMCompile[_  z:{_, _, -1}] := f[z, {2, 1}] f[{s_, a_, _}, {ra_, rb_}] := Flatten[{i[3], dr[ra, -1], dr[3, 3], i[ra], i[ra], dr[3, -2], If[a  1, i[ra], {}], i[3], dr[rb, 5], i[rb], dr[3, -1], dr[rb, 1], dm[rb, {s, 0}], dr[rb, -6], i[rb], dr[3, -1], dr[rb, 1], dm[rb, {s, 1}]}] A blank initial tape for the Turing machine corresponds to initial conditions {1, {0, 0, 0}} for the register machine. (Assuming that the Turing machine starts in state 1, with a 0 under its head, other initial conditions can be encoded just by taking the values of cells on the left and right to give the digits of the numbers that are initially in the first two registers.) Given the list of successive configurations of the register machine, the steps that correspond to successive steps of Turing machine evolution can be obtained from (Flatten[Partition[Complement[#, #-1], 1, 2]]&)[ Position[list, {_,{_,_,0}}]] The program given above works for Turing machines with any number of states, but it requires some simple extensions to handle more than two possible colors for each cell.

[Turing] machine 596440 For any list of initial colors init , it turns out that successive rows in the first t steps of the compressed evolution pattern turn out to be given by NestList[Join[{0}, Mod[1 + Rest[FoldList[Plus, 0, #]], 2], {{0}, {1, 1, 0}} 〚 Mod[Apply[Plus, #], 2] + 1] 〛 &, init, t] Inside the right-hand part of this pattern the cell values can then be obtained from an upside-down version of the rule 60 additive cellular automaton, and starting from a sequence of 1 's the picture below shows that a typical rule 60 nested pattern can be produced, at least in a limited region.

Particularly dramatic are the concatenation systems discussed on page 913 , as well as successive rows in nested patterns such as Flatten[IntegerDigits[NestList[BitXor[#, 2 #] &, 1, 500], 2]] and sequences based on numbers such as Flatten[Table[If[GCD[i, j]  0, 1, 0], {i, 1000}, {j, i}]] (see page 613 ).

Many of the maps I will consider can be expressed in terms of standard mathematical functions, but in general all that is needed is that the map take any possible number between 0 and 1 and yield some definite number that is also between 0 and 1. … On the first of the next two pages all the examples start with the number 1/2 —which has a simple digit sequence. … For on the first page it just yields 0's.

Each of these functions has definite finite values only in a limited region of the complex plane, and on the boundary of this region they exhibit singularities at every single rational point. … Nested behavior is also found for example in EllipticTheta[3, 0, z] , which is given essentially by Sum[z n 2 , {n, ∞ }] .

Generalized aggregation models One can in general have rules in which new cells can be added only at positions whose neighborhoods match specific templates (compare page 213 ). … (The rule numbers here follow the scheme on page 927 with offsets {{-1, 0}, {0, -1}, {0, 1}, {1, 0}} ). … An extreme case is rule 2, where only a single neighborhood with a single black cell is allowed, so that growth occurs along a single line.

String overlaps The total numbers of strings with length n and k colors that cannot overlap themselves are given by a[0] = 1; a[n_] := k a[n - 1] - If[EvenQ[n], a[n/2], 0] Up to reversal and interchange of A and B , the first few overlap-free strings with 2 colors are A , AB , AAB , AAAB , AABB . The shortest pairs of strings of 2 elements with no self- or mutual overlaps are {"A", "B"} , {"AABB", "AABAB"} , {"AABB", "ABABB"} ; there are a total of 13 such pairs with strings up to length 5, and 85 with strings up to length 6.

In Cantor's theory ω + 1 is still larger (though 1 + ω is not), as are 2 ω , ω 2 and ω ω . … The first one that cannot is ε 0 , given by the limit ω ω ω ... , or effectively Nest[ ω # &, ω , ω ] . ε 0 is the smallest solution to ω ε  ε . … As Cantor noted, however, even this only allows one to reach the lowest class of transfinite numbers—in effect those corresponding to sets whose size corresponds to the cardinal number ℵ 0 .