And in the context of network models it could be that spin is somehow related to something as simple as the number of places at which the core of a particle is attached to the rest of the network.

The transitions between these states have probabilities given by m[Map[Length, list]] where m[s_] := With[{q = FoldList[Plus, 0, s]}, ReplacePart[ RotateRight[IdentityMatrix[Last[q]], {0, 1}], 1/Length[s], Flatten[Outer[List, Rest[q], Drop[q, -1] + 1], 1]]] The average spectrum of sequences generated according to these probabilities can be obtained by computing the correlation function for elements a distance r apart ξ [list_, r_] := With[{w = (# - Apply[Plus, #]/Length[#] &)[ Flatten[list]]}, w .

This is similar to picture (c) on page 131 , and is a digit-by-digit version of FoldList[Plus, 0, Table[Apply[Plus, 2 Rest[IntegerDigits[i, 2]] - 1], {i, n}]] Note that although the picture above has a nested structure, the original concatenation sequences are not nested, and so cannot be generated by substitution systems.

Nonlinear feedback shift registers Linear feedback shift registers of the kind discussed on page 974 can be generalized to allow any function f (note the slight analogy with cyclic tag systems): NLFSRStep[f_, taps_, list_] := Append[Rest[list], f[list 〚 taps 〛 ]] With the choice f=IntegerDigits[s, 2, 8] 〚 8 - # . {4, 2, 1} 〛 & and taps = {1, 2, 3} this is essentially a rule s elementary cellular automaton.

Sierpiński pattern Other ways to generate step n of the pattern shown here in various orientations include: • Mod[Array[Binomial, {2, 2} n , 0], 2] (see pages 611 and 870 ) • 1 - Sign[Array[BitAnd, {2, 2} n , 0]] (see pages 608 and 871 ) • NestList[Mod[RotateLeft[#] + #, 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • NestList[Mod[ListConvolve[{1, 1}, #, -1], 2] &, PadLeft[{1}, 2 n ], 2 n - 1] (see page 870 ) • IntegerDigits[NestList[BitXor[2#, #] &, 1, 2 n - 1], 2, 2 n ] (see page 906 ) • NestList[Mod[Rest[FoldList[Plus, 0, #]], 2] &, Table[1, {2 n }], 2 n - 1] (see page 1034 ) • Table[PadRight[ Mod[CoefficientList[(1 + x) t - 1 , x], 2], 2 n - 1], {t, 2 n }] (see pages 870 and 951 ) • Reverse[Mod[CoefficientList[Series[1/(1 - (1 + x)y), {x, 0, 2 n - 1}, {y, 0, 2 n - 1}], {x, y}], 2]] (see page 1091 ) • Nest[Apply[Join, MapThread[ Join, {{#, #}, {0 #, #}}, 2]] &, {{1}}, n] (compare page 1073 ) The positions of black squares can be found from: • Nest[Flatten[2# /.

Continued fractions The first n terms in the continued fraction representation for a number x can be found from the built-in Mathematica function ContinuedFraction , or from Floor[NestList[1/Mod[#, 1]&, x, n - 1]] A rational approximation to the number x can be reconstructed from the continued fraction using FromContinuedFraction or by Fold[(1/#1 + #2 )&, Last[list], Rest[Reverse[list]]] The pictures below show the digit sequences of successive iterates obtained from NestList[1/Mod[#, 1]&, x, n] for several numbers x . … In analogy to digits in a concatenation sequence the terms in the sequence Flatten[Table[Rest[ContinuedFraction[a/b]], {b, 2, n}, {a, b - 1}]] are known to occur with the same frequencies as they would in the continued fraction representation for a randomly chosen number.

Any real number x can be represented as a set of integers using for example Rest[FoldList[Plus, 1, ContinuedFraction[x]]] but except when x is rational this list is not finite.

Hump m in the picture of sequence (c) shown is given by FoldList[Plus, 0, Flatten[Nest[Delete[NestList[Rest, #, Length[#] - 1], 2]&, Append[Table[1, {m}], 0], m]] - 1/2] The first 2 m elements in the sequence can also be generated in terms of reordered base 2 digit sequences by FoldList[Plus, 1, Map[Last[Last[#]]&, Sort[Table[{Length[#], Apply[Plus, #], 1 - #}& [ IntegerDigits[i, 2]], {i, 2 m }]]]] Note that the positive and negative fluctuations in sequence (f) are not completely random: although the probability for individual fluctuations in each direction seems to be the same, the probability for two positive fluctuations in a row is smaller than for two negative fluctuations in a row.

(The two bodies at the bottom are initially at rest; the body at the top is given progressively larger rightward velocities.)

In a typical case, each cell is updated using LFSRStep[list_] := Append[Rest[list], Mod[list 〚 1 〛 + list 〚 2 〛 , 2]] with a step of cellular automaton evolution corresponding to the result of updating all cells in the register. … In general, linear feedback shift registers can have "taps" at any list of positions on the register, so that their evolution is given by LFSRStep[taps_List, list_] := Append[Rest[list], Mod[Apply[Plus, list 〚 taps 〛 ], 2]] (With taps specified by the positions of 1's in a vector of 0's, the inside of the Mod can be replaced by vec . list as on page 1087 .) … Such generators are directly related to linear feedback shift registers, since with a list of length q , each step is simply Append[Rest[list], Mod[list 〚 1 〛 + list 〚 q - p + 1 〛 , 2 k ]] Cryptographic generators.