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But for multiway systems where each rule p  q is accompanied by its reverse q  p , and such pairs are represented say by "AAB" ↔ "BBAA" , an equivalent operator system can immediately be obtained either from Apply[Equal, Map[Fold[#2[#1] &, x, Characters[#]] &, rules, {2}], {1}] or from (compare page 1172 ) Append[Apply[Equal, Map[(Fold[f, First[#], Rest[#]] &)[Characters[#]] &, rules, {2}], {1}], f[f[a, b], c]  f[a, f[b, c]]] where now objects like "A" and "B" are treated as constants—essentially functions with zero arguments.
But in practice very different shapes can probably have almost identical energies, so that in as much as a given protein always takes on the same shape this must be associated with the dynamics of the process by which the protein folds when it is assembled. … (Biological evolution may conceivably have selected for proteins that fold reliably or are more robust with respect to changes in single amino acids, but there is currently no clear evidence for this.)
I also discuss the definition of life on pages 823 and 1178 , as well as mentioning protein folding and structure on pages 1003 and 1184 .
The first m rules (which yield far more than m elements of the original sequence) are obtained for any h that is not a rational number from the continued fraction form (see page 914 ) of h by Map[(({0  Join[#, {1}], 1  Join[#, {1, 0}]} &)[Table[0, {# - 1}]]) &, Reverse[Rest[ContinuedFraction[h, m]]]] Given these rules, the original sequence is given by Floor[h] + Fold[Flatten[#1 /. #2] &, {0}, rules] If h is the solution to a quadratic equation, then the continued fraction form is repetitive, and so there are a limited number of different substitution rules. … (The presence of nested structure is particularly evident in FoldList[Plus, 0, Table[Mod[h n, 1] - 1/2, {n, max}]] .)
[Generating sequences with] unequal probabilities Given a sequence a of n equally probable 0's and 1's, the following generates a single 0 or 1 with probabilities approximating {1 - p, p} to n digits: Fold[({BitAnd, BitOr} 〚 1 + First[#2] 〛 [#1, Last[#2]]) &, 0, Reverse[Transpose[{First[RealDigits[p, 2, n, -1]], a}]]] This can be generalized to allow a whole sequence to be generated with as little as an average of two input digits being used for each output digit.
Rest[list]/#1) &, Apply[ ExtendedGCD, Drop[list, -1]]]}, {Mod[ α , #], #} &[ Fold[GCD[#1, If[#1  0, #2, Mod[#2, #1]]] &, 0, ListCorrelate[{ α , -1}, list]]]] With slightly more effort both x and {a, m} can be found just from First[IntegerDigits[list, 2, p]] .
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]]
EvenQ] := Partition[ Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Floor[Range[1, n + 2/3, 1/3]]], 2] Networks obtained in this way are usually connected, but will almost always contain self-loops and multiple edges.
The quantity FoldList[Plus, 0, Table[MoebiusMu[i], {i, n}]] behaves very much like a random walk.
Cases in which the underlying mechanism is probably more associated with folding of fixed amounts of material include human fingerprint patterns and patterns in ferrofluids consisting of suspensions of magnetic particles.
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