However, in an actual shell material can only be added on the outside of what already exists, and this can be represented by restricting θ to run over only part of the range - π to π .

One can count the number of occurrences of each of the k b possible blocks of length b in a given state using BC[list_] := With[{z = Map[FromDigits[#, k] &, Partition[list, b, 1, 1]]}, Map[Count[z, #] &, Range[0, k b - 1]]] Conserved quantities of the kind discussed here are then of the form q .

And in general the ideas and methods of this book seem to yield an unending stream of important questions of a remarkable range of different kinds. … But in the new kind of science that I describe in this book I believe that at least at first there will be opportunities for a much broader range of people to make contributions.

To emulate cellular automaton evolution one starts by encoding a list of cell values by the single combinator p[num[Length[list]]][ Fold[p[Nest[s, k, #2]][#1] &, p[k][k], list]] //. crules where num[n_] := Nest[inc, s[k], n] inc = s[s[k[s]][k]] One can recover the original list by using Extract[expr, Map[Reverse[IntegerDigits[#, 2]] &, 3 + 59(16^Range[Depth[expr[s[k]][s][k] //. crules] - 1, 1, -1] - 1)/ 15)]]/.

But the Principle of Computational Equivalence implies that in fact there are a huge range of other formal systems, equivalent in their ultimate richness, but different in their details, and in the questions to which they naturally lead.

The Riemann Hypothesis is also equivalent to the statement that a bound of order √ n Log[n] 2 exists on Abs[Log[Apply[LCM, Range[n]]] - n] .

[Structures in] the Game of Life The 2D cellular automaton described on page 949 supports a whole range of persistent structures, many of which have been given quaint names by its enthusiasts.

But despite the vast range of phenomena in nature that have never successfully been described in mathematical terms, it has become quite universally assumed that, as David Hilbert put it in 1900, "mathematics is the foundation of all exact knowledge of natural phenomena".

Note that if the rule for the finite automaton is represented for example as {{1, 2}, {2, 1}} where each sublist corresponds to a particular state, and the elements of the sublist give the successor states with inputs Range[0, k - 1] , then the n th element in the output sequence can be obtained from Fold[rule 〚 #1, #2 〛 &, 1, IntegerDigits[n - 1, k] + 1] - 1 while the first k m elements can be obtained from Nest[Flatten[rule 〚 # 〛 ] &, 1, m] - 1 To treat examples such as case (c) where elements can subdivide into blocks of several different lengths one must generalize the notion of digit sequences.

The following will update triples of cells in the specified order by using the function f : OrderedUpdate[f_, a_, order_]:= Fold[ReplacePart[ #1, f[Take[#1, {#2 - 1, #2 + 1}]], #2] &, a, order] A random ordering of n cells corresponds to a random permutation of the form Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Range[n]]