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111 - 115 of 115 for IntegerDigits

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 . … A way to find candidates for q is to compute
NullSpace[Table[With[{u = Table[Random[Integer, {0, k - 1}], {m}]}, BC[CAStep[u]] - BC[u]], {s}]]
for progressively larger m and s , and to see what lists continue to appear.

An example due to Gregory Chaitin is the digits of the fraction Ω of initial conditions for which a universal system halts (essentially a compressed version—with various subtleties about limits—of the sequence from page 1127 giving the outcome for each initial condition). As emphasized by Chaitin, it is possible to ask questions purely in arithmetic (say about sequences of values of a parameter that yield infinite numbers of solutions to an integer equation) whose answers would correspond to algorithmically random sequences.

In the 1940s it also became popular to use frequency modulation (FM) Sin[(1 + s[t]) ω t] , and in the 1970s pulse code modulation (PCM) (pulse trains for IntegerDigits[s[t], 2] ).

(a_ s_) (rtab 〚 i k + a + 1 〛 k 2r (s - 1) + 1 + Mod[i k + a, k 2r ]), {i, 0, k 2r - 1}]&, net], 1]
where here elementary rule 126 is specified for example by {2, 1, Reverse[IntegerDigits[126, 2, 8]]} .

It also has a polar plot of the positions of 14 pulsars relative to the Sun, with the pulsars specified by giving their periods as base 2 integers—but with trailing zeros inserted to cover inadequate precision. … At the left-hand end is a version of the pattern of digits from page 117 —but distorted so it has no obvious nested structure.