The values are obtained by applying to cells at each position one of the unary operations (endomorphisms) σ that satisfy σ [a ⊕ b]  σ [a] ⊕ σ [b] for individual cell values a and b . … But one can also imagine setting up systems whose states are continuous functions of position. ϕ then defines a mapping from one such function to another.

The element at position n in the first sequence discussed above can however be obtained in about Log[n] steps using ((IntegerDigits[#3 + Quotient[#1, #2], 2] 〚 Mod[#1, #2] + 1 〛 &)[n - (# - 2)2 # - 1 - 2, #, 2 # - 1 ]&)[NestWhile[# + 1&, 0, (# - 1)2 # + 1 < n &]] where the result of the NestWhile can be expressed as Ceiling[1 + ProductLog[1/2(n - 1)Log[2]]/Log[2]] Following work by Maxim Rytin in the late 1990s about k n+1 digits of a concatenation sequence can be found fairly efficiently from k/(k - 1) 2 - (k - 1) Sum[k (k s - 1) ((1 + s - s k)/(k - 1)) (1/((k - 1) (k s - 1) 2 ) - k/((k - 1) (k s + 1 - 1) 2 ) + 1/(k s + 1 - 1)), {s, n}] Concatenation sequences can also be generated by joining together digits from other representations of numbers; the picture below shows results for the Gray code representation from page 901 .

By the mid-1980s, however, it began to be clear that the whole game-theoretical idea of thinking of markets as collections of rational entities that optimize their positions on the basis of complete information was quite inadequate.

Tap positions {1, 2, 3, 4} were among those studied, but nothing like the pictures below were apparently ever explicitly generated—and nearly three decades passed before I noticed the remarkable behavior of the rule 30 cellular automaton.

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.

And by about 200 BC the development of gears had made it possible to create devices (such as the Antikythera device from perhaps around 90 BC) in which the positions of wheels would correspond to positions of astronomical objects.

There is a very direct mapping of positions on the retina to regions in the visual cortex.

In the 1890s Henri Poincaré studied so-called return maps giving for example positions of objects on successive orbits.

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 ).

As discovered by Jeffrey Shallit in 1979, numbers of the form Sum[1/k 2 i , {i, 0, ∞ }] that have nonzero digits in base k only at positions 2 i turn out to have continued fractions with terms of limited size, and with a nested structure that can be found using a substitution system according to {0, k - 1, k + 2, k, k, k - 2, k, k + 2, k - 2, k} 〚 Nest[Flatten[{{1, 2}, {3, 4}, {5, 6}, {7, 8}, {5, 6}, {3, 4}, {9, 10}, {7, 8}, {9, 10}, {3, 4}} 〚 # 〛 ]&, 1, n] 〛 The continued fractions for square roots are always periodic; for higher roots they never appear to show any significant regularities. The first million terms in the continued fraction for 2 1/3 contain 414,983 1's, have geometric mean 2.68505, and have largest term 4,156,269 at position 484,709.