In medieval times there were some doubts about the utility of mathematics in science, and in the late 1200s, for example, Albertus Magnus made the statement that "many of the geometer's figures are not found in natural bodies, and many natural figures, particularly those of animals and plants, are not determinable by the art of geometry".

And in general, the probabilities for all 8 possible combinations of 3 cells are given by probs = Apply[Times, Table[IntegerDigits[8 - i, 2, 3], {i, 8}] /. {1  p, 0  1 - p}, {1}] In terms of these probabilities the density at the next step in the evolution of cellular automaton with rule number m is then given by Simplify[probs .

At times there have been hopes of so-called dynamical symmetry breaking giving the same effective results as the Higgs mechanism, but without an explicit Higgs field—perhaps through something similar to various phenomena in condensed matter physics.

This can be achieved by taking e n to be Nest[inc, zero, n] where zero = s[k] inc = s[s[k[s]][k]] With this setup one then finds plus = s[k[s]][s[k[s[k[s]]]][s[k[k]]]] times = s[k[s]][k] power = s[k[s[s[k][k]]]][k] (Note that power[x][y]//.crules is y[x] , and that by analogy x[x[y]] corresponds to y x 2 , x[y[x]] to x x y , x[y][x] to x y x , and so on.) … And from this it follows that Nest[s, k, n] can be converted to the Church numeral for n by applying s[s[s[s[s[k][k]][k[s[s[k[s]][k]][s[k][k]]]]][ k[s[s[k[s]][k]][s[s[k[s]][k]][s[k][k]]]]]][s[s[k[s]][ s[s[k[s]][s[k[s[s[s[s[s[s[s[k][k]][k[s]]][k[k]]][k[s[s[ k[s]][k]][s[k][k]]]]][k[s[s[k[s]][k]][s[s[k[s]][k]][s[k][ k]]]]]][k[s[s[s[s[k][k]][k[s[s[k[s]][s[k[s[s[k][k]]]][s[ k[k]][s[k[s[s[k[s]][k]]]][s[s[k][k]][k[k]]]]]]][s[k[k]][s[ s[k][k]][k[k]]]]]]][k[s[s[s[k][k]][k[s[k]]]][k[s[k]]]]]][ k[s[k]]]]]]]][s[k[k]][s[s[s[k][k]][k[s[s[k[s]][k]][s[k][ k]]]]][k[s[s[k[s]][k]][s[s[k[s]][k]][s[k][k]]]]]]]]][ k[s[k[k]][s[s[k[s]][k]]]]]]][k[s[k][k]]]]][k[s[k]]] Using this one can find from the corresponding results for Church numerals combinator expressions for plus , times and power —with sizes 377, 378 and 382 respectively.

Several times since the 1940s it has been suggested that models of computation should be closer to traditional continuous mathematics, and should look at real numbers as a whole, not in terms of their digit or other representations.

As discussed in the note below, this can be viewed as a consequence of the fact that the probability distribution in a random walk depends only on Sum[Outer[Times, e 〚 s 〛 , e 〚 s 〛 ], {s, Length[e]}] and not on products of more of the e 〚 s 〛 .

The series has an accumulation of poles on the circle Abs[a] 2  1 ; the coefficient of x m turns out to have denominator 2^(m - DigitCount[m, 2, 1]) Apply[Times, Table[Cyclotomic[s, a]^Floor[(m - 1)/s], {s, m - 1}]] For other iterated maps general formulas also seem rare.

History of universality In Greek times it was noted as a philosophical matter that any single human language can be used to describe the same basic range of facts and processes.

The first 2 m elements in the sequence can be obtained from (see page 1081 ) (CoefficientList[Product[1 - z 2 s , {s, 0, m - 1}], z] + 1)/2 The first n elements can also be obtained from (see page 1092 ) Mod[CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/ (2(1 + x)), {x, 0, n - 1}], x], 2] The sequence occurs many times in this book; it can for example be derived from a column of values in the rule 150 cellular automaton pattern discussed on page 885 .

Hardy and John Littlewood in 1922 to be proportional to 2n Apply[Times, Map[(# - 1)/(# - 2)&, Map[First, Rest[FactorInteger[n]]]]]/Log[n] 2 It was proved in 1937 by Ivan Vinogradov that any large odd integer can be expressed as a sum of three primes.