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Most randomly selected primitive recursive functions show very simple behavior—either constant or linearly increasing when fed successive integers as arguments. The smallest examples that show other behavior are: • r[z, r[s, s]] , which is 1/2#(# + 1)& , with quadratic growth • r[z, r[s, c[s, s]]] , which is 2 # + 1 - # - 2 & , with exponential growth • r[z, r[s, p[2]]] , which is 2^Ceiling[Log[2, # + 2]] - # - 2 & , which shows very simple nesting • r[z, r[c[s, z], z]] , which is Mod[#, 2]& , with repetitive behavior • r[z, r[s, r[s, s]]] which is Fold[1/2#1(# + 1) + #2 &, 0, Range[#]]& , growing like 2 2 x . r[z, r[s, r[s, r[s, p[2]]]]] is the first function to show significantly more complex behavior, and indeed as the picture below indicates, it already shows remarkable randomness. … But by reducing the results modulo 2 one gets a function that does not grow—and has seemingly quite random behavior—yet is presumably again not primitive recursive.
Starting in the 1980s fast algorithms based on randomized methods (see page 1192 ) have also become popular.
In the limit, such sequences contain with equal frequency all possible blocks of any given length, but as shown on page 597 , they exhibit other obvious deviations from randomness.
But no clear understanding has yet emerged, and indeed most of the analysis that is done—which tends to be largely statistical in nature—is not likely to shed much light on the general question of why there is so much apparent randomness in turbulence.
Just as for all sorts of other systems with complex behavior, some idea of overall properties of Diophantine equations can be found on the basis of an approximation of perfect randomness. Writing equations in the form p[x 1 , x 2 , …, x n ]  0 the distribution of values of p will in general be complicated (see page 1161 ), but as a first approximation one can try taking it to be purely random. … The assumption of perfect randomness then suggests that for d < n , more and more cases with p  0 will be seen as x increases, so that the equation will have an infinite number of solutions.
In his 1952 paper Turing used a finite difference approximation to a pair of diffusion equations to show that starting from a random distribution of concentration values dappled regions could develop in which one or the other chemical was dominant.
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.
And assuming this variation is somehow random, the contributions of these nearby configurations will tend to cancel out. … In the quantum case, a sign of confinement would be exponential decrease with spacetime area of the average phase of color flux through so-called Wilson loops—and this is achieved if there is in a sense maximal randomness in field configurations.
An example studied by John Hopfield in 1981 is a symmetric matrix w with neuron values being updated sequentially in a random order rather than in parallel.
Then in the 1990s so-called combinatorial chemistry became popular, in which—somewhat in imitation of the immune system—large numbers of possible compounds were created by successively adding at random several different possible amino acids or other units.
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