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The so-called Mertens Conjecture from 1897 stated that the magnitude of this quantity is less than √ n .
Density oscillations in rule 73
Although there are always some fluctuations, most rules yield densities that converge more or less uniformly to their final values.
A slightly less straightforward mechanism is illustrated in the third set of pictures below.
But what we will discover in this chapter is that by thinking in terms of simple programs, the fundamental origins of this behavior become much less mysterious.
Note that actual algorithms for pointer-based encoding will typically find a slightly less regular pattern of repeats than is shown in the pictures here.
But for complicated molecules of the kind that are common in living systems, this principle becomes much less useful.
And this means that no known algorithm can be expected to solve this problem exactly for a size n array (say with given boundaries) in much less than 2 n steps (see page 1145 ).
But I suspect that in fact a LeafCount less than about 20 is enough.
Note that Abs[u] is plotted in the second picture, while for the last equation a common less symmetrical form replaces the last term by u[t, x] ∂ x u[t, x] .)
But by changing the procedure just slightly, one can get much less regular growth.