This will happen whenever FractionalPart[Log[b, a[n]]] is uniformly distributed, which, as discussed on page 903 , is known to be true for sequences such as r n (with Log[b, r] irrational), n n , n!

The pictures at the top of the next page show how averages of successively larger collections of uniformly distributed numbers converge to a Gaussian distribution.

Measurements suggest that these values are uniformly distributed in the range 0 to 1, but despite a fair amount of mathematical work since the 1940s, there has been no substantial progress towards proving this.

Voronoi diagrams for irregularly distributed points have found many applications.

And in fact, all that is really necessary is that the hashing procedure generate enough randomness that even though there may be regularities in the original data, the hash codes that are produced still end up being distributed roughly uniformly across all possibilities.

(Since the 1960s, so-called mu-law companding has often been used, in which these levels are distributed exponentially in amplitude.)

In some cases I have run programs for many days or weeks, sometimes distributed via MathLink across a few hundred computers in my company's network.

There is also exchange of DNA between paternal and maternal chromosomes, typically with a few crossovers per chromosome, at positions that seem more or less randomly distributed among many possibilities (the details affect regions of repeating DNA used for example in DNA fingerprinting).

But if one uses instead s = {1, 2} then starts with {1} and {2} one gets any of {{}, {1}, {2}, {1, 2}} and in general with s = Range[n] one gets any of the 2 n elements in the powerset Distribute[Map[{{}, {#}} &, s], List, List, List, Join] But applying Complement[s, Intersection[a, b]] to these elements still always produces the same equivalences as with a ⊼ b .

The statistical distribution of zeros was studied by Andrew Odlyzko and others starting in the late 1970s (following ideas of David Hilbert and George Pólya in the early 1900s), and it was found that to a good approximation, the spacings between zeros are distributed like the spacings between eigenvalues of random unitary matrices (see page 977 ).