A typical collection of tests described by Donald Knuth in 1968 includes: (1) frequency or equidistribution test (possible elements should occur with equal frequency); (2) serial test (pairs of elements should be equally likely to be in descending and ascending order); (3) gap test (runs of elements all greater or less than some fixed value should have lengths that follow a binomial distribution); (4) poker test (blocks corresponding to possible poker hands should occur with appropriate frequencies); (5) coupon collector's test (runs before complete sets of values are found should have lengths that follow a definite distribution); (6) permutation test (in blocks of elements possible orderings of values should occur equally often); (7) runs up test (runs of monotonically increasing elements should have lengths that follow a definite distribution); (8) maximum-of-t test (maximum values in blocks of elements should follow a power-law distribution).

But seeing the increasing emphasis on information rather than mechanical work in human affairs this now seems much less clear.)

Particularly in its less strict form x 3 + y 3 + z 3  a with x , y , z positive or negative the equation was mentioned in the 1800s and again in the mid-1900s; computer searches for solutions were begun in the 1960s, and by the mid-1990s solutions such as {283059965, 2218888517, 2220422932} for the case a= -30 had been found.

There has been much discussion of the idea that Hawking radiation somehow shows pure quantum states spontaneously turning into mixed ones, more or less as in quantum measurements.

There are some alternative approaches to defining dimension in which some of these issues at least become less explicit.

The validity of Church's Thesis has long been taken more or less for granted by computer scientists, but among physicists there are still nagging doubts, mostly revolving around the perfect continua assumed in space and quantum mechanics in the traditional formalism of theoretical physics (see page 730 ).

This is a k = 8 2D cellular automaton in which toppling of sand above a critical slope is captured by updating an array of relative sand heights s according to the rule SandStep[s_]:= s + ListConvolve[ {{0, 1, 0}, {1, -4, 1}, {0, 1, 0}}, UnitStep[s - 4], 2, 0] Starting from any initial condition, the rule eventually yields a fixed configuration with all values less than 4, as in the picture below.

Already in 1724 Stephen Hales looked at the motion of grids of marks on fig leaves, and noted that growth seemed to occur more or less uniformly throughout the leaf.

And largely as a result of this, there has tended to be less interest in ideas like simplicial complexes.

Such circles are no longer flat, but instead are like caps on the sphere—with a circle of radius r containing all points that are geodesic (great circle) distance less than r from its center.