(Sums of squares of moments of given order in general provide rotationally invariant measures of anisotropy—equal to pair correlations weighted with LegendreP or GegenbauerC functions.)

All four possible pairs of successive elements occur, though not with equal frequency.

Circle packings Hexagonal packing of equal circles has been known since early antiquity (e.g. the fourth picture on page 43 ).

A convenient way to tell whether expressions are equal is to evaluate them with random numerical values for variables.

Boltzmann then showed that his equation implied the so-called H Theorem, which states that a quantity equal to entropy in equilibrium must always increase with time.

One might imagine that with such a computer it would be easy to solve the NP-complete problem of satisfiability from page 768 : one would just start off with a superposition in which all 2 n possible inputs have equal amplitude, then look at whether the spin representing the output from the function has any amplitude to be in a particular configuration.

Each zero is immediately followed by a maximum equal to x , and as picture below shows, values tend to accumulate for example on lines of the form ± x/2 u ± (2m + 1)2 v .

But whenever the evolution is ergodic, so that all states of a given energy are visited with equal frequency, the average behavior obtained will at least eventually correspond to the average over all states discussed above.

Note that if the period is equal to its absolute maximum of m , then every possible n is always visited, whatever n one starts from.