For sequences involving only two distinct integers flat spectra are rare; with ± 1 those equivalent to {1, 1, 1, -1} seem to be the only examples. ( {r 2 , r s, s 2 , -r s} works for any r and s , as do all lists obtained working modulo x n - 1 from p[x]/p[1/x] where p[x] is any invertible polynomial.)

Charles Spearman suggested in 1904 that there might be a general intelligence factor (usually called g ) associated with all intellectual tasks.

But as Turing pointed out, the overwhelming majority of all possible real numbers are not computable.

By the 1960s undecidability was being found in all sorts of systems, but most of the examples were too complicated to seem of much relevance in practical mathematics or computing.

The picture below shows in each of the axiom systems from page 808 the lengths of the shortest proofs found by a version of Waldmeister (see page 1158 ) for all 582 equivalences (see page 818 ) that involve two variables and up to 3 Nand s on either side.

Note that if one uses base 6 rather than base 2, then as shown on page 614 powers of 3 still yield a complicated pattern, but all operations are strictly local, and the system corresponds to a cellular automaton with 6 possible colors for each cell and rule {a_, b_, c_}  3 Mod[b, 2] + Floor[c/2] (see page 1093 ).

Equation for the background [in my PDEs] If u[t, x] is independent of x , as it is sufficiently far away from the main pattern, then the partial differential equation on page 165 reduces to the ordinary differential equation u''[t]  (1 - u[t] 2 )(1 + a u[t]) u[0]  u'[0]  0 For a = 0 , the solution to this equation can be written in terms of Jacobi elliptic functions as ( √ 3 JacobiSN[t/3 1/4 , 1/2] 2 ) / (1 + JacobiCN[t/3 1/4 , 1/2] 2 ) In general the solution is (b d JacobiSN[r t, s] 2 )/(b - d JacobiCN[r t, s] 2 ) where r = -Sqrt[1/8 a c (b - d)] s = (d (c - b))/(c (d - b)) and b , c , d are determined by the equation (x - b)(x - c)(x - d)  -(12 + 6 a x - 4 x 2 - 3 a x 3 )/(3a) In all cases (except when -8/3 < a < -1/ √ 6 ), the solution is periodic and non-singular.

Michael Beeler in 1973 used a computer at MIT to investigate all 1296 possible worms with rules of the simplest type on a hexagonal grid, and he found several with fairly complex behavior.

In cases where all strings that appear both in rules and initial conditions are sorted—so that for example A 's appear before B 's—any string generated will also be sorted, so it can be specified just by giving a list of how many A 's and how many B 's appear in it.

In the lattice version in physics one typically considers what happens to averages over all possible configurations of a system if one does a so-called blocking transformation that replaces blocks of elements by individual elements.