A number is said to be "normal" in a particular base if every digit and every block of digits of any length occur with equal frequency.

For a random walk (see page 977 ) in which ± 1 occur with equal probability the spectrum is Csc[ π ω ] 2 /2 , or roughly 1/ ω 2 .

In the limit, such sequences contain with equal frequency all possible blocks of any given length, but as shown on page 597 , they exhibit other obvious deviations from randomness.

(With k = 2 , for b = 1 , {1, 1} represents conservation of the total number of cells, regardless of color, while for b = 2 , {1, 1, 1, 1} represents the same thing, while {0, 1, -1, 0} represents the fact that in going along in any state the number of black-to-white transitions must equal the number of white-to-black ones.)

Given a particular such semigroup satisfying axioms derived from a multiway system, one can see whether the operator representations of particular strings are equal—and if they are not, then it follows that the strings can never be reached from each other through evolution of the multiway system.

The largest individual term is the 432th one, which is equal to 20,776. … In a few known cases simple formulas yield numbers that are close but not equal to integers.

Particle masses The measured masses of known elementary particles in units of GeV (roughly equal to the proton mass) are: photon: 0, electron: 0.000510998902; muon: 0.1056583569; τ lepton: 1.77705; W : 80.4; Z : 91.19.

If one draws a circle of radius r on a page, then the smaller r is, the more curved the circle will be—and one can define the circle to have a constant curvature equal to 1/r . … (The parts of the Riemann tensor not captured by the Ricci tensor correspond to the so-called Weyl tensor; for d = 2 the Ricci tensor has only one independent component, equal to the negative of the Gaussian curvature.)

The word problem then asks if a given product of such generators is equal to the identity element.

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).