With a list of length n , Nest[NLFSRStep[f, taps, #] &, list, n] gives one step in the evolution of the cellular automaton in a register of width n , with a certain kind of spiral boundary condition. … And as noted by Nicolaas de Bruijn in 1946 there are 2 2 n - 1 -n such paths with length 2 n , and thus this number of functions f out of the 2 2 n possible must yield sequences of maximal length.

By inserting k = 6 Ceiling[Length[subs]/6] in the definition of TS1ToCT from page 1113 one can construct a cyclic tag system of this kind to emulate any one-element-dependence tag system.

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). With appropriate values of parameters, these tests in practice tend to be at least somewhat independent, although in principle, if sufficient data were available, they could all be subsumed into basic block frequency and run-length tests.

After n steps the total length of all stems is given by Apply[Plus, Abs[b]] n .

Cylinder volumes In any d -dimensional space, the volume of a cylinder of length x and radius r whose direction is defined by a unit vector v turns out to be given by s[d - 1] r d - 1 x (1 - (d - 1)/(d + 1)(RicciScalar-RicciTensor . v . v) r 2 + …) Note that what determines the volume of the cylinder is curvature orthogonal to its direction—and this is what leads to the combination of Ricci scalar and tensor that appears.

This discovery led to some confusion in early interpretations of the Second Law, but the huge length of time involved in a Poincaré recurrence makes it completely irrelevant in practice.

And as a generalization of this one can consider cases in which negation can be any operation that preserves lengths of strings.

This can be done for blocks up to length n in a 1D cellular automaton with k colors using ReversibleQ[rule_, k_, n_] := Catch[Do[ If[Length[Union[Table[CAStep[rule, IntegerDigits[i, k, m]], {i, 0, k m - 1}]]] ≠ k m , Throw[False]], {m, n}]; True] For k = 2 , r = 1 it turns out that it suffices to test only up to n = 4 (128 out of the 256 rules fail at n = 1 , 64 at n = 2 , 44 at n = 3 and 14 at n = 4 ); for k = 2 , r = 2 it suffices to test up to n = 15 , and for k = 3 , r = 1 , up to n = 9 .

For class 1 and 2 cellular automata, there are typically only a limited number of possible sequences of any length allowed. And when the length is large, the sequences are almost always either just uniform or repetitive. For class 3 cellular automata, however, the number of sequences of length n typically grows rapidly with n .

Each repeating block of digits typically seems quite random, and has properties such as all possible subblocks of digits up to a certain length appearing (see page 1084 ).