But whereas cellular automata always evolve only in discrete steps, differential equations instead go through a continuous process of evolution in which time appears just as a parameter.

For rational a this always repeats, cycling through Denominator[a] possible values (compare page 255 ).

As mentioned on page 938 , through major mathematical effort, a complete classification of all finite so-called simple groups that in effect have no factors is known.

In general, whenever the possible sequences correspond to paths through a finite network, it follows that the Cantor set obtained has a nested structure.

Origins of randomness [in animal patterning] The model in the main text assumes that randomness enters through initial conditions.

If one looks not just at specific sequences, but instead at all 2 n possible sequences of length n , one can ask how many cellular automaton rules (say with k = 2 , r = 2 ) one has to go through in order to generate every one of these. … Since some different rules generate the same sequences (see page 956 ) one needs to go through somewhat more than 2 n rules to get every sequence of length n .

NetCAStep above in general produces a non-deterministic finite automaton (NDFA) for which a particular sequence of values does not determine a unique path through the network.

And by the late 1990s, similar ideas were being used for example in materials science: in a typical case an array of different combinations of substances is made by successively spraying through an appropriate sequence of masks, with some physical or chemical test then applied to all the samples. … And in recent years—notably in the building of Mathematica—optimal algorithms for operations such as function evaluation and numerical integration have sometimes been found through searches.

One is furrowing or tearing of tissue through a loss of adhesion between cells.

Examples (o) through (u) show seven ways of drawing the same network, in this case the so-called Petersen network.