To model the pouring of sand into a pile one can consider a series of cycles, in which at each cycle one first adds 4 to the value of the center cell, then repeatedly applies the rule until a new fixed configuration FixedPoint[SandStep, s] is obtained.

Map[ ∂ # f &, p]), p]/Sqrt[Det[g]] In general the series in r may not converge, but it is known that at least in most cases only flat space can give a result that shows no correction to the basic r d form.

Following my 1984 study of minimal sizes of finite automata capable of reproducing states in cellular automaton evolution (see page 276 ) a whole series of definitions were developed based on minimal sizes of descriptions in terms of deterministic and probabilistic finite automata (see page 1084 ).

In 1876 William Thomson (Kelvin) constructed a so-called harmonic analyzer, in which an assembly of disks were used to sum trigonometric series and thus to predict tides.

And by the 1960s a long series of successes in the predictions of QED had led to the almost universal belief that its assumption of point-like electrons must be correct.

And for it to all have been even remotely possible has required a series of particular personal circumstances.

And sometimes the idea was to work out terms in a symbolic perturbation series approximation, say in quantum field theory or celestial mechanics.

Starting in the late 1800s and continuing ever since a series of progressively more sophisticated geometric and algebraic views of Diophantine equations have developed.