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One can imagine weighting different network distances differently, but usually I have found that equal weightings work best.
And in the full Einstein equations the Ricci tensor need not be zero; instead it is specified at every point in space as being equal to a certain combination of energy and momentum density for matter at that point.
If all blocks occur with an equal probability of 2 -b , then h takes on its maximum possible value of b .
A somewhat better approximation is LogIntegral[n] , equal to Integrate[1/Log[t], {t, 2, n}] .
With just two piles one player can force the other to lose by arranging that after each of their moves the two piles have equal heights.
(Such trees have been studied since at least the early 1900s, with typical examples of concepts being Horton stream order, equal to Depth for trees given as Mathematica expressions.)
Local conservation laws
Whenever a system like a cellular automaton (or PDE) has a global conserved quantity there must always be a local conservation law which expresses the fact that every point in the system the total flux of the conserved quantity into a particular region must equal the rate of increase of the quantity inside it.
As early as 1851, for example, Eugène Prouhet showed that if sequences of integers were partitioned according to sequence (b) on page 83 , then sums of powers of these integers would be equal: thus Apply[Plus, Flatten[Position[s, i]] k ] is equal for i = 0 and i = 1 if s is a sequence of the form (b) on page 83 with length 2 m , m > k .
Unlike for the case of rule 60, the period is usually equal to q[n] (and is assumed so for the picture on page 260 ), with the first exception occurring at n=37 .
Indeed, in general with operators Implies , And and Or one gets to 2 n - 1 elements, while with operators Xor and Equal one gets to 2^(2Floor[n/2]) elements.