Tables of primes No explicit tables of primes appear to have survived from antiquity, but it seems likely that all primes up to somewhere between 5000 and 10000 were known.

/n all yield 1 .

In this case all the various kinds of patterns that we saw in Chapter 5 can occur.

Computational fluid dynamics From its inception in the mid-1940s until the invention of cellular automaton fluids in the 1980s, essentially all computational fluid dynamics involved taking the continuum Navier–Stokes equations and then approximating these equations using some form of discrete mesh in space and time, and arguing that when the mesh becomes small enough, correct results would be obtained.

Substitution systems in which all replacements are done that are found to fit in a left-to-right scan can be implemented as follows GSSEvolveList[rule_, s_, n_] := NestList[GSSStep[rule, #] &, s, n] GSSStep[rule_, s_] := g[rule, s, f[StringPosition[s, Map[First, rule]]]] f[{ }] = { }; f[s_] := Fold[If[Last[Last[#1]] ≥ First[#2], #1, Append[#1, #2]]&, {First[s]}, Rest[s]] g[rule_, s_, { }] := s; g[rule_, s_, pos_] := StringReplacePart[ s, Map[StringTake[s, #] &, pos] /. rule, pos] with rules given as {"ABA"  "BAAB", "BBBB"  "AA"} .

(In 2D it is straightforward to decide equivalence, and in 3D it is known that only some fundamental groups can be obtained—roughly because not all networks can be embedded in 2D—and it is expected that it will ultimately be possible to decide equivalence.)

Reproducing quantum phenomena Given molecular dynamics it is much easier to see how to reproduce fluid mechanics than rigid-body mechanics—since to get rigid bodies with only a few degrees of freedom requires taking all sorts of limits of correlations between underlying molecules.

The values of f[r] for all other r will depend on the pattern of connections in the network.

The pictures below show the distributions of numbers of Nand operations needed for all 2 2 n n -input Boolean functions. For n = 2 , the largest number of such operations is 6, achieved by Nor ; for n = 3 , it is 14, achieved by Xor (rule 150); for n = 4 , it is 27, achieved by rule 5737, which is Not[Xor[##]] & except when all inputs are True .

Starting with integers and then applying arithmetic operations and fractional powers one can readily reproduce all algebraic numbers up to degree 4, but not beyond. … Multiple integrals of rational functions can be more complicated, as in Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}]  HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan and presumably often cannot be expressed at all in terms of standard mathematical functions.