In the first 200 billion digits, the frequencies of 0 through 9 differ from 20 billion by {30841, -85289, 136978, 69393, -78309, -82947, -118485, -32406, 291044, -130820} An early approximation to π was 4 Sum[(-1) k /(2k + 1), {k, 0, m}] 30 digits were obtained with 2 Apply[Times, 2/Rest[NestList[Sqrt[2 + #]&, 0, m]]] An efficient way to compute π to n digits of precision is (# 〚 2 〛 2 /# 〚 3 〛 )& [NestWhile[Apply[Function[{a, b, c, d}, {(a + b)/2, Sqrt[a b], c - d (a - b) 2 , 2 d}], #]&, {1, 1/Sqrt[N[2, n]], 1/4, 1/4}, # 〚 2 〛 ≠ # 〚 2 〛 &]] This requires about Log[2, n] steps, or a total of roughly n Log[n] 2 operations (see page 1134 ).

The first examples were graphs of functions: the curve on page 918 was discussed by Bernhard Riemann in 1861 and by Karl Weierstrass in 1872.

Continued fractions The first n terms in the continued fraction representation for a number x can be found from the built-in Mathematica function ContinuedFraction , or from Floor[NestList[1/Mod[#, 1]&, x, n - 1]] A rational approximation to the number x can be reconstructed from the continued fraction using FromContinuedFraction or by Fold[(1/#1 + #2 )&, Last[list], Rest[Reverse[list]]] The pictures below show the digit sequences of successive iterates obtained from NestList[1/Mod[#, 1]&, x, n] for several numbers x . … Numbers whose continued fraction terms are polynomials in n can presumably also be represented in terms of suitably generalized hypergeometric functions. … The pictures below show as a function of n the quantity With[{r = FromContinuedFraction[ContinuedFraction[x, n]]}, -Log[Denominator[r], Abs[x - r]]] which gives a measure of the closeness of successive rational approximations to x .

= {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes. … To obtain such trimmed networks one can apply the function TrimNet[net_] := With[{m = Apply[Intersection, Map[FixedPoint[ Union[#, Flatten[Map[Last, net 〚 # 〛 , {2}]]]&, #]&, Map[List, Range[Length[net]]]]]}, net 〚 m 〛 /.

History [of Boolean functions] Logic has been used as an abstraction of arguments in ordinary language since antiquity.

In general the pattern of probabilities for changes can be thought of as being somewhat like a Green's function in mathematical physics—though the nonadditivity of most cellular automata makes this analogy less useful.

Note that the Weierstrass function of page 918 yields a 1/f spectrum, and presumably suitable averages of spectra from any substitution system should also have 1/f α forms (compare page 586 ).

(Some physicists might argue for a somewhat narrower definition that allows only discontinuities in the so-called partition function of equilibrium statistical mechanics, but for many of the most interesting applications, the definition I use is the appropriate one.)

In general one can imagine characterizing the power of any axiom system by giving a transfinite number κ which specifies the first function  [ κ ] (see note above ) whose termination cannot be proved in that axiom system (or similarly how rapidly the first example of y must grow with x to prevent ∃ y p[x, y] from being provable). … And in general I suspect that there are a vast number of functions with simple definitions whose termination cannot be proved not just because they grow too quickly but instead for the more fundamental reason that their behavior is in a sense too complicated.

Another approach, illustrated in picture (c), is to set up a continuous function with minima at the existing points, and then to search for the closest minimum.