And while at first this might seem to suggest that the rich history of biology, civilization and technology needed to reach this point would somehow be wasted, what I believe instead is that this just highlights the extent to which such history is what is ultimately the defining feature of the human condition.

How this could be consistent with God having infinite power was not clear, although around 420 AD Augustine suggested that while God might have infinite knowledge of the future we as humans could not—yielding what can be viewed as a very rough analog of my explanation for free will. … But while this has some similarity to the ideas of computational irreducibility in this book it appears never to have been widely studied.

So notation like a + b and a × b , while convenient for interpretation, could equally well be replaced by more generic forms such as f[a, b] or g[a, b] without affecting any of the actual operation of the axioms. … Note that while  in basic logic is used in the underlying construction of rules,  in predicate logic is just an abstract operator with properties defined by the last two axioms given.

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 ).

For a = 1 , the period is about 4.01; for a = 2 , it is about 3.62; while for a = 4 , it is about 3.18.

There are only 8 rules (the inequivalent ones being 16740555 and 3327051468) where s > s , and in each case s = 6 while s = 5 .

Note that while many of the people listed are well known, extensive research (often through personal contacts, as well as institutional and government records) was required to track down quite a few of them.

Select[PM[s], Count[#, 1] > 1 &], 2]] while blocks of length n (and at most one error) can be decoded with Drop[(If[#  0, data, MapAt[1 - # &, data, #]] &)[ FromDigits[Mod[data .

Starting with an ordinary base 2 digit sequence, one prepends a unary specification of its length, then a specification of that length specification, and so on: (Flatten[{Sign[-Range[1 - Length[#], 0]], #}] &)[ Map[Rest, IntegerDigits[Rest[Reverse[NestWhileList[ Floor[Log[2, #] &, n + 1, # > 1 &]]],2]]] (d) Binary-coded base 3.

The fact that there are a million nerve fibers going from the eye to the brain, but only about 30,000 going from the ear to the brain means that while it takes several million bits per second to transmit video of acceptable quality, a few tens of thousands of bits are adequate for audio (NTSC television is 5 MHz; audio CDs 22 kHz; telephone 8 kHz).