Of the sequences on page 594 , (a) through (d) as well as (f) fail every single one of the tests, (e) fails only the serial test, while (g) and (h) pass all the tests.

The so-called Drake equation gives a straightforward upper bound on the number of cases of extraterrestrial intelligence that could have arisen in our galaxy through the same basic chain of circumstances as humans.

For in particular it ends up being almost impossible to distinguish whatever genuine instability and apparent randomness may be implied by the Navier–Stokes equations from artifacts that get introduced through the discretization procedure used in solving the equations on a computer.

The undecidability of PCP can be seen to follow from the undecidability of the halting problem through the fact that the question of whether a tag system of the kind on page 93 with initial sequence s ever reaches a halting state (where none of its rules apply) is equivalent to the question of whether there is a way to satisfy the PCP constraint TSToPCP[{n_, rule_}, s_] := Map[Flatten[IntegerDigits[#, 2, 2]] &, Module[{f}, f[u_] := Flatten[Map[{1, #} &, 3u]]; Join[Map[{f[Last[#]], RotateLeft[f[First[#]]]} &, rule], {{f[s], {1}}}, Flatten[ Table[{{1, 2}, Append[RotateLeft[f[IntegerDigits[j, 2, i]]], 2]}, {i, 0, n - 1}, {j, 0, 2 i - 1}], 1]]], {2}] Any PCP constraint can also immediately be related to the evolution of a multiway tag system of the kind discussed in the note below. … In cases (m) through (s) there is ultimately no solution, but as the pictures indicate, in these specific PCP systems there are always strings that agree as far as they have gone—it is just that they never end up the same length.

The reason for this is that if c changes only slightly, then the pattern to a first approximation deforms only slightly, so that the part seen through the peephole just shifts, and in a small region of c values the peephole in effect simply scans over different parts of the pattern.

The first two blocks of pictures above show all the diagrams for Compton scattering that involve 2 or 3 photons—and contribute through order α 3 .

In these circuits, lumps of electric charge (in 2001 about half a million electrons each) flow through channels which cross to form various kinds of gates.

If one steps through frequencies with an AM radio scanner, one sometimes hears intelligible speech—from radio or TV broadcasts, or two-way radio communication.

In the mid-1960s, notably through the work of Steve Smale , proofs were given that there could be differential equations in which such sensitivity is generic.

The best-known way in which cellular automata were introduced (and which eventually led to their name) was through work by John von Neumann in trying to develop an abstract model of self-reproduction in biology—a topic which had emerged from investigations in cybernetics. … Largely through popularization in Scientific American by Martin Gardner , Life became widely known. … In the early 1960s there was for a time interest in iterative arrays in which data would be run repeatedly through such systems.