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When a water drop hits a water surface, at first a symmetrical crater forms. … If the original drop was moving quickly, a whole hemisphere of water then closes in above. But in any case a peak appears at the center, sometimes with a spherical drop at the top.
Indeed, the simple pegboard shown below exhibits the same phenomenon, with balls dropped at even infinitesimally different initial positions eventually following very different trajectories. The details of these trajectories cannot be deduced quite as directly as before from the digit sequences of initial positions, but Paths followed by four idealized balls dropped from initial positions differing by one part in a thousand into an array of identical circular pegs.
Difference tables and polynomials A common mathematical approach to analyzing sequences is to form a difference table by repeatedly evaluating d[list_] := Drop[list, 1] - Drop[list, -1] .
The combination Drop[list, -1] + 2 Drop[list, 1] of the result from CA2EvolveList corresponds to evolution according to a first-order k = 4 , r = 1 rule.
Spectra [of sequences] The spectra shown are given by Abs[Fourier[data]] , where the symmetrical second half of this list is dropped in the pictures.
And by being able to drop details that have little or no perceptible effect one can often achieve much higher levels of compression.
Note that the process of dropping nodes that become disconnected is analogous to so-called "garbage collection" for data structures.
Rest[list]/#1) &, Apply[ ExtendedGCD, Drop[list, -1]]]}, {Mod[ α , #], #} &[ Fold[GCD[#1, If[#1  0, #2, Mod[#2, #1]]] &, 0, ListCorrelate[{ α , -1}, list]]]] With slightly more effort both x and {a, m} can be found just from First[IntegerDigits[list, 2, p]] .
Note that on every step the rightmost element is always dropped, since no rule is given for how to replace it.
(For example, in {{0, 0, _}, {0, _, 1}, {_, 0, 0}} the first prime implicant is covered by the others, and can therefore be dropped.) Given the original list s and the complete prime implicant list p the so-called Quine–McCluskey procedure can be used to find a minimal list of prime implicants, and thus a minimal DNF: QM[s_, p_] := First[Sort[Map[p 〚 # 〛 &, h[{}, Range[Length[s]], Outer[MatchQ, s, p, 1]]]]] h[i_, r_, t_] := Flatten[Map[h[Join[i, r 〚 # 〛 ], Drop[r, #], Delete[Drop[t, {}, #], Position[t 〚 All, # 〛 ], {True}]]] &, First[Sort[Position[#, True] &, t]]]], 1] h[i_, _, {}] := {i} The number of steps required in this procedure can increase exponentially with the length of p .