And the way that we hear sounds is by such vibrations being transmitted to the array of hair cells in our inner ear. The mechanics of the inner ear are set up so that each row of hair cells ends up being particularly sensitive to vibrations at some specific frequency.

During the evolution the rule can apply only to the inner part FixedPoint[Replace[#, ℯ [x_]  x] &, expr] of an expression. The depth of this inner part for initial condition ℯ [ ℯ ][ ℯ ][ ℯ ][ ℯ ][ ℯ ] is shown below.

Auditory system Sound is detected by the motion it causes in hair cells in the cochlea of the inner ear.

In a rotating but perfectly spherical black hole (Kerr metric), the situation is nevertheless different, and in this case the whole infinite future of the outside universe can indeed in principle be seen in the finite time between crossing the outer and inner event horizons.

These include one-of-a-kind beings that have nothing to communicate with, as well as beings with inner intellectual activity but no effect on the outside world.

In general, the pattern produced by evolution for t steps is given by NestList[ Inner[f, Prepend[#, 0], Append[#, 0], List] &, {a}, t] so that the first few steps yield {a}, {f[0, a], f[a, 0]}, {f[0, f[0, a]], f[f[0, a], f[a, 0]], f[f[a, 0], 0]}, {f[0, f[0, f[0, a]]], f[f[0, f[0, a]], f[f[0, a], f[a, 0]]], f[f[f[0, a], f[a, 0]], f[f[a, 0], 0]], f[f[f[a, 0], 0], 0]} If f is Flat , however, then the last two lines here become {f[0, 0, a], f[0, a, a, 0], f[a, 0, 0]}, {f[0, 0, 0, a], f[0, 0, a, 0, a, a, 0], f[0, a, a, 0, a, 0, 0], f[a, 0, 0, 0]} and in general the number of a 's that appear in a particular element is given as in Pascal's triangle by a binomial coefficient.

Inverse[#2], RotateLeft[ Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]] (Inverse[g] .