These numbers can also be obtained as the coefficients of x n in the series expansion of x ∂ x Log[ ζ [m, x]] , with the so-called zeta function, which is always a rational function of x , given by ζ [m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x] and corresponds to the product over all cycles of 1/(1 - x n ) .

One can start from the fact that the volume density in any space is given in terms of the metric by Sqrt[Det[g]] . … Inverse[#2], RotateLeft[ Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]] (Inverse[g] . Map[ ∂ # f &, p]), p]/Sqrt[Det[g]] In general the series in r may not converge, but it is known that at least in most cases only flat space can give a result that shows no correction to the basic r d form.

Among such computations are Plus , Times , Divide , Det and LinearSolve for integers, as well as determining outcomes in additive cellular automata (see page 609 ).

Another way to state the Einstein equations—already discussed by David Hilbert in 1915—is as the constraint that the integral of RicciScalar Sqrt[Det[g]] (the so-called Einstein–Hilbert action) be an extremum. (An idealized soap film or other minimal surface extremizes the integral of the intrinsic volume element Sqrt[Det[g]] , without a RicciScalar factor.)

(Note that given explicit coordinates, one can check whether one is in d or more dimensions by asking for all possible points Det[Table[(x[i] - x[j]) .