In ordinary quantum theory, a straightforward calculation implies that the expected value of the product of the two measured spin values will be -Cos[ θ ] . … Now the expected value of the product of the two measured spin values is found just by averaging over ϕ as Integrate[f[ ϕ ] f[ θ - ϕ ], { ϕ , 0, 2 π }]/(2 π ) A version of Bell's inequalities is then that this integral can decrease with θ no faster than θ /(2 π ) - 1 —as achieved when f = Sign . … But at least in axiomatic quantum field theory it is typically assumed that one can somehow measure expectation values of any suitably smeared product of field operators.

One way to do this is by using the Gödel number Product[Prime[i]^list 〚 i 〛 , {i, Length[list]}] .

(The groups can be written as products of cyclic ones whose orders correspond to the possible factors of n .)

In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.

If one computes the product of Exp[  (j 1 + j 2 - j 3 )] for all triangles, then it turns out for example that this quantity is extremized exactly when the whole surface is flat. … And in 1968 Tullio Regge and Giorgio Ponzano suggested—almost as an afterthought in technical work on 6j symbols—that the quantum probability amplitude for any form of space might perhaps be given by the product of 6j symbols for the spins on each tetrahedron. … And from this it turns out that limits of products of 6j symbols correspond essentially to Exp[  s] , where s is the discrete form of the Einstein–Hilbert action—extremized by flat 3D space.

The basic idea is to encode the list of values of all the registers in the multiregister machine in the single number given by RMEncode[list_] := Product[Prime[j]^list 〚 j 〛 , {j, Length[list]}] and then to have this number be the value at appropriate steps of the first register in the 2-register machine.

An example is the algorithm of Anatolii Karatsuba from 1961 for finding products of n -digit numbers (with n = 2 s ) by operating on their digits in the nested pattern of page 608 (see also page 1093 ) according to First[f[IntegerDigits[x, 2, n], IntegerDigits[y, 2, n], n/2]] f[x_, y_, n_] := If[n < 1, x y, g[Partition[x, n], Partition[y, n], n]] g[{x1_, x0_}, {y1_, y0_}, n_] := With[{z1 = f[x1, y1, n/2], z0 = f[x0, y0, n/2]}, z1 2 2n + (f[x0 + x1, y0 + y1, n/2] - z1 - z0)2 n + z0] Other examples include the fast Fourier transform (page 1074 ) and related algorithms for ListConvolve , the quicksort algorithm for Sort , and many algorithms in fields such as computational geometry.

Given two numbers x and y their product can be computed in base k by ( FromDigits does the carries) FromDigits[ListConvolve[IntegerDigits[x, k], IntegerDigits[y, k], {1, -1}, 0], k] For numbers with n digits direct evaluation of the convolution would take about n 2 steps.

The element at position n in the first sequence discussed above can however be obtained in about Log[n] steps using ((IntegerDigits[#3 + Quotient[#1, #2], 2] 〚 Mod[#1, #2] + 1 〛 &)[n - (# - 2)2 # - 1 - 2, #, 2 # - 1 ]&)[NestWhile[# + 1&, 0, (# - 1)2 # + 1 < n &]] where the result of the NestWhile can be expressed as Ceiling[1 + ProductLog[1/2(n - 1)Log[2]]/Log[2]] Following work by Maxim Rytin in the late 1990s about k n+1 digits of a concatenation sequence can be found fairly efficiently from k/(k - 1) 2 - (k - 1) Sum[k (k s - 1) ((1 + s - s k)/(k - 1)) (1/((k - 1) (k s - 1) 2 ) - k/((k - 1) (k s + 1 - 1) 2 ) + 1/(k s + 1 - 1)), {s, n}] Concatenation sequences can also be generated by joining together digits from other representations of numbers; the picture below shows results for the Gray code representation from page 901 .

Most arose first as solutions to specific differential equations, typically in physics and astronomy; some arose as products, sums of series or inverses of other functions.