It is also possible to set up emulations where this equality does not hold—and indeed some of the cases listed in the main text and shown in the picture above are of this type.

For a problem can be thought of as an infinite list of solutions for successive possible inputs.

(In Mathematica, the explicit form of such a tensor can be represented as a nested list for which TensorRank[list]  4 .) … If p is a list of coordinate parameters that appear in a d -dimensional metric g , then Riemann = Table[ ∂ p 〚 j 〛 Γ 〚 i, k 〛 - ∂ p 〚 i 〛 Γ 〚 j, k 〛 + Γ 〚 i, k 〛 .

In 1900, as one of his list of 23 important mathematical problems, David Hilbert posed the problem of finding a single finite procedure that could systematically determine whether a solution exists to any specified Diophantine equation.

Flatten[{1, 0, CTList[{{1, 0, 0, 1}, {0, 1, 1, 0}}, {0, 1}, t]}] gives for example the Thue–Morse substitution system {1  {1, 0}, 0  {0, 1}} .

Sequential substitution systems [from cellular automata] Given a sequential substitution system with rules in the form used on page 893 , the rules for a cellular automaton which emulates it can be obtained from SSSToCA[rules_] := Flatten[{{v[_, _, _], u, _}  u, {_, v[rn_, x_, _], u}  r[rn + 1, x], {_, v[_, x_, _], _}  x, MapIndexed[ With[{r n = #2 〚 1 〛 , rs = #1 〚 1 〛 , rr = #1 〚 2 〛 }, {If[Length[rs]  1, {u, r[rn, First[rs]], _}  q[0, rr], {u, r[rn, First[rs]], _}  v[rn, First[rs], Take[rs, 1]]], {u, r[rn, x_], _}  v[rn, x, {}], {v[rn, _, Drop[rs, -1]], Last[rs], _}  q[Length[rs] - 1, rr], Table[{v[rn, _, Flatten[{___, Take[rs, i - 1]}]], rs 〚 i 〛 , _}  v[ rn, rs 〚 i 〛 , Take[rs, i]], {i, Length[rs] - 1, 1, -1}], {v[rn, _, _], y_, _}  v[rn, y, {}]}] & , rules /. s  List], {_, q[0, {x__, _}], _}  q[0, {x}], {_, q[0, {x_}], _}  r[1, x], {_, q[0, {}], x_}  r[1, x], {_, q[_, {___, x_}], _}  x, {_, q[_, {}], x_}  x, {_, x_, q[0, _]}  x, {_, _, q[n_, {}]}  q[n - 1, {}], {_, _, q[n_, {x___, _}]}  q[n - 1, {x}], {q[_, {}], _, _}  w, {q[0, {__, x_}], p[y_, _], _}  p[x, y], {q[0, {__, x_}], y_, _}  p[x, y], {p[_, x_], p[y_, _], _}  p[x, y], {p[_, x_], u, _}  x, {p[_, x_], y_, _}  p[x, y], {_, p[x_, _], _}  x, {w, u, _}  u, {w, x_, _}  w, {_, w, x_}  x, {_, r[rn_, x_], _}  x, {_, u, r[_, _]}  u, {_, x_, r[rn_, _]}  r[rn, x], {_, x_, _}  x}] The initial condition is obtained by applying the rule s[x_, y__]  {r[1, x], y} and then padding with u 's.

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.

Quadratic residue sequences As an outgrowth of ideas related to RSA cryptography it was shown in 1982 by Lenore Blum , Manuel Blum and Michael Shub that the sequence Mod[NestList[Mod[# 2 , m] &, x0, n], 2] discussed on page 975 has the property that if m=p q with p and q primes (congruent to 3 modulo 4) then any systematic regularities detected in the sequence can eventually be used to discover factors of m .

The pattern obtained after t steps is then given by NestList[f[RotateRight[#], #]&, init, t] The pictures below show results with f being Times , and cells having values (a) {1, -1} , (b) the unit complex numbers {1,  , -1, -  } , (c) the unit quaternions.

Then given a list s[i] of the values of one set of neurons, one finds the values of another set using s[i + 1] = u[w . s[i]] , where in early models u = Sign was usually chosen, and now u = Tanh is more common, and w is a rectangular matrix which gives weights—normally assumed to be continuous numbers, often between -1 and +1—for the synaptic connections between the neurons in each set.