For rule 18 it appears that h μ tx = 1 , while for rule 22, h μ tx and for rule 54 h μ tx .

For ordinary massive particles in d -dimensional space the group is Spin( d ), while for massless particles it is E ( d -1) (the Euclidean group).

CTToR110[rules_ /; Select[rules, Mod[Length[#], 6] ≠ 0 &]  {}, init_] := Module[{g1, g2, g3, nr = 0, x1, y1, sp}, g1 = Flatten[ Map[If[#1 === {}, {{{2}}}, {{{1, 3, 5 - First[#1]}}, Table[ {4, 5 - # 〚 n 〛 }, {n, 2, Length[#]}]}] &, rules] /. a_Integer  Map[({d[# 〚 1 〛 , # 〚 2 〛 ], s[# 〚 3 〛 ]}) &, Partition[c[a], 3]], 4]; g2 = g1 = MapThread[If[#1 === #2 === {d[22, 11], s3}, {d[ 20, 8], s3}, #1] &, {g1, RotateRight[g1, 6]}]; While[Mod[ Apply[Plus, Map[# 〚 1, 2 〛 &, g2, 30] ≠ 0, nr++; g2 = Join[ g2, g1]]; y1 = g2 〚 1, 1, 2 〛 - 11; If[y1 < 0, y1 += 30]; Cases[ Last[g2] 〚 2 〛 , s[d[x_, y1], _, _, a_]  (x1 = x + Length[a])]; g3 = Fold[sadd, {d[x1, y1], {}}, g2]; sp = Ceiling[5 Length[ g3 〚 2 〛 ]/(28 nr) + 2]; {Join[Fold[sadd, {d[17, 1], {}}, Flatten[Table[{{d[sp 28 + 6, 1], s[5]}, {d[398, 1], s[5]}, { d[342, 1], s[5]}, {d[370, 1], s[5]}}, {3}], 1]] 〚 2 〛 , bg[ 4, 11]], Flatten[Join[Table[bgi, {sp 2 + 1 + 24 Length[init]}], init /. {0  init0, 1  init1}, bg[1, 9], bg[6, 60 - g2 〚 1, 1, 1 〛 + g3 〚 1, 1 〛 + If[g2 〚 1, 1, 2 〛 < g3 〚 1, 2 〛 , 8, 0]]]], g3 〚 2 〛 }] s[1] = struct[{3, 0, 1, 10, 4, 8}, 2]; s[2] = struct[{3, 0, 1, 1, 619, 15}, 2]; s[3] = struct[{3, 0, 1, 10, 4956, 18}, 2]; s[4] = struct[{0, 0, 9, 10, 4, 8}]; s[5] = struct[{5, 0, 9, 14, 1, 1}]; {c[1], c[2]} = Map[Join[{22, 11, 3, 39, 3, 1}, #] &, {{63, 12, 2, 48, 5, 4, 29, 26, 4, 43, 26, 4, 23, 3, 4, 47, 4, 4}, {87, 6, 2, 32, 2, 4, 13, 23, 4, 27, 16, 4}}]; {c[3], c[4], c[5]} = Map[Join[#, {4, 17, 22, 4, 39, 27, 4, 47, 4, 4}] &, {{17, 22, 4, 23, 24, 4, 31, 29}, {17, 22, 4, 47, 18, 4, 15, 19}, {41, 16, 4, 47, 18, 4, 15, 19}}] {init0, init1} = Map[IntegerDigits[216 (# + 432 10 49 ), 2] &, {246005560154658471735510051750569922628065067661, 1043746165489466852897089830441756550889834709645}] bgi = IntegerDigits[9976, 2] bg[s_, n_] := Array[bgi 〚 1 + Mod[# - 1, 14] 〛 &, n, s] ev[s[d[x_, y_], pl_, pr_, b_]] := Module[{r, pl1, pr1}, r = Sign[BitAnd[2^ListConvolve[{1, 2, 4}, Join[bg[pl - 2, 2], b, bg[pr, 2]]], 110]]; pl1 = (Position[r - bg[pl + 3, Length[r]], 1 | -1] /. {}  {{Length[r]}}) 〚 1, 1 〛 ; pr1 = Max[pl1, (Position[r - bg[pr + 5 - Length[r], Length[r]], 1 | -1] /. {}  {{1}}) 〚 -1, 1 〛 ]; s[d[x + pl1 - 2, y + 1], pl1 + Mod[pl + 2, 14], 1 + Mod[pr + 4, 14] + pr1 - Length[r], Take[r, {pl1, pr1}]]] struct[{x_, y_, pl_, pr_, b_, bl_}, p_Integer : 1] := Module[ {gr = s[d[x, y], pl, pr, IntegerDigits[b, 2, bl]], p2 = p + 1}, Drop[NestWhile[Append[#, ev[Last[#]]] &, {gr}, If[Rest[Last[#]] === Rest[gr], p2--]; p2 > 0 &], -1]] sadd[{d[x_, y_], b_}, {d[dx_, dy_], st_}] := Module[{x1 = dx - x, y1 = dy - y, b2, x2, y2}, While[y1 > 0, {x1, y1} += If[Length[st]  30, {8, -30}, {-2, -3}]]; b2 = First[Cases[st, s[d[x3_, -y1], pl_, _, sb_]  Join[bg[pl - x1 - x3, x1 + x3], x2 = x3 + Length[sb]; y2 = -y1; sb]]]; {d[x2, y2], Join[b, b2]}] CTToR110[{{}}, {1}] yields blocks of lengths {7204, 1873, 7088} .

The method of averaging to correct for what were assumed to be random errors of observation began to be used, primarily in astronomy, in the mid-1700s, while least squares fitting and the notion of probability distributions became established around 1800.

Indeed, in general with operators Implies , And and Or one gets to 2 n - 1 elements, while with operators Xor and Equal one gets to 2^(2Floor[n/2]) elements.

(Polynomials with 42 variables and degree 5, and 10 variables and degree 10 45 , are also known to work, while it is known that one with 2 variables cannot.)

The first one on the bottom (with 63 comparisons) has a nested structure and uses the method invented by Kenneth Batcher in 1964: Flatten[Reverse[Flatten[With[{m = Ceiling[Log[2, n]] - 1}, Table[With[{d = If[i  m, 2 t , 2 i + 1 - 2 t ]}, Map[ {0, d} + # &, Select[Range[n - d], BitAnd[# - 1, 2 t ]  If[i  m, 0, 2 t ] &]]], {t, 0, m}, {i, t, m}]], 1]], 1] The second one on the bottom also uses 63 comparisons, while the last one is the smallest known for n = 16 : it uses 60 comparisons and was invented by Milton Green in 1969.

But while this was a convenient and powerful notion, it also involved a new level of abstraction, and it brought with it considerable confusion about fundamental issues.

Elliptic equations such as the Laplace equation need boundary values, while hyperbolic and parabolic equations such as the wave equation and diffusion equation need initial values.

In the early 1900s, common sense again for a while became dominant.