An example studied since antiquity involves finding lengths or angles using a ruler and compass (i.e. as intersections between lines and circles). … Linkages consisting of rods of integer lengths always trace out algebraic curves (or algebraic surfaces in 3D) and in general allow any algebraic number (as represented by Root ) to be constructed.

Current image compression standards include: FAX CCITT 3 (run-length encoding, with codewords determined by Huffman coding from a definite distribution of run lengths); GIF (LZW); JPEG (lossy discrete cosine transform, then Huffman or arithmetic coding); BMP (run-length encoding, etc.); TIFF (FAX, JPEG, GIF, etc.).

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.

These are related to the autocorrelation function according to Fourier[list] 2  Fourier[ListConvolve[list, list, {1, 1}]]/Sqrt[Length[list]] (See also page 1074 .)

• Circumference: the length of the longest cycle in the network. … (Up to 8 nodes, all 8 trivalent networks have this property; up to 10 nodes 25 of 27 do.) • Girth: the length of the shortest cycle in the network.

Indeed, for period p , the length of blocks required is at most 2 2p (or 2 2 p r for range r rules). … Within each row a gray bar indicates that a particular period can be obtained with blocks of some length. … To find them one considers all possible blocks of length 2 p r + 1 and picks out those that after p steps evolve so that their center cell ends up the same color as it was originally.

With this setup, the evolution of any register machine can be implemented using the functions (a typical initial condition is {1, {0, 0}} ) RMStep[prog_, {n_Integer, list_List}] := If[n > Length[prog], {n, list}, RMExecute[prog 〚 n 〛 , {n, list}]] RMExecute[i[r_], {n_, list_}] := {n + 1, MapAt[(# + 1)&, list, r]} RMExecute[d[r_, m_], {n_, list_}] := If[list 〚 r 〛 > 0, {m, MapAt[(# - 1)&, list, r]}, {n + 1, list}] RMEvolveList[prog_, init:{_Integer, _List}, t_Integer] := NestList[RMStep[prog, #]&, init, t] The total number of possible programs of length n using k registers is (k (1 + n)) n .

If all letter probabilities are equal, then words will simply be ranked by length, with all k m words of length m occurring with frequency p m .

In each case a row of initial black cells of the specified length was used.

Digit count sequences Starting say with {1} repeatedly replace list by Join[list, IntegerDigits[Apply[Plus, list], 2]] The resulting sequences grow in length roughly like n Log[n] .