An example with 8 registers and 41 instructions is: or {d[4, 40], i[5], d[3, 9], i[3], d[7, 4], d[5, 14], i[6], d[3, 3], i[7], d[6, 2], i[6], d[5, 11], d[6, 3], d[4, 35], d[6, 15], i[4], d[8, 16], d[5, 21], i[1], d[3, 1], d[5, 25], i[2], d[3, 1], i[6], d[5, 32], d[1, 28], d[3, 1], d[4, 28], i[4], d[6, 29], d[3, 1], d[5, 24], d[2, 28], d[3, 1], i[8], i[6], d[5, 36], i[6], d[3, 3], d[6, 40], d[4, 3]} Given any register machine, one first applies the function RMToRM2 from page 1114 , then takes the resulting program and initial condition and finds an initial condition for the URM using R2ToURM[prog_, init_] := Join[init, With[ {n = Length[prog]}, {1 + LE[Reverse[prog] /.

Following the work of Lars Onsager around 1944, it turns out that an exact solution in terms of traditional mathematical functions can be found in this case. … (Another analytical result is that for e~e 0 correlations between pairs of spins can be expressed in terms of Painlevé functions.) … The pictures at the top of the next page show the values of m[s] (densities of +1 cells) after 0, 10, 100 and 1000 steps for a 500*500 system as a function of the initial values of m[s] and e[s] .

Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .

[Mollusc] shell model The center of the opening of a shell is taken to trace out a helix whose {x, y, z} coordinates are given as a function of the total angle of revolution t by a t {Cos[t], Sin[t], b} .

Many detailed mathematical analyses of linear reaction-diffusion equations have been done since the 1970s; numerical solutions to linear and occasionally nonlinear such equations have also often been found, and in recent years explicit pictures of patterns—rather than just curves of related functions—have commonly been generated.

Here are examples of how some of the basic Mathematica constructs used in the notes in this book work: • Iteration Nest[f, x, 3] ⟶ f[f[f[x]]] NestList[f, x, 3] ⟶ {x, f[x], f[f[x]], f[f[f[x]]]} Fold[f, x, {1, 2}] ⟶ f[f[x, 1], 2] FoldList[f, x, {1, 2}] ⟶ {x, f[x, 1], f[f[x, 1], 2]} • Functional operations Function[x, x + k][a] ⟶ a + k (# + k&)[a] ⟶ a + k (r[#1] + s[#2]&)[a, b] ⟶ r[a] + s[b] Map[f, {a, b, c}] ⟶ {f[a], f[b], f[c]} Apply[f, {a, b, c}] ⟶ f[a, b, c] Select[{1, 2, 3, 4, 5}, EvenQ] ⟶ {2, 4} MapIndexed[f, {a, b, c}] ⟶ {f[a, {1}], f[b, {2}], f[c, {3}]} • List manipulation {a, b, c, d} 〚 3 〛 ⟶ c {a, b, c, d} 〚 {2, 4, 3, 2} 〛 ⟶ {b, d, c, b} Take[{a, b, c, d, e}, 2] ⟶ {a, b} Drop[{a, b, c, d, e}, -2] ⟶ {a, b, c} Rest[{a, b, c, d}] ⟶ {b, c, d} ReplacePart[{a, b, c, d}, x, 3] ⟶ {a, b, x, d} Length[{a, b, c}] ⟶ 3 Range[5] ⟶ {1, 2, 3, 4, 5} Table[f[i], {i, 4}] ⟶ {f[1], f[2], f[3], f[4]} Table[f[i, j], {i, 2}, {j, 3}] ⟶ {{f[1, 1], f[1, 2], f[1, 3]}, {f[2, 1], f[2, 2], f[2, 3]}} Array[f, {2, 2}] ⟶ {{f[1, 1], f[1, 2]}, {f[2, 1], f[2, 2]}} Flatten[{{a, b}, {c}, {d, e}}] ⟶ {a, b, c, d, e} Flatten[{{a, {b, c}}, {{d}, e}}, 1] ⟶ {a, {b, c}, {d}, e} Partition[{a, b, c, d}, 2, 1] ⟶ {{a, b}, {b, c}, {c, d}} Split[{a, a, a, b, b, a, a}] ⟶ {{a, a, a}, {b, b}, {a, a}} ListConvolve[{a, b}, {1, 2, 3, 4, 5}] ⟶ {2a + b, 3a + 2b, 4a + 3b, 5a + 4b} Position[{a, b, c, a, a}, a] ⟶ {{1}, {4}, {5}} RotateLeft[{a, b, c, d, e}, 2] ⟶ {c, d, e, a, b} Join[{a, b, c}, {d, b}] ⟶ {a, b, c, d, b} Union[{a, a, c, b, b}] ⟶ {a, b, c} • Transformation rules {a, b, c, d} /. b  p ⟶ {a, p, c, d} {f[a], f[b], f[c]} /. f[a]  p ⟶ {p, f[b], f[c]} {f[a], f[b], f[c]} /. f[x_]  p[x] ⟶ {p[a], p[b], p[c]} {f[1], f[b], f[2]} /. f[x_Integer]  p[x] ⟶ {p[1], f[b], p[2]} {f[1, 2], f[3], f[4, 5]} /. f[x_, y_]  x + y ⟶ {3, f[3], 9} {f[1], g[2], f[2], g[3]} /. f[1] | g[_]  p ⟶ {p, p, f[2], p} • Numerical functions Quotient[207, 10] ⟶ 20 Mod[207, 10] ⟶ 7 Floor[1.45] ⟶ 1 Ceiling[1.45] ⟶ 2 IntegerDigits[13, 2] ⟶ {1, 1, 0, 1} IntegerDigits[13, 2, 6] ⟶ {0, 0, 1, 1, 0, 1} DigitCount[13, 2, 1] ⟶ 3 FromDigits[{1, 1, 0, 1}, 2] ⟶ 13 The Mathematica programs in these notes are formatted in Mathematica StandardForm .

The discovery from the mid-1800s to the mid-1900s of all sorts of elaborate chemical processes in living systems led biologists often to view life as defined by its ability to maintain fixed overall structure while achieving chemical functions such as metabolism.

But in the 1980s it began to be popular to try so-called rational design in which molecules were created that could at least to some extent specifically be computed to have relevant shapes and chemical functions.

And this means for example that it is in general undecidable whether expressions that involve both algebraic and trigonometric functions can be zero for real values of variables, or what the values of integrals are in which such expressions appear as denominators (compare page 916 ).

 [i, j, k] || If[EvenQ[n + i - j],  [i, j], False] ||  [i + 1, j, k], {i, 0, t - 2}, {j, Max[1, n - i], n + i}, {k, 0, ktot - 1}], Table[Map[Function[ z, Outer[!