One way to do this is by using the Gödel number Product[Prime[i]^list 〚 i 〛 , {i, Length[list]}] . An alternative is to use the Chinese Remainder Theorem. … Based on this LE[list_] := Module[{n = Length[list], i = Max[MapIndexed[ #1 - #2 &, PrimePi[list]]] + 1}, CRT[PadRight[ list, n + i], Join[Array[Prime[i + #] &, n], Array[Prime, i]]]] will yield a number x that can be decoded into a list of length n using essentially the so-called Gödel β function Mod[x, Prime[Rest[NestList[NestWhile[# + 1 &, # + 1, Mod[x, Prime[#]]  0 &] &, 0, n]]]]

The density of white squares is asymptotically 6/ π 2 ≃ 0.61 . … (This follows from the Chinese Remainder Theorem.) … The densities of such blocks are respectively about 0.002, 2 × 10 -6 and 10 -14.

This yields a chord such as Play[Evaluate[Apply[Plus, Flatten[Map[Sin[1000 # t] &, N[2 1/12 ]^Position[list, 1]]]]], {t, 0, 0.2}] A sequence of such chords can sometimes provide a useful representation of cellular automaton evolution.

In plants, cells typically expand—normally through intake of water—only for a limited period, after which the cellulose in their walls crystallizes to make them quite rigid. … Often the very tip of a stem consists of a single cell in the shape of an inverted tetrahedron, and in lower plants such as mosses this is essentially the only cell that divides. In flowering plants, cell division normally occurs around the edge of a region of size 0.2-1 mm containing many tens of cells.

The generating function for the sequence (with 0 replaced by -1) satisfies f[z]  (1 - z) f[z 2 ] , so that f[z]  Product[1 - z 2 n , {n, 0, ∞ }] . (Z transform or generating function methods can be applied directly only for substitution systems with rules such as {1  list, 0  1 - list} .) … It is related to the product of sawtooth functions given by Product[Abs[Mod[2 s ω , 2, -1]], {s, t}] .

Arithmetic systems [emulating register machines] Given the program for a register machine with nr registers in the form on page 896 , an arithmetic system which emulates it can be obtained from RMToAS[prog_, nr_] := With[{p = Length[prog], g = Product[Prime[j], {j, nr}]}, {p g, Sort[Flatten[MapIndexed[ With[{n = First[#2] - 1}, #1 /. {i[r_]  Table[n + j p  (1 + n Prime[r] (-n + #) &), {j, 0, g - 1}], d[r_, k_]  Table[n + j p  If[Mod[j, Prime[r]]  0, -1 + k + (-n + #)/Prime[r] &, # + 1 &], {j, 0, g - 1}]}] &, prog]]]}] The rules for the arithmetic system are represented so that the system from page 122 becomes for example {2, {0  (3 #/2 &), 1  (3 (# + 1)/2 &)}} . … The evolution of the arithmetic system is given by ASEvolveList[{n_, rules_}, init_, t_] := NestList[(Mod[#, n] /. rules)[#] &, init, t] Given a value m obtained in the evolution of the arithmetic system, the state of the register machine to which it corresponds is {Mod[m, p] + 1, Map[Last, FactorInteger[ Product[Prime[i], {i, nr}] Quotient[m, p]]] - 1} Note that it is possible to have each successive step involve only multiplication, with no addition, at the cost of using considerably larger numbers overall.

Cells at distances 2 and 3 enter with negative weights— -0.4 per cell for the first rule, and -0.2 for the second.

Particle masses The measured masses of known elementary particles in units of GeV (roughly equal to the proton mass) are: photon: 0, electron: 0.000510998902; muon: 0.1056583569; τ lepton: 1.77705; W : 80.4; Z : 91.19. … For all of them their confinement contributes perhaps 0.3 GeV of effective mass. Then there is also a direct mass: gluons 0; u : ~0.005; d ~0.01; s : ~0.2; c : 1.3; b : 4.4; t : 176 GeV.

Rate equations In standard chemical kinetics one assumes that molecules are uniformly distributed in space, so that the rates for particular reactions are proportional to the products of the densities of the molecules that react in them. … (For the cellular automaton on page 339 the simple condition for equilibrium is p  p 2 (3 - 2p) , which correctly implies that 0, 1/2 and 1 are possible equilibrium densities.)

This was for example done by Julia Robinson in 1949 with Δ (or a + 1 ) and Mod[a, b]  0 . And in the 1990s Ivan Korec and others showed that it could be done just with Mod[Binomial[a + b, a], k] with k = 6 or any product of primes—and that it could not be done with k a prime or prime power. … Korec showed that finding elements in the nested pattern produced by the k = 3 cellular automaton with rule {{1, 1, 3}, {2, 2, 1}, {3, 3, 2}} 〚 #1, #2 〛 & (compare page 886 ) was also enough.