Mobile automata [from cellular automata] Given a mobile automaton with rules in the form used on page 887 , a cellular automaton which emulates it can be constructed using MAToCA[rules_] := Append[Flatten[Map[g, rules]], {_, _, x_, _, _}  x] g[{a_, b_, c_}  {d_, e_}] := {{_, a, b + 2, c, _}  d, If[e  1, {a, b + 2, c, _, _}  c + 2, {_, _, a, b + 2, c}  a + 2]} This specific definition assumes that the mobile automaton has two possible colors for each cell; it yields a cellular automaton with four possible colors for each cell.

The patterns below show where BitXor[x, y]  t for successive t and correspond to steps in the "munching squares" program studied on the PDP-1 computer in 1962.

However, the 1D version of the problem is not NP-complete, and in fact there is a specific rather efficient algorithm described on page 954 for solving it.

The chapters were written roughly as follows: Chapter 1: 1991, 1999, 2001; Chapter 2: 1991-2; Chapter 3: 1992; Chapter 4: 1992-3; Chapter 5: 1993; Chapter 6: 1992-3; Chapter 7: 1994-6; Chapter 8: 1994-5, 1997; Chapter 9: 1995-8, 2001; Chapter 10: 1998-9; Chapter 11: 1995; Chapter 12: 1999-2001.

Equivalential calculus Expressions with variables vars are equivalent if they give the same results for Mod[Map[Count[expr, #, {-1}] &, vars], 2] With n variables, there are thus 2 n equivalence classes of expressions (compared to 2 2 n for ordinary logic).

The longest tautology at step t is Nest[(# ⊼ #) ⊼ (# ⊼ p t ) & , p ⊼ (p ⊼ p), t - 1] whose LeafCount grows like 3 t .

(The deterministic Kardar–Parisi–Zhang equation ∂ t u[t, x]  a ∂ xx u[t, x] + 1/2b( ∂ x u[t, x]) 2 yields behavior like Burger's equation, but symmetrical.

Sets of so-called Möbius transformations of the form z  (a z + b)/(c z + d) always yield such patterns (and correspond to so-called modular groups when a d - b c  1 ).

(For a triangular lattice the critical density is exactly 1/2.)

The sound wave turns out to travel at a fraction 1/ √ 2 of the microscopic particle speed.