If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution Exp[-(Log[x] - μ ) 2 /(2 σ 2 )]/(Sqrt[2 π ] x σ ) For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution Exp[(x - μ )/ β ] Exp[-Exp[(x - μ )/ β ]]/ β related to the Weibull distribution used in reliability analysis. For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law 2Sqrt[1 - x 2 ] UnitStep[1 - x 2 ]/ π while the distribution of spacings between tends to 1/2( π x)Exp[1/4(- π )x 2 ] The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.

In 1909 Axel Thue showed that any equation of the form p[x, y]  a , where p[x, y] is a homogeneous irreducible polynomial of degree at least 3 (such as x 3 + x y 2 + y 3 ) can have only a finite number of integer solutions. … Over the history of number theory the sophistication of equations for which proofs of no solutions can be given has gradually increased—though even now it is state of the art to show say that x  y  1 is the only solution to x 2  3 y 4 - 2 . … Writing equations in the form p[x 1 , x 2 , …, x n ]  0 the distribution of values of p will in general be complicated (see page 1161 ), but as a first approximation one can try taking it to be purely random.

Multivalued logic As noted by Jan Łukasiewicz and Emil Post in the early 1920s, it is possible to generalize ordinary logic to allow k values Range[0, 1, 1/(k - 1)] , say with 0 being False , and 1 being True . … Axiom systems can be set up for multivalued logic, but they are presumably more complicated than for ordinary k = 2 logic. … Often—as in the work of George Boole in 1847—a continuum of values between 0 and 1 are taken to represent probabilities of events, and this is the basis for the field of fuzzy logic popular since the 1980s.

A particular case on which many studies have been done is the so-called iterated Prisoner's Dilemma, in which two players make a sequence of choices a and b to "cooperate" ( 1 ) or "defect" ( 2 ), each trying to maximize their score m 〚 a, b 〛 with m = {{1, -1}, {2, 0}} . … In the late 1980s similar studies were done on processes such as auctions (cf page 1015 ), and in the late 1990s on games such as Rock, Paper, Scissors (RoShamBo) (with m = {{0, -1, 1}, {1, 0, -1}, {-1, 1, 0}} ).

3D class 4 [cellular automaton] rules With a cubic lattice of the type shown on page 183 , and with updating rules of the form LifeStep3D[{p_, q_, r_}, a_List] := MapThread[If[ #1  1 && p ≤ #2 ≤ q || #2  r, 1, 0]&, {a, Sum[RotateLeft[ a, {i, j, k}], {i, -1, 1}, {j, -1, 1}, {k, -1, 1}] - a}, 3] Carter Bays discovered between 1986 and 1990 the three examples {5, 7, 6} , {4, 5, 5} , and {5, 6, 5} .

Generating functions [for nested patterns] A convenient algebraic way to describe a sequence of numbers a[n] is to give a generating function Sum[a[n] x n , {n, 0, ∞ }] . 1/(1 - x) thus corresponds to the constant sequence and 1/(1 - x - x 2 ) to the Fibonacci sequence (see page 890 ). A 2D array can be described by Sum[a[t, n] x n y t , {n, - ∞ , ∞ }, {t, - ∞ , ∞ }] . The array for rule 60 is then 1/(1- (1 + x) y) , for rule 90 1/(1 - (1/x + x) y) , for rule 150 1/(1 - (1/x + 1 + x) y) and for second-order reversible rule 150 (see page 439 ) 1/(1 - (1/x + 1 + x) y - y 2 ) .

If the coefficients inside all the sine functions are rational, then going from t = 0 to t = 2 π Apply[LCM, Map[Denominator, list]] yields a closed curve.

Cyclic tag systems which allow any value for each element can be obtained by adding the rule CTStep[{{r_, s___}, {n_, a___}}] := {{s, r}, Flatten[{a, Table[r, {n}]}]} The leading elements in this case can be obtained using CTListStep[{rules_, list_}] := {RotateLeft[rules, Length[list]], With[{n = Length[rules]}, Flatten[Apply[Table[#1, {#2}] &, Map[Transpose[ {rules, #}] &, Partition[list, n, n, 1, 0]], {2}]]]}

However, with the rule n  If[EvenQ[n], 3n/2, Round[3n/4]] it is always possible to go backwards by the rule n  If[Mod[n,3]  0, 2n/3, Round[4n/3]] The picture shows the number of base 10 digits in numbers obtained by backward and forward evolution from n = 8 . … But apart from these cycles, the numbers produced always seem to grow without bound at an average rate of 3/(2 √ 2 ) in the forward direction, and 2 4 1/3 /3 in the backward direction (at least all numbers up to 10,000 grow to above 10 100 ).

For rule 126, the density after many steps is still 1/2. For rule 22, it is approximately 0.35095. … In the algebraic representation discussed on page 869 , rule 22 is Mod[p + q + r + p q r, 2] , rule 126 is Mod[(p + q)(q + r) + (p + r), 2] , rule 150 is Mod[p + q + r, 2] and rule 182 is Mod[p r (1 + q) + (p + q + r), 2] .