The theorems are respectively: (1), (2) idempotence (laws of tautology) of And and Or, (3), (4) commutativity of And and Or, (5) law of double negation, (6), (7) absorption (redundancy) laws, (8) law of noncontradiction (definition of False), (9) law of excluded middle (definition of True), (10) de Morgan's law, (11), (12) associativity of And and Or, (13), (14) distributive laws.

For rule 170, it is 1 for both and . For rule 150, it is 1 for and , with all computations done modulo 2. … What the values for these blocks should be can be found by solving a system of linear equations; that a solution must exist can be seen by looking at the de Bruijn network (see page 941 ), with nodes labelled by size b + 2r - 1 blocks, and connections by value differences between size b blocks at the center of the possible size b + 2r blocks.

To see the consequences of such constraints consider breaking a sequence of colors into blocks of length n , with each block overlapping by n - 1 cells with its predecessor, as in Partition[list, n, 1] . … The possible sequences of length n blocks that can occur are conveniently represented by possible paths by so-called de Bruijn networks, of the kind shown for k = 2 and n = 2 through 5 below. … But the crucial point is that since there are only k n - 1 nodes in the network, then if any infinite path is possible, there must be such a path that visits the same node and thus repeats itself after at most k n - 1 cells.

Semigroups were defined by Jean-Armand de Séguier in 1904, and beginning in the late 1920s a variety of algebraic results about them were found.

Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables. When s max  ∞ , f[s] = a s Sin[s] yields 2D shapes that are basically nested, with pieces overlapping for Abs[a] < 1 .

The first one on the bottom (with 63 comparisons) has a nested structure and uses the method invented by Kenneth Batcher in 1964: Flatten[Reverse[Flatten[With[{m = Ceiling[Log[2, n]] - 1}, Table[With[{d = If[i  m, 2 t , 2 i + 1 - 2 t ]}, Map[ {0, d} + # &, Select[Range[n - d], BitAnd[# - 1, 2 t ]  If[i  m, 0, 2 t ] &]]], {t, 0, m}, {i, t, m}]], 1]], 1] The second one on the bottom also uses 63 comparisons, while the last one is the smallest known for n = 16 : it uses 60 comparisons and was invented by Milton Green in 1969. For n ≤ 16 the smallest numbers of comparisons known to work are {0, 1, 3, 5, 9, 12, 16, 19, 25, 29, 35, 39, 45, 51, 56, 60} . … Various structures such as de Bruijn and Cayley graphs can be used as the basis for sorting networks, though it is my guess that typically the smallest networks for given n will have no obvious regularity.

Nonlinear feedback shift registers Linear feedback shift registers of the kind discussed on page 974 can be generalized to allow any function f (note the slight analogy with cyclic tag systems): NLFSRStep[f_, taps_, list_] := Append[Rest[list], f[list 〚 taps 〛 ]] With the choice f=IntegerDigits[s, 2, 8] 〚 8 - # . {4, 2, 1} 〛 & and taps = {1, 2, 3} this is essentially a rule s elementary cellular automaton. … And as noted by Nicolaas de Bruijn in 1946 there are 2 2 n - 1 -n such paths with length 2 n , and thus this number of functions f out of the 2 2 n possible must yield sequences of maximal length. … k n - 1 /k n .)

The formula for the Gaussian distribution was derived by Abraham de Moivre around 1733 in connection with theoretical studies of gambling.

Ciphers of the type shown on page 599 were introduced in the 1500s, notably by Blaise de Vigenère ; systematic methods for their cryptanalysis were developed in the mid-1800s and early 1900s.

And this means that if black is represented by 1 and white by 0, one can then give an explicit formula for the color of the square at position x on row y : it is simply (1 - (-1)^Binomial[y, x])/2 . … This can be determined either from Mod[a, 2] or equivalently from (1 - (-1) a )/2 or Sin[ π /2 a] 2 . The succession of polynomials above can be obtained by expanding the generating functions 1/(1 - (1 + x) y) and 1/(1 - (1 + x + x 2 ) y) .