The total number of replacements that can be used in the rules of a network mobile automaton and which involve clusters with up to four nodes and have from 1 to 4 dangling connections is {14, 10, 2727, 781} .

Class 1 behavior can never occur in a reversible system.

Case (a) uses the same constraints as in the previous picture, (b) requires every black square and every white square to have exactly two adjacent black squares, and (c) requires every black square to have 3 adjacent black squares and 1 white square, and every white square to have 4 adjacent white squares.

And in fact in the case shown below there are roughly a factor 1.22 more nodes on each successive row—corresponding to overall approximate exponential growth.

In quantum theory the quantization of particle spin implies that any photon hitting a polarizing filter will always either just go through or be absorbed—so that in effect its spin measured relative to the orientation of the polarizer is either +1 or -1. … Now the expected value of the product of the two measured spin values is found just by averaging over ϕ as Integrate[f[ ϕ ] f[ θ - ϕ ], { ϕ , 0, 2 π }]/(2 π ) A version of Bell's inequalities is then that this integral can decrease with θ no faster than θ /(2 π ) - 1 —as achieved when f = Sign . … But for triples of particles, it was noticed in the late 1980s that they can give constraints that force probabilities to be 0 or 1, implying that with the assumptions made, certain configurations of measurement results are simply impossible.

The central region grows by 1 cell every 2 steps on the left and 22 cells every 340 steps on the right.

Like other elliptic modular functions, ModularLambda satisfies f[z]  f[(a + b z)/(c + d z)] with a , b , c , d integers such that a c - b d  1 .

For n > 4 the largest results are 2 n + 2 - 3 , achieved for x = 2 n - 1 with machines 378 and 1351.

But for Sin[a x] + Sin[b x] + Sin[c x] , there are usually zeros off the real axis (even say for a = 1 , b = 3/2 , c = 5/3 ), as shown in the pictures below.

Hashing Given data in the form of sequences of numbers between 0 and k - 1 , a very simple hashing scheme is just to compute FromDigits[Take[list, n], k] . … If m = k s -1 then it turns out that interchanging a pair of adjacent length s blocks in list never affects the result.