String overlaps The total numbers of strings with length n and k colors that cannot overlap themselves are given by a[0] = 1; a[n_] := k a[n - 1] - If[EvenQ[n], a[n/2], 0] Up to reversal and interchange of A and B , the first few overlap-free strings with 2 colors are A , AB , AAB , AAAB , AABB .

Finding layouts [for networks] One way to lay out a network g so that network distances in it come as close as possible to ordinary distances in d -dimensional space, is just to search for values of the x[i, k] which minimize a quantity such as With[{n = Length[g]}, Apply[Plus, Flatten[(Table[Distance[g, {i, j}], {i, n}, {j, n}] 2 - Table[ Sum[(x[i, k] - x[j, k]) 2 , {k, d}], {i, n}, {j, n}]) 2 ]]] using for example FindMinimum starting say with x[1, _]  0 and all the other x[_, _]  Random[] . … If one ignores all constraints beyond network distance 1, then one is in effect just trying to build the network out of identical rigid rods. It turns out that this is almost always possible even in 2D (though not in 1D); the only exception is the tetrahedron network.

Undecidability in cellular automata For 1D cellular automata, almost all questions about ultimate limiting behavior are undecidable, even ones that ask about average properties such as density and entropy. … Questions about behavior after a finite number of steps, even with infinite initial conditions, tend to be decidable for 1D cellular automata, and related to regular languages (see page 957 ).

For prime k , each cycle (except all 0's) corresponds to a term in the product Factor[x k n - 1 - 1, Modulus  k] .

The specific form of the continuous generalization of the modulo 2 function used is λ [x_] := Exp[-10 (x - 1) 2 ] + Exp[-10 (x - 3) 2 ] Each cell in the system is then updated according to λ [a + c] for rule 90, and λ [a + b + c + b c] for rule 30. Perturbations of size δ are then added using v + Sign[v - 1/2] Random[] δ .

If it has 1, 2 or 4 black neighbors then it stays the same color as it was before, and if it has 5 or more black neighbors, then it becomes white on the next step (code number 746).

Typical compression ratios currently achieved for text are around 3:1, for line diagrams and text images around 3:1, and for photographic images around 2:1 lossless, and 20:1 lossy.

It is known that Exp[n] and Log[n] for whole numbers n (except 0 and 1 respectively) are transcendental. It is also known for example that Gamma[1/3] and BesselJ[0, n] are transcendental. … One based on gradual extension of work by Richard Stoneham from 1971 is that numbers of the form Sum[1/(p n b p n ), {n, ∞ }] for prime p > 2 are normal in base b (for GCD[b, p]  1 ), and are transcendental.

The amount of this time dilation is given by the classic relativistic formula 1/Sqrt[1-v 2 /c 2 ] , where v/c is the ratio of the speed of the clock to the speed of light.

Thus, for example, one might compute the fraction of squares in a given sequence that are black, and compare this to 1/2 . Or one might compute the frequency with which more than two consecutive black squares occur together, and compare this with the value 1/4 obtained by averaging over all possible sequences.