The fractal dimension of the (d + 1) -dimensional structure formed from all black cells is Log[2, 1 + 2d] .

[Underlying] structure of Mathematica Beneath all the sophisticated capabilities of Mathematica lies a remarkably simple basic structure.

But because such axioms can be described by schemas they must all have similar forms, so that even though the definition in the main text suggests that each corresponds to an interesting theorem these theorems are not in a sense independently interesting.

And as Garrett Birkhoff showed in 1935, any equivalence between expressions that holds for all possible forms of operator must have a finite proof using just these rules.

In many places in the book—especially these notes—I discuss all sorts of specific problems and issues of direct relevance to current computer science.

The result of all this has been that remarkably few truly meaningful computer experiments have ended up ever being done.

And indeed, by now I have come to trust the correctness of conclusions based on simple systematic computer experiments much more than I trust all but the simplest proofs.

These numbers can also be obtained as the coefficients of x n in the series expansion of x ∂ x Log[ ζ [m, x]] , with the so-called zeta function, which is always a rational function of x , given by ζ [m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x] and corresponds to the product over all cycles of 1/(1 - x n ) .

Random causal networks If one assumes that there are events at random positions in continuous spacetime, then one can construct an effective causal network for them by setting up connections between each event and all events in its future light cone—then deleting connections that are redundant in the sense that they just provide shortcuts to events that could otherwise be reached by following multiple connections.

It also implies that all 2 t possible single columns of t cells can be generated from some initial condition. Not all 4 t pairs of adjacent columns can occur, however. … Given two complete adjacent columns page 601 shows how all columns any distance to the left can be found.