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If all 2 b possible blocks of length b occur with equal probability, then the Huffman codewords will consist of blocks equivalent to the original ones.

Note (a) for Statistical Analysis

And in the late 1980s, building on work of mine from 1984 (described on page 276 ), James Crutchfield made a study of such models in which he defined the complexity of a model to be equal to -p Log[p] summed over all connections in the network.

Note (a) for Relations to Other Areas

The notes to almost all chapters of the book contain a great many new mathematical results, mostly emerging from my analysis of some of the simpler behavior considered in the book.

Note (g) for Continuous Cellular Automata

In all cases the rules have been at least slightly more complicated than the ones I consider here, and behavior starting from simple initial conditions does not appear to have been studied before.

Note (f) for Continuous Cellular Automata

On page 160 the effect is much larger, and almost all the pictures would be completely wrong—with the notable exception of the one that shows localized structures.

Note (d) for Partial Differential Equations

If all equations are of the form a'[t]  f[a[t], b[t],…] , etc. then it is known for example that it is necessary to have at least three equations in order to get behavior that is not ultimately fixed or repetitive.

Note (b) for Chaos Theory and Randomness from Initial Conditions

But the problem is that it is hard to be sure that the system really is in the same state—and that there are not all sorts of large differences that do not happen to have been observed.)

Note (a) for Space, Time and Relativity

But its predictions like E = m c 2 were immediately applied for example to radioactivity, and soon it came to be assumed that the theory would work for any system at all—unless it involved gravity.

How the Discoveries in This Chapter Were Made

And it is this that means that even if we do not know all the details of what is inside some specific system in nature, we can still potentially make fundamental statements about its overall behavior.

Relations to Other Areas

One consequence is a dramatic broadening of the domain to which computational ideas can be applied—in particular to include all sorts of fundamental questions about nature and about mathematics.