• Is there a path shorter than some given length that visits all of some set of points in the plane?

The first of these points means that one is concerned with so-called pure group theory—and with finding results valid for all possible groups.

In the simplest case, the cells are all identical, and are laid out in the same orientation in a repetitive array.

If f is commutative ( Orderless ) then all that can ever matter to the value of an element is its number of a 's.

In connection with early work on chaos theory, it was noted that there are some systems that act like "full shifts", in the sense that the set of sequences they generate includes all possibilities—and corresponds to what one would get by starting with any possible number, then successively shifting digits to the left, and at each step picking off the leading digit.

As mentioned in the main text, there were by the late 1950s already all sorts of general-purpose computers on which simulations of cellular automata would have been easy to perform. … Quite disconnected from all this, even in the 1950s, specific types of 2D and 1D cellular automata were already being used in various electronic devices and special-purpose computers.

But in general, even though geodesics are not straight lines their lengths can still be used to define a so-called geodesic distance—which turns out to have all the various properties of a distance discussed on page 1030 . … In higher-dimensional spaces, there are more components, but in general they are all part of the so-called Riemann tensor—a rank-4 tensor introduced by Bernhard Riemann in 1854.

The new kind of science in this book connects in all sorts of ways with mathematics and the existing sciences—and it can be used at an educational level to place some of the fundamental ideas in these areas in a clearer context.

Present-day publicly available cryptographic systems are almost all based on variants of either DES (such as the IDEA system of PGP), linear feedback shift registers or RSA.

And with this definition all standard mathematical functions are computable—even those from chaos theory that excavate digits rapidly.