Random number generators A fairly small number of different types of random number generators have been used in practice, so it is possible to describe all the major ones here. … Page 962 showed diagrams that represent the evolution for all possible starting values of n . … Essentially all the other generators discussed here have certain linearity properties which allow for fairly complete analysis using traditional mathematical methods.

And in the late 1970s, with online storage of text files becoming common, software compression programs began to be developed, almost all based on adaptive Huffman coding.

For n = 3 this polytope comes close to filling the region of all possible colors, but for no n can it completely fill it—which is why practical displays and printing processes can produce only limited ranges of colors.

All too often, however, inadequate amounts of data have ended up being subjected to elaborate statistical analyses whose results are then blindly assumed to represent definitive scientific conclusions.

(Sometimes it is also possible to recognize microscopic features characteristic of particular kinds of use or wear—and it is conceivable that in the future analysis of trillions of atomic-scale features could reveal all sorts of details of the history of an object.)

For traditional ideas lead to the notion that in this case all our actions must somehow be thought of as the direct result of whatever external causes (over which we have no control) are responsible for the underlying rules in our brains and the environment in which we find ourselves.

But particularly from the discoveries in this book, it seems likely that the very fastest algorithms for many kinds of problems will not in the end have the type of regular structure that characterizes almost all algorithms currently used.

Set theory and predicate logic were proposed as ultimate foundations for all of mathematics (see note below ).

Typical human experience makes small positive integers and simple shapes familiar—so that all human brains are at least well adapted to such constructs.

(If one considers for example theorems about computational issues such as whether Turing machines halt, then it becomes inevitable that to cover more Turing machines one needs more axioms—and to cover all possible machines one needs an infinite set of axioms, that cannot even be generated by any finite set of rules.)