Generating textures As discussed on page 217 , it is in general difficult to find 2D patterns which at all points match some definite set of templates. … (All the examples shown turn out to correspond to ordinary, sequential and reversible cellular automata seen elsewhere in this book.)

For s = 2 , k = 2 the largest results for all inputs of sizes 0 to 4 are {7, 17, 31, 49, 71} , all obtained with machine 1447.

But I suspect that essentially all of the various phenomena that we have observed in discrete systems in the past several chapters can in fact also be found even in continuous systems with fairly simple rules.

In the examples above, two variables, p and q , are used, and in each case the behavior obtained with all four possible combinations of values for p and q are shown.

Proofs of axiom systems One way to prove that an axiom system can reproduce all equivalences for a given operator is to show that its axioms can be used to transform any expression to and from a unique standard form. … Given a particular axiom system that one knows reproduces all equivalences for a given operator one can tell whether a new axiom system will also work by seeing whether it can be used to derive the axioms in the original system. … (Note that even if it works for all finite k this does not establish its validity.)

This is nonzero when all prime factors of n of the form 4k + 3 appear with even exponents. … Seven cubes appear to suffice for all but 17 numbers, the last of which is 455; four cubes may suffice for all but 113936676 numbers, the last of which is 7373170279850. … (e) Goldbach's Conjecture has been verified for all even numbers up to 4*10 14.

Scope of existing sciences One might imagine that physics would for example concern itself with all aspects of physical systems, biology with all aspects of biological systems, and so on.

As mentioned on page 938 , through major mathematical effort, a complete classification of all finite so-called simple groups that in effect have no factors is known. … But this classification does not immediately provide a practical way to enumerate all possible groups.

If all blocks occur with an equal probability of 2 -b , then h takes on its maximum possible value of b . … For certainly h does not in general give the length of the shortest possible description of the data; all it does is to give the shortest length of description that is obtained by treating successive blocks as if they occur with independent probabilities. … Huffman coding with a large number of codewords will approach this if all the p[i] are powers of 1/2.

If all events occurred in a definite sequence in time, this would define a total linear ordering for them. … The collection of all events that will lead to a given set of events (the union of their past light cones) is known as the filter of that set. … There can also be sets of elements between which no ordering relations at all are defined, and these are called antichains.