But while this axiom is convenient in simplifying work in set theory it has not been found generally useful in mathematics, and is normally considered optional at best.

The number of operations m[n] needed to multiply them increases just slightly more rapidly than n (see page 1093 ). … Roots of polynomials can thus almost always be found with NSolve in about Log[n] m[n] operations. … The best-known algorithms for evaluating Zeta[1/2 +  x] (see page 918 ) to fixed precision take roughly √ x operations—or 2 n/2 operations if x is an n -digit integer.

The best current experiments could successfully detect radio emission at the level now produced on Earth only from about 10 light years away—or from about the nearest 10 stars. … Many SETI experiments look for signals in the so-called "water hole" between the 1420 MHz frequency associated with the 21 cm line of hydrogen and the 1720 MHz frequency associated with hydroxyl (OH).

But it was not until the 1980s—perhaps particularly following some of my work—that it began to be more widely realized that Church's Thesis should best be considered a statement about nature and about the kinds of computations that can be done in our universe.

Nevertheless, a number of interesting systems did emerge, the best known being the idealized sandpile model from the 1987 work of Per Bak , Chao Tang and Kurt Wiesenfeld . … With a total initial s value of m , the number of steps before a fixed point is reached seems to increase roughly like m 2 .

He argued that if the time evolution of a single state were to visit all other states in the ensemble—the so-called ergodic hypothesis—then averaged over a sufficiently long time a single state would behave in a way that was typical of the ensemble.

If one draws a more general curve on a page, one can define its curvature at every point by seeing what size of circle fits it best at that point—or equivalently what the coefficients are in a quadratic approximation. … One is based on looking at infinitesimal vectors u , v and w and asking how much w differs when transported two ways around the edges of a parallelogram, from x to x + u + v via x + u and via x + v . In ordinary flat space there is no difference, but in general the difference is a vector that is defined to be Riemann . u . v. w .

The repetition period for a generator with rule n  Mod[a n, m] is given (for a and m relatively prime) by MultiplicativeOrder[a, m] . … In general the maximum period is CarmichaelLambda[m] , where the value m - 1 can be achieved for prime m . … Note that if the period is equal to its absolute maximum of m , then every possible n is always visited, whatever n one starts from.

The best-known way in which cellular automata were introduced (and which eventually led to their name) was through work by John von Neumann in trying to develop an abstract model of self-reproduction in biology—a topic which had emerged from investigations in cybernetics.