Particularly with all the successes of science, there has been a great desire to capture the essence of the human condition in abstract scientific

Different current methods asymptotically require slightly different numbers of steps—but all typically at least Exp[Sqrt[Log[n]]] .

Boolean formulas A Boolean function of n variables can always be specified by an explicit table giving values for all 2 n possible inputs.

The patterns are exactly repetitive only when Tan[ θ ]  u/v , where u and v are elements of a primitive Pythagorean triple (so that u , v and Sqrt[u 2 + v 2 ] are all integers, and GCD[u, v]  1 ).

Up to limited m nested sequences can contain all k m possible blocks, and can do so with asymptotically equal frequencies.

And so for example all the state lotteries in the U.S. are currently based on mixing between 10 and 54 balls.

(Note that if the surfaces minimize area like soap films they are slightly curved in all these cases.

Of all possible such networks, most large ones end up being connected.

However, almost all the relevant ideas generated have remained forever controversial, and almost none have become concrete enough to be applied in science or technology. … Logic developed somewhat in medieval times, and in the late 1600s Gottfried Leibniz tried to use it as the foundation for a universal language to capture all systematic thinking. … But by the 1970s it was becoming clear that in almost all cases where programs were successful (notable examples being chess, algebra and autonomous control), they typically worked by following definite algorithms not closely related to general human thinking.

(Especially in antiquity, all sorts of seemingly random phenomena have been used as a basis for fortune telling.) … There are all sorts of situations where in the absence of anything better it is good to use randomness. … In cryptography randomness is used to make messages look typical of all possibilities (see page 598 ).