Note that the length of a minimal DNF representation cannot be considered a reliable measure of the complexity of a function, since among other things, just exchanging the role of black and white can substantially change this length (as in the case of rule 126 versus rule 129).

Examples where I have seen evidence of repeatable randomness as a function of time in published experimental data include temperature differences in thermal convection in closed cells of liquid helium, reaction rates in oxidation of carbon monoxide on catalytic surfaces, and output voltages from firings of excited single nerve cells.

Originally intended as an idealized way to represent structures of functions defined in logic, combinators were actually first introduced in 1920—sixteen years before Turing machines.

And this setup is now directly analogous to the search-and-replace function of a typical text editor.

The songs sometimes seem to be associated with attracting mates, and sometimes with defining territory—but often their function is unclear, even when one bird seems to sing in response to another.

(Attempts were made in the 1980s to study phase transitions as a function of s in analogy to ones in percolation and spin glasses.)

For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law 2Sqrt[1 - x 2 ] UnitStep[1 - x 2 ]/ π while the distribution of spacings between tends to 1/2( π x)Exp[1/4(- π )x 2 ] The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.

The number of such strings containing 2n characters is the n th Catalan number Binomial[2n, n]/(n + 1) (as obtained from the generating function (1 - Sqrt[1 - 4x])/(2x) ).

(The system is also universal with one two-argument function or two one-argument functions.)

In 1887, however, Heinrich Bruns showed that there could be no such quantities expressible as algebraic functions of the positions and velocities of the bodies (in standard Cartesian coordinates). … And even now it remains conceivable that the three-body problem could be solved in terms of more sophisticated standard mathematical functions.