If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution Exp[-(Log[x] - μ ) 2 /(2 σ 2 )]/(Sqrt[2 π ] x σ ) For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution Exp[(x - μ )/ β ] Exp[-Exp[(x - μ )/ β ]]/ β related to the Weibull distribution used in reliability analysis. 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.

Planck length Even in existing particle physics it is generally assumed that the traditional simple continuum description of space must break down at least below about the Planck length Sqrt[ℏ G/c 3 ] ≃ 2 × 10 -35 meters—since at this scale dimensional analysis suggests that quantum effects should be comparable in magnitude to gravitational ones.

However, as was shown in the 1800s, this method can yield only numbers formed by operating on rationals with combinations of Plus , Times and Sqrt . (Thus it is impossible with ruler and compass to construct π and "square the circle" but it is possible to construct 17-gons or other n -gons for which FunctionExpand[Sin[ π /n]] contains only Plus , Times and Sqrt .)

These are related to the autocorrelation function according to Fourier[list] 2  Fourier[ListConvolve[list, list, {1, 1}]]/Sqrt[Length[list]] (See also page 1074 .)

For the Klein–Gordon equation, however, there is an exact solution: u[t, x] = If[x 2 > t 2 , 0, BesselJ[0, Sqrt[t 2 - x 2 ]]]

And in many cases these functions end up trying to prove theorems; so for example FullSimplify[(a + b)/2 ≥ Sqrt[a b], a > 0 && b > 0] must in effect prove a theorem to get the result True .

The pattern generated by rule 150R has fractal dimension Log[2, 3 + Sqrt[17]] - 1 or about 1.83.

Lorentzian spaces In ordinary Euclidean space distance is given by Sqrt[x 2 + y 2 + z 2 ] . In setting up relativity theory it is convenient (see page 1042 ) to define an analog of distance (so-called proper time) in 4D spacetime by Sqrt[c 2 t 2 - x 2 - y 2 - z 2 ] .

The 2D analog of rule 150 yields the patterns below; the fractal dimension of the structure in this case is Log[2, (1 + Sqrt[1 + 4/d]) d] .

In each case the n th element appears at coordinates Sqrt[n] {Cos[n θ],Sin[n θ]} .