Mechanisms that generally seem able to give α ≃ 1 include random walks with exponential waiting times, power-law distributions of step sizes (Lévy flights), or white noise variations of parameters, as well as random processes with exponentially distributed relaxation times (as from Boltzmann factors for uniformly distributed barrier heights), fractional integration of white noise, intermittency at transitions to chaos, and random substitution systems.

Intrinsically defined curves With curvature given by a function f[s] of the arc length s , explicit coordinates {x[s], y[s]} of points are obtained from (compare page 1048 ) NDSolve[{x'[s]  Cos[ θ [s]], y'[s]  Sin[ θ [s]], θ '[s]  f[s], x[0]  y[0]  θ [0]  0}, {x, y, θ }, {s, 0, s max }] For various choices of f[s] , formulas for {x[s], y[s]} can be found using DSolve : f[s] = 1: {Sin[ θ ], Cos[ θ ]} f[s] = s: {FresnelS[ θ ], FresnelC[ θ ]} f[s] = 1/ √ s : √ θ {Sin[ √ θ ], Cos[ √ θ ]} f[s] = 1/s: θ {Cos[Log[ θ ]], Sin[Log[ θ ]]} f[s] = 1/s 2 : θ {Sin[1/ θ ], Cos[1/ θ ]} f[s] = s n : result involves Gamma[1/n, ±  θ n/n ] f[s] = Sin[s] : result involves Integrate[Sin[Sin[ θ ]], θ ] , expressible in terms of generalized Kampé de Fériet hypergeometric functions of two variables.

The perceived color of light with a given wavelength distribution is basically determined by the three numbers obtained by integrating these responses.

And in recent years—notably in the building of Mathematica—optimal algorithms for operations such as function evaluation and numerical integration have sometimes been found through searches.

In the 1970s they also became widely used for high-dimensional numerical integration, notably for Feynman diagram evaluation in quantum electrodynamics.

Now the expected value of the product of the two measured spin values is found just by averaging over ϕ as Integrate[f[ ϕ ] f[ θ - ϕ ], { ϕ , 0, 2 π }]/(2 π ) A version of Bell's inequalities is then that this integral can decrease with θ no faster than θ /(2 π ) - 1 —as achieved when f = Sign .

If the coordinates along a path are given by an expression s (such as {t, 1 + t, t 2 } ) that depends on a parameter t , and the metric at position p is g[p] , then the length of a path turns out to be Integrate[Sqrt[ ∂ t s . g[s] . ∂ t s], {t, t 1 , t 2 }] and geodesics then correspond to paths that extremize this quantity.