For at least fifty years it has now been quite routine to test many hundreds or thousands of substances in looking, say, for catalysts or drugs with particular functions. … In some compilers searches are occasionally done for optimal sequences of instructions to implement particular simple functions. 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.

History of iterated maps Newton's method from the late 1600s for finding roots of polynomials (already used in specific cases in antiquity) can be thought of as a smooth iterated map (see page 920 ) in which a rational function is repeatedly applied (see page 1101 ). Questions of convergence led in the late 1800s and early 1900s to interest in iteration theory, particularly for rational functions in the complex plane (see page 933 ). … In the 1950s Paul Stein and Stanislaw Ulam did an extensive computer study of various iterated maps of nonlinear functions.

A typical example of the first approach is the Ising model for spin systems in which relative probabilities of sequences are found by multiplying together the results of applying a simple function to blocks of nearby elements in the sequence.

The function of the song is quite unclear.

The pictures below show results obtained as a function of n for various choices of h .

(A corollary to Gödel's proof had in fact already supplied another explicit undecidable problem by implying that no finite procedure based on recursive functions could decide whether a given primitive recursive function is identically 0.)

Equation for the background [in my PDEs] If u[t, x] is independent of x , as it is sufficiently far away from the main pattern, then the partial differential equation on page 165 reduces to the ordinary differential equation u''[t]  (1 - u[t] 2 )(1 + a u[t]) u[0]  u'[0]  0 For a = 0 , the solution to this equation can be written in terms of Jacobi elliptic functions as ( √ 3 JacobiSN[t/3 1/4 , 1/2] 2 ) / (1 + JacobiCN[t/3 1/4 , 1/2] 2 ) In general the solution is (b d JacobiSN[r t, s] 2 )/(b - d JacobiCN[r t, s] 2 ) where r = -Sqrt[1/8 a c (b - d)] s = (d (c - b))/(c (d - b)) and b , c , d are determined by the equation (x - b)(x - c)(x - d)  -(12 + 6 a x - 4 x 2 - 3 a x 3 )/(3a) In all cases (except when -8/3 < a < -1/ √ 6 ), the solution is periodic and non-singular. … For a = 8/3 , the solution can be written without Jacobi elliptic functions, and is given by 3 Sin[Sqrt[5/6] t] 2 /(2 + 3 Cos[Sqrt[5/6] t] 2 )

Examples include cryptography (pages 598 – 606 ), Boolean functions (pages 616 – 619 and 806 – 814 ), user interfaces (page 1102 ) and quantum computing (page 1147 ).

(The function σ above can for example be used to specify the order in which to sample elements in RealDigits[list] ). The total number of possible functions of real numbers is 2 2 ℵ 0 ; the number of continuous such functions (which can always be represented by a list of coefficients for a series) is however only 2 ℵ 0 .

Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2, Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3]) Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2 Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}] (As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .) … Page 147 showed how Sin[x] + Sin[ √ 2 x] has nested features, and these are reflected in the distribution of eigenvalues for ODEs containing such functions.