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Willans' formula is a prime-generating formula due to Willan (1964) that is defined as follows. Let F(j) = |_cos^2[pi((j-1)!+1)/j]_| (1) = {1 for j=1 or j prime; 0 otherwise ...

Wolfram's iteration is an algorithm for computing the square root of a rational number 1<=r<4 using properties of the binary representation of r. The algorithm begins with ...

The h-statistic h_r is the unique symmetric unbiased estimator for a central moment of a distribution <h_r>=mu_r. (1) In addition, the variance var(h_r)=<(h_r-mu_r)^2> (2) is ...

A q-analog of Zeilberger's algorithm.

An array is a "list of lists" with the length of each level of list the same. The size (sometimes called the "shape") of a d-dimensional array is then indicated as ...

One of the Zermelo-Fraenkel axioms, also known as the axiom of regularity (Rubin 1967, Suppes 1972). In the formal language of set theory, it states that x!=emptyset=> exists ...

Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points. The intersections of two circles determine a line known as the radical ...

Clebsch-Gordan coefficients are mathematical symbol used to integrate products of three spherical harmonics. Clebsch-Gordan coefficients commonly arise in applications ...

The probability that a random integer between 1 and x will have its greatest prime factor <=x^alpha approaches a limiting value F(alpha) as x->infty, where F(alpha)=1 for ...

Let the elliptic modulus k satisfy 0<k^2<1, and the Jacobi amplitude be given by phi=amu with -pi/2<phi<pi/2. The incomplete elliptic integral of the first kind is then ...