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The spherical harmonics Y_l^m(theta,phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. Some ...

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres ...

The uniform polyhedra are polyhedra consisting of regular (possibly polygrammic) faces of equal edge length whose polyhedron vertices are all symmetrically equivalent. The ...

In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the ...

The so-called explicit formula psi(x)=x-sum_(rho)(x^rho)/rho-ln(2pi)-1/2ln(1-x^(-2)) gives an explicit relation between prime numbers and Riemann zeta function zeros for x>1 ...

rho_(2s)(n)=(pi^s)/(Gamma(s))n^(s-1)sum_(p,q)((S_(p,q))/q)^(2s)e^(2nppii/q), where S_(p,q) is a Gaussian sum, and Gamma(s) is the gamma function.

e is transcendental.

In the American system, 10^(51).

Define E(x;q,a)=psi(x;q,a)-x/(phi(q)), (1) where psi(x;q,a)=sum_(n<=x; n=a (mod q))Lambda(n) (2) (Davenport 1980, p. 121), Lambda(n) is the Mangoldt function, and phi(q) is ...

The largest cube dividing a positive integer n. For n=1, 2, ..., the first few are 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, ... (OEIS A008834).