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Inversion is the process of transforming points P to a corresponding set of points P^' known as their inverse points. Two points P and P^' are said to be inverses with ...

Given a group of n men arranged in a circle under the edict that every mth man will be executed going around the circle until only one remains, find the position L(n,m) in ...

In 1611, Kepler proposed that close packing (either cubic or hexagonal close packing, both of which have maximum densities of pi/(3sqrt(2)) approx 74.048%) is the densest ...

The Kirchhoff index Kf, also simply called the resistance and denoted R (Lukovits et al. 1999), of a connected graph G on n nodes is defined by ...

The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix ...

A symmetry of a knot K is a homeomorphism of R^3 which maps K onto itself. More succinctly, a knot symmetry is a homeomorphism of the pair of spaces (R^3,K). Hoste et al. ...

The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions ...

The M_(22) graph, also known as the 77-graph, is a strongly regular graph on 77 nodes related to the Mathieu group M_(22) and to the Witt design. It is illustrated above in ...

Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as x=(u,v,h(u,v)) therefore satisfies Lagrange's equation, ...

Multiply all the digits of a number n by each other, repeating with the product until a single digit is obtained. The number of steps required is known as the multiplicative ...