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Given a Poisson distribution with rate of change lambda, the distribution of waiting times between successive changes (with k=0) is D(x) = P(X<=x) (1) = 1-P(X>x) (2) = ...

The polynomials G_n(x;a,b) given by the associated Sheffer sequence with f(t)=e^(at)(e^(bt)-1), (1) where b!=0. The inverse function (and therefore generating function) ...

The Hochberg-O'Donnell fish graph is the unit-distance graph on 23 nodes and 42 edges illustrated above. It is triangle-free and is the smallest example of a unit-distance ...

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements f, g, and h of a Lie algebra satisfy [f,f]=0 (1) [f+g,h]=[f,h]+[g,h], (2) and ...

A transformation of the form w=f(z)=(az+b)/(cz+d), (1) where a, b, c, d in C and ad-bc!=0, (2) is a conformal mapping called a linear fractional transformation. The ...

A set A of integers is said to be one-one reducible to a set B (A<<_1B) if there is a one-one recursive function f such that for every x, x in A=>f(x) in B (1) and f(x) in ...

The radical circle of the Stammler circles has center at the nine-point center N, which is Kimberling center X_5. The radius is given by R_S = sqrt(R^2+ON^2) (1) = ...

If x_0 is an ordinary point of the ordinary differential equation, expand y in a Taylor series about x_0. Commonly, the expansion point can be taken as x_0=0, resulting in ...

Compass and straightedge geometric constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, ...

A busy beaver is an n-state, 2-color Turing machine which writes a maximum number Sigma(n) of 1s before halting (Rado 1962; Lin and Rado 1965; Shallit 1998). Alternatively, ...