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An (n,k) fountain is an arrangement of n coins in rows such that exactly k coins are in the bottom row and each coin in the (i+1)st row touches exactly two in the ith row. ...

A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds: 1. If alpha is surjective, and beta and delta are injective, then ...

Let Omega be an open, bounded, and connected subset of R^d for some d and let dx denote d-dimensional Lebesgue measure on R^d. In functional analysis, the Friedrichs ...

A set of algebraic invariants for a quantic such that any invariant of the quantic is expressible as a polynomial in members of the set. Gordan (1868) proved the existence of ...

At rational arguments p/q, the digamma function psi_0(p/q) is given by psi_0(p/q)=-gamma-ln(2q)-1/2picot(p/qpi) +2sum_(k=1)^([q/2]-1)cos((2pipk)/q)ln[sin((pik)/q)] (1) for ...

The generalized diameter is the greatest distance between any two points on the boundary of a closed figure. The diameter of a subset E of a Euclidean space R^n is therefore ...

A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. ...

Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any ...

The golden angle is the angle that divides a full angle in a golden ratio (but measured in the opposite direction so that it measures less than 180 degrees), i.e., GA = ...

A general quartic surface defined by x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c=0 (1) (Gray 1997, p. 314). The above two images correspond to (a,b,c)=(0,0,-1), and ...