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The falling factorial (x)_n, sometimes also denoted x^(n__) (Graham et al. 1994, p. 48), is defined by (x)_n=x(x-1)...(x-(n-1)) (1) for n>=0. Is also known as the binomial ...

A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight ...

The unique (modulo rotations) scalene triangle formed from three vertices of a regular heptagon, having vertex angles pi/7, 2pi/7, and 4pi/7. There are a number of amazing ...

Maximize the distance a Jeep can penetrate into the desert using a given quantity of fuel. The Jeep is allowed to go forward, unload some fuel, and then return to its base ...

Let kappa_1 and kappa_2 be the principal curvatures, then their mean H=1/2(kappa_1+kappa_2) (1) is called the mean curvature. Let R_1 and R_2 be the radii corresponding to ...

The nine-point center N (sometimes instead denoted F) is the center of the nine-point circle. It has equivalent triangle center functions alpha_5 = cos(B-C) (1) alpha_5 = ...

The line joining the three collinear points of intersection of the extensions of corresponding sides in perspective triangles, also called the perspective axis or homology ...

Quicksort is the fastest known comparison-based sorting algorithm (on average, and for a large number of elements), requiring O(nlgn) steps. Quicksort is a recursive ...

Given a simplex of unit content in Euclidean d-space, pick n>=d+1 points uniformly and independently at random, and denote the expected content of their convex hull by ...

A triangle center (sometimes simply called a center) is a point whose trilinear coordinates are defined in terms of the side lengths and angles of a triangle and for which a ...