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A binary bracketing is a bracketing built up entirely of binary operations. The number of binary bracketings of n letters (Catalan's problem) are given by the Catalan numbers ...

A quaternion with complex coefficients. The algebra of biquaternions is isomorphic to a full matrix ring over the complex number field (van der Waerden 1985).

Consider the probability Q_1(n,d) that no two people out of a group of n will have matching birthdays out of d equally possible birthdays. Start with an arbitrary person's ...

A divisor d of a positive integer n is biunitary if the greatest common unitary divisor of d and n/d is 1. For a prime power p^y, the biunitary divisors are the powers 1, p, ...

An extremal graph in which the forced triangles are all the same color. Call R the number of red monochromatic forced triangles and B the number of blue monochromatic forced ...

The finite zeros of the derivative r^'(z) of a nonconstant rational function r(z) that are not multiple zeros of r(z) are the positions of equilibrium in the field of force ...

An n-bridge knot is a knot with bridge number n. The set of 2-bridge knots is identical to the set of rational knots. If L is a 2-bridge knot, then the BLM/Ho polynomial Q ...

The area of the shadow of a body on a plane, also called the "outer quermass."

A Tschirnhausen transformation can be used to algebraically transform a general quintic equation to the form z^5+c_1z+c_0=0. (1) In practice, the general quintic is first ...

Any continuous function G:B^n->B^n has a fixed point, where B^n={x in R^n:x_1^2+...+x_n^2<=1} is the unit n-ball.