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Let V!=(0) be a finite dimensional vector space over the complex numbers, and let A be a linear operator on V. Then V can be expressed as a direct sum of cyclic subspaces.

The four planes determined by the four altitudes of a tetrahedron and the orthocenters of the corresponding faces pass through the Monge point of the tetrahedron.

The middle levels conjecture, also known as revolving door conjecture, posits that the middle layer graph has a Hamilton cycle for every n>=1. The conjecture was proved by ...

A necessary and sufficient condition that [alpha^'] should be comparable with [alpha] for all positive values of the a is that one of (alpha^') and (alpha) should be ...

A surface which is homeomorphic to a finite collection of spheres, each with a finite number of handles, cross-handles, cross-caps, and perforations. A preliminary version of ...

The portion of a surface left when an open disk is removed from it.

The polyhedral formula generalized to a surface of genus g, V-E+F=chi(g) where V is the number of polyhedron vertices, E is the number of polyhedron edges, F is the number of ...

Polynomial identities involving sums and differences of like powers include x^2-y^2 = (x-y)(x+y) (1) x^3-y^3 = (x-y)(x^2+xy+y^2) (2) x^3+y^3 = (x+y)(x^2-xy+y^2) (3) x^4-y^4 = ...

The volume of a prismatoid is equal to the sum of the volumes of a pyramid, a wedge, and a parallelepiped.

A proposition is a mathematical statement such as "3 is greater than 4," "an infinite set exists," or "7 is prime." An axiom is a proposition that is assumed to be true. With ...