1 Hence a butterfly inscribed in a quadrilateral satisfies the same relation (1) as a butterfly inscribed in a circle. Equivalently, the conclusion of the theorem indicates that the ratio of the ratios, (AM/IM)/(CN/IN), is the same as the ratio IA/IC, or that the harmonic mean of IC and IM equals the harmonic mean of IA and IN. In either case, if IC = IA, we have IM = IN thereby the analog of the usual butterfly theorem for quadrilaterals.

Matrices with very few non-zero entries cannot have large rank. On the other hand matrices without any zero entries can have rank as low as 1. These simple observations lead us to our main question.… Expand

There is a homeomorphicism between the space of bimatrix games and their equilibrium correspondence that preserves rank, a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component.Expand

Introduction Given an n X n matrix A, it can be useful to have a monic polynomial p(x) of low degree that annihilates A; that is, for which p(A) = 0. For example, higher degree polynomials in A can… Expand

Editor's note. This article illustrates the diversity of geometric techniques that can be brought to bear on a single problem. The author was prompted to examine his ample collection of historical… Expand