A completely integrable partial differential equation is one which has a Lax representation, or, more precisely, can be solved via a linear integral equation of Gel'fand-Levitan type, the classic example being the Korteweg-de Vries equation. An ordinary differential equation is of Painlev type if the only singularities of its solutions in the complex plane are poles. It is shown that, under cer...

This paper presents the solution to the following optimization problem: What is the shape of the region that minimizes the average Lp distance between all pairs of points if the area of this region is held fixed? (The Lp-distance between two points x = (x1, x2) and y = (y1, y2) in < 2 is (|x1 − y1| p + |x2 − y2| p).) We use variational methods to determine the shape of the region that minimizes...