A five-square theorem

Generation of the Ahlfors Five Islands Theorem

A local Tb Theorem for square functions

We prove a “local” Tb Theorem for square functions, in which we assume only Lq control of the pseudo-accretive system, with q > 1. We then give an application to variable coefficient layer potentials for divergence form elliptic operators with bounded measurable non-symmetric coefficients.

A "Four Integers" Theorem and a "Five Integers" Theorem

The recent exciting results by Bhargava, Conway, Hanke, Kaplansky, Rouse, and Schneeberger concerning the representabiltity of integers by positive integral quadratic forms in any number of variables are presented. These results build on the earlier work of Dickson, Halmos, Ramanujan, and Willerding on quadratic forms. Two results of this type for positive diagonal ternary forms are proved. The...

A Three and Five Color Theorem

Let / be a face of a plane graph G. The Three and Five Color Theorem proved here states that the vertices of G can be colored with five colors, and using at most three colors on the boundary of /. With this result the well-known Five Color Theorem for planar graphs can be strengthened, and a relative coloring conjecture of Kainen can be settled except for a single case which happens to be a par...

Dissecting the square into five congruent parts

We give an affirmative answer to an old conjecture proposed by Ludwig Danzer: there is a unique dissection of the square into five congruent convex tiles. 2010 Mathematics Subject Classification: 52C20 Introduction and notation In the eighties of the last century, Ludwig Danzer conjectured in several conferences that there is a unique dissection of the square into five congruent parts—see Figur...