In this paper we give a necessary condition and a sufficient condition for a graph to be a pairwise compatibility graph (PCG). Let G be a graph and let Gc be the complement of G. We show that if Gc has two disjoint chordless cycles then G is not a PCG. On the other hand, if Gc has no cycle then G is a PCG. Our conditions are the first necessary condition and the first sufficient condition for p...

We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ǫ. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit ε = 0 is always no-slip. We give in this paper error estimates for this homo...

Sweating is a normal and important mechanism of thermoregulation which is essential for survival. When sweating becomes excessive, the resulting condition is called hyperhidrosis. While hyperhidrosis is not a fatal condition, it can greatly affect one’s quality of life due to its psychological and social impact. To understand this condition, it is necessary to explore the biology of sweat gland...

Ozeki and Takeuchi [12, I] introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and Münzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and Münzner c...