Let ⊂ R n be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equationwith the boundary condition u = g 0 on ∂ , where f 0 ∈ C( ) and g 0 ∈ C(∂ ) are given functions and M is the singular integral operator given by

The size of a metric disc embedded in a compact non-orientable hyperbolic surface is bounded by some constant depending only on the genus g ≥ 3. We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case g = 3, we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.

We have investigated the surface electronic structures of transition-metal diboride (0001) surfaces by angleresolved ultraviolet photoelectron spectroscopy (ARUPS) and ab-initio calculations based on the slab model with 13 layers. In the spectra of NbB2 and TaB2, the peaks near −1 eV blow EF are derived from the π band of their surface graphitic boron layer. As compared with graphene, the dispersion curves of the π bonds are greatly modulated by the hybridization of B-π orbits with the metal d orbits. On the other hand, the similar peaks were not observed in spectra of ZrB2, of which the surface is terminated with Zr atoms.

A compact Riemann surface of genus g b 2 is said to be extremal if it admits an extremal disk, a disk of the maximal radius determined by g. If g ¼ 2 or g b 4, it is known that how many extremal disks an extremal surface of genus g can admit. In the present paper we deal with the case of g ¼ 3. Considering the side-pairing patterns of the fundamental polygons, we show that extremal surfaces of genus 3 admit at most two extremal disks and that 16 surfaces admit exactly two. Also we describe the group of automorphisms and hyperelliptic surfaces.

Abstract. A compact hyperbolic surface of genus g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined by g. We know how many extremal discs are embedded in a non-orientable extremal surface of genus g = 3 or g > 6. We show in the present paper that there exist 144 non-orientable extremal surfaces of genus 4, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of g = 3.