Mosses new to Hong Kong (4)

Sixteen moss species - Eurhynchium asperisetum (C. Muell.) Tak.; Rhynchostegium pallidifolium (Mitt.) Jaeg.; Bryum argenteum Hedw.; Bryum caespiticium Hedw.; Bryum capillare Hedw.; Platyhynidium riarioides (Hedw.) Dix.; Dicranella varia (Hedw.) Schimp.;Entodon virudulus Card.; Fissidens strictulus C. Muell.; Ectropothecium obtusulum (Card.) Iwats.; Caduciella guangdongensis Enroth.; Plagiomnium cuspidatum (Hedw.) T. Kop.; Plagiomnium vesicatum (Besch.) T. Kop.; Pyrrhobryum spiniforme (Hedw.) Mitt., Taxithelium nepalense (Schwaegr.) Broth. and Claopodium aciculum (Broth.) Broth. are reported new to Hong Kong. Among them, four are new to Guangdong Province of China

On Euler characteristics for large Kronecker quivers

We study Euler characteristics of moduli spaces of stable representations of m-Kronecker quivers for m>>0.Comment: submitted versio

Optimal Order Convergence Implies Numerical Smoothness

It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability