A compact (orientable or non-orientable) surface of genus g is said to be extremal if it contains an extremal disc, that is, a disc of the largest radius determined only by g. The present paper concerns non-orientable extremal surfaces of genus 5. We represent the surfaces as side-pairing patterns of a hyperbolic regular 24-gon, that is, a generic fundamental region of an NEC group uniformizing...

This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In particular, this approach enables us to find explicit equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are indeed defined over Q. Th...

Binary extremal self-dual codes of type II and their automorphisms Wolfgang Willems Otto-von-Guericke-Universität Magdeburg Extremal type II codes are for several reasons of particular interest. However only for small lengths such codes have been contructed. In order to find larger examples ’symmetries’ or in other words ’non-trivial automorphisms’ may be helpful. In this spirit the talk deals ...

We discuss the two-dimensional (2D), ǫ < 2 extremal ground states of charged black hole. Here ǫ is the dilaton coupling parameter for the Maxwell term. The complete analysis of stability is carried out for all these extremal black holes. It is found that they are all unstable. To understand this instability, we study the non-extremal charged black hole with two (inner and outer) horizons. The e...

In this paper we investigate the extremal richness of the multiplier algebra M(A) and the corona algebra M(A)/A, for a simple C∗-algebra A with real rank zero and stable rank one. We show that the space of extremal quasitraces and the scale of A contain enough information to determine whether M(A)/A is extremally rich. In detail, if the scale is finite, then M(A)/A is extremally rich. In import...

In 1956 Shiffman [14] proved that every minimally immersed annulus in 3 bounded by convex curves in parallel planes is embedded. He proved this theorem by showing that the minimal annulus was foliated by convex curves in parallel planes. We are able to prove a related embeddedness theorem for extremal convex planar curves. Recall that a subset of 3 is extremal if it is contained on the boundary...

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

Extremal Combinatorics is an area in Discrete Mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive...

Through the ages, humankind has struggled to come to terms with its own existence, a thorny issue that will no doubt keep philosophers tossing and turning in their beds for aeons to come. The ‘how’ of our existence, though still fiercely debated in some corners, is now reasonably well-understood, but it is the ‘why’ that continues to plague us. Why are we here? What purpose are we meant to serv...

Unlike the extremal Reissner-Nordström black hole in ordinary spacetime, the one in anti-de Sitter spacetime is a minimum of action and has zero entropy if quantization is carried out after extremaliza-tion. However, if extremalization is carried out after quantization, then the entropy is a quarter of the area as in the usual case. While the entropy of ordinary (non-extremal) black holes has b...