1. We refer to a region £2 of the extended z-plane as a (parallel) slit domain if =o £Í2, and if the components of the boundary, dß, are either points, or segments ("slits") parallel to a common line, which without loss of generality will be assumed to be the y-axis 'z = x-\-iy). It was originally conjectured by Koebe that if two slit domains ßi and Í22 are conformally equivalent, that is, if t...

We show that for each cusp on the boundary of Maskit’s embedding M ⊂ H of the Teichmüller space of punctured tori there is a sequence of parameters in the complement ofM converging to the cusp such that the parameters correspond to discrete groups with elliptic elements. Using Tukia’s version of Marden’s isomorphism theorem we identify them as cusps on the boundary of certain deformation spaces...

Two open subsets of $${\mathbb {R}}^n$$ are called Schwartz equivalent if there exists a diffeomorphism between them that induces an isomorphism Fréchet spaces their functions. In this paper we use tools from quasiconformal geometry in order to prove the equivalence few families planar domains. We all quasidiscs equivalent. also any non-simply-connected domain whose boundary is quasicircle comp...

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-frac...

The touching graph of balls is a graph that admits a representation by non-intersecting balls in the space (of prescribed dimension), so that its edges correspond to touching pairs of balls. By a classical result of Koebe [?], the disc touching graphs are exactly the planar graphs. This paper deals with a recognition of unit-ball touching graphs. The 2– dimensional case was proved to be NP-hard...

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realize...

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realize...

We show that, given a non-degenerate, finitely connected domain D, its boundary, and the number of its boundary components, it is possible to compute a conformal mapping of D onto a circular domain without prior knowledge of the circular domain. We do so by computing a suitable bound on the error in the Koebe construction (but, again, without knowing the circular domain in advance). Recent resu...