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...

These are well known and important theorems of Nehari, Ahlfors and Weill, and Krauss. We refer to Lehto's book [8] for a discussion of these results and for the properties of the Schwarzian that we shall need. The constants 2 in (1.1) and 6 in (1.3) are sharp. An example for the latter is the Koebe function k(z) = z(l — z)~ which has Schwarzian Sk(z) = — 6/(1 —z). We also remark that the class ...

Colin de Verdi ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n ? 4, and if it is planar, then (G) n ? 5. We give a full characterization of maximal planar graphs with (G) = n ? 5. In the opposite di...

Abstract A semigroup generated by two dimensional C contracting maps is considered. We call a such semigroup regular if the maximum K of the conformal dilatations of generators, the maximum l of the norms of the derivatives of generators and the smoothness α of the generators satisfy a compatibility condition K < 1/l. We prove that the shape of the image of the core of a ball under any element ...

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...

A mesh M with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh M d such that corresponding edges of M and M d lie on parallel lines of constant distance d. The edges emanating from a vertex of M lie on a right circular cone. Viewing M as set of these vertex cones, we show that the image of M under any Laguerre transformation is again an EO mesh. ...

The critical curve C on which Im τ̂ = 0, τ̂ = aD/a, determines hyperbolic domains whose Poincaré metric is constructed in terms of aD and a. We describe C in a parametric form related to a Schwarzian equation and prove new relations for N = 2 Super SU(2) Yang-Mills. In particular, using the Koebe 1/4-theorem and Schwarz’s lemma, we obtain inequalities involving u, aD and a, which seem related to ...