We consider the class of univalent log-harmonic mappings on unit disk. Firstly, we present general idea constructing Koebe mappings, right half-plane and two-slits then show precise ranges these mappings. Moreover, coefficient estimates for starlike are obtained. Growth distortion theorems certain special subclasses studied. Finally, propose two conjectures, namely, covering conjectures.

We study C-algebras Oλ which arise in dynamics of the interval exchange transformations and measured foliations on compact surfaces. Using Koebe-Morse coding of geodesic lines, we establish a bijection between Bratteli diagrams of such algebras and measured foliations. This approach allows us to apply K-theory of operator algebras to prove strict ergodicity criterion and Keane’s conjecture for ...

The Koebe-Andreev-Thurston Circle Packing Theorem states that every triangulated planar graph has a circle-contact representation. The theorem has been generalized in various ways. The arguably most prominent generalization assures the existence of a primal-dual circle representation for every 3-connected planar graph. The aim of this note is to give a streamlined proof of this result.

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 develop the K-theory of a C∗–algebra Oλ which represents the leaf space of measured foliations studied by Novikov, Masur, Thurston and Veech. The K-theory construction is based on the coding of geodesic lines due to Koebe and Morse. This method allows to calculate the range of Elliott group (K0, K + 0 , [1]) of Oλ, to establish a condition of strict ergodicity of the interval exchange transf...