Convolution of two harmonic mappings in the right-half plane

This paper is to give a univalent criterion and a geometric property of the convolution of two right half-plane harmonic mappings f0(z) and f (z), where f0(z) is canonical and the second complex dilatation w(z) of f (z) is of the form w(z) = - z-a/1-az z-b/1-bz.

Asymptotic conformality of the barycentric extension of quasiconformal maps

We first remark that the complex dilatation of a quasiconformal homeomorphism of a hyperbolic Riemann surface R obtained by the barycentric extension due to Douady-Earle vanishes at any cusp of R. Then we give a new proof, without using the Bers embedding, of a fact that the quasiconformal homeomorphism obtained by the barycentric extension from an integrable Beltrami coefficient on R is asymptotically conformal if R satisfies a certain geometric condition.

On Geodesics in Asymptotic Universal Teichmüller Space

Let AT(Δ) be the asymptotic universal Teichmüller space, viewed as the space of all asymptotic Teichmüller equivalence classes [[μ]]. We show that ifμis asymptotically extremal in AT(Δ) andhp([[μ]]) <h([[μ]]) for some boundary pointpofΔ, then there are infinitely many geodesics joining [[0]] and [[μ]] in AT(Δ). As a corollary, a necessary condition for a complex dilatation to be uniquely extremal in AT(Δ) is given.

Dynamics of quasiregular mappings with constant complex dilatation

The study of quadratic polynomials is a foundational part of modern complex dynamics. In this article, we study quasiregular counterparts to these in the plane. More specifically, let $h:\mathbb{C}\rightarrow \mathbb{C}$ be an $\mathbb{R}$-linear map and consider the quasiregular mapping $H=g\circ h$, where $g$ is a quadratic polynomial. By studying $H$ and via the Böttcher-type coordinate constructed in A. Fletcher and R. Fryer [On Böttcher coordinates and quasiregular maps. Contemp. Math.575 (2012), 53–76], we are able to obtain results on the dynamics of any degree-two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in a neighbourhood of infinity. We also show that in most cases, two such mappings are not quasiconformally conjugate on a neighbourhood of infinity.

Ultrastructural alterations of choroid plexuses of lateral ventricles of rats (Rattus norvegicus) submitted to experimental chronic alcoholism

Adult male rats (Wistar lineage) were alcoholized with sugar cane liquor diluted at 30(0) GL during 300 days and sacrificed every 60 days in 5 stages. Samples of choroid plexuses of lateral ventricles were collected and examined at transmission electronic microscope to detect possible ultrastructural alterations and to raise possible pathological correlations. Gradual changes were observed in these animals during all the experiment: dilatation and enlargement of cisternae of Golgi complex, dilatation of RER, presence of digestive vacuoles and a large amount of pinocytic vesicles as well as vesicles with electronlucent content throughout cytoplasm, as well as an enlargement of intercellular space between basolateral interdigitation of the cells and of the connective tissue. The changes observed in the epithelium and connective tissue of choroid plexuses specially in 240 and 300 days of treatment are presumably due to a disturbance in hydroelectrolitic homeostasis, contributing to several morpho-functional disturbs of central nervous system. No changes were observed in the control group animals.