Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Evaluation of a boundary integral representation for the conformal mapping of the unit disk onto a simply-connected domain

A representation of the conformal mapping g of the interior or exterior of the unit circle onto a simply-connected domain Ω as a boundary integral in terms of?|_?Ω is obtained, where? :=g ^-l. A product integration scheme for the approximation of the boundary integral is described and analysed. An ill-conditioning problem related to the domain geometry is discussed. Numerical examples confirm the conclusions of this discussion and support the analysis of the quadrature scheme.     

On the approximate conformal mapping of the unit disk on a simply connected domain

We present a method of constructing the analytic function realizing an approximate conformal mapping of the unit disk on an arbitrary simply connected domain with the given smooth parametrically defined boundary. The method is based on a new boundary parameterization. Solution of the problem is reduced to the Fredholm integral equation of the second kind. We present the examples of three ways to solve the integral equation.     

A domain decomposition method for conformal mapping onto a rectangle

Letg be the function which maps conformally a rectangleR onto a simply connected domainG so that the four vertices ofR are mapped respectively onto four specified pointsz ₁,z ₂,z ₃,z ₄ onG. This paper is concerned with the study of a domain decomposition method for computing approximations tog and to an associated domain functional in cases where: (i)G is bounded by two parallel straight lines and two Jordan arcs. (ii) The four pointsz ₁,z ₂,z ₃,Z ₄, are the corners where the two straight lines meet the two arcs.Communicated by Dieter Gaier.     

Some extremal properties of conformal mappings of a multiply connected domain onto canonical p-sheeted surfaces

Some theorems of Kühnau on the mutual position of the boundary components of images of a multiply connected domain under single-sheeted conformal mappings are generalized to p-sheeted conformal mappings of a multiply connected domain with given singularities at given points of the domain. Concrete estimates for certain functionals are obtained for an annulus.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 261–270, February, 1978.     

Linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits

In this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region Ω onto a disk with circular slits. The method is based on some uniquely solvable boundary integral equations with classical adjoint and generalized Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.     

On a domain decomposition method for the computation of conformal modules

We consider a domain decomposition method for approximating the conformal modules of a certain class of long quadrilaterals.     

On the polynomial approximation of a conformal mapping of a domain with nonzero corner

Let G be a bounded domain with a Jordan boundary that is smooth at all points except a single point at which it forms a nonzero corner. We prove Korevaar’s conjecture on the order of polynomial approximation of a conformal mapping of this domain into a disk. We also obtain a pointwise estimate for the error of approximation.     

An isomorphism criterion for modules over a principal ideal domain

Let U and V be modules of finite composition length over a principal ideal domain D. Then, denoting the composition length of the D-module Horn (U,V) by 〈U,V〉, it is shown that 〈U,V〉²≤〈U,U〉〈V,V〉 and that equality holds iff U and V are isomorphic to direct sums of copies of some common module. This gives an isomorphism criterion for U and V based entirely on composition lengths. As a special case, the theorem gives a strong version of a criterion of M. A. Gauger and C. 1. Byrnes for the similarity of two matrices.     

On the theory and application of a domain decomposition method for computing conformal modules

We consider the theory and application of a domain decomposition method for computing the conformal modules of long quadrilaterals. The method has been studied already by us and also by Gaier and Hayman. Our main purpose here is to extend its area of application and, in the same time, improve some of our earlier error estimates.     

On essential conformal groups and a conformal invariant

We show that various conformal groups, including classical conformal diffeomorphism groups are essential. The essentiality is shown to be equivalent to the non-vanishing of a conformal cohomological invariant.     

On the conformal mappings in the unit disk fixing arbitrarily many boundary points

In this paper, we give a method of constructing conformal mappings defined in the unit disk which can fix arbitrarily many points on the unit circle.