Related papers: Computing boundary extensions of conformal maps
It is shown that there is a computable conformal map of the unit disk onto a domain $D$ that has a computable extension to the closure of the unit disk even though the boundary of $D$ is not effectively locally connected. The proof encodes…
Consider being given a mapping \phi from the unit sphere S^{d-1}, d>2, to the smooth boundary of a simply-connected region \Omega in R^d. We consider the problem of constructing an extension \Phi from the unit ball B_d to \Omega. The…
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 \emph{without} prior knowledge of the…
In this paper, we consider mappings on uniform domains with exponentially integrable distortion whose Jacobian determinants are integrable. We show that such mappings can be extended to the boundary and moreover these extensions are…
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of…
The method of boundary curve reparametrization is applied to construction of the approximate analytical conformal mapping of the unit disk onto an arbitrary given finite domain with a boundary smooth at every point but fininte number of…
In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The…
We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. The…
We prove necessary and sufficient integral conditions involving extremal distance for a conformal mapping of the unit disk to belong to the Hardy or weighted Bergman spaces. We also give characterizations for the Hardy number and the…
Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be…
We show that the computational complexity of Riemann mappings can be bounded by the complexity needed to compute conformal mappings locally at boundary points. As a consequence we get first formally proven upper bounds for…
We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.
We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain $D \subset \hat{C}$ admits exhaustions, i.e., increasing sequences of finitely connected subdomains $D_j$…
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by…
The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $\mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the…
The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this…
By means of the property of effective local connectivity, the computability of finding the Carath\'eodory extension of a conformal map of a Jordan domain onto the unit disk is demonstrated.
Theorems on continuous extension on boundary for one class of open discrete mappings between Riemannian manifolds are obtained. In particular, there is proved that, open discrete ring $Q$-mappings $f:D\rightarrow D^{\,\prime}$ are extend to…