Numerical conformal mapping of multiply connected domains to regions with circular boundaries

We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.     

The degree of a mapping in some problems of combinatorial geometry

The problem of dividing a unit mass that is continuously distributed inR ⁿ into various ratios by a pencil ofn+1 solid angles is connected with the question of the possibility of homotopies between certain coverings of the sphereS ^n–1. We prove that it is possible to inscribe a simplex homothetic to a given simplex in any convex solid region with a smooth boundary. In all the proofs we use the concept of the degree of special mappings onto a canonical simplex and its boundary. One figure.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 62–66.     

Fast conformal mapping of an ellipse to a simply connected region

The iterative method of Wegmann (1978) for conformal mapping of a region E to a region G is reformulated in terms of a conjugation operator. If the boundary of G is sufficiently smooth, the method converges quadratically in a Sobolev space W. To overcome the crowding an elongated canonical region E should be chosen if the image region G is elongated. For ellipses E the operator of conjugation can be expressed in a very simple way in terms of Fourier series. Numerically, this can be performed very efficiently by fast Fourier transform (FFT).     

A pair of Fibonacci-like polynomials arising from a special mapping

We study here a pair of sequences of polynomials that arise from a particular iterated mapping on the plane. We show how these sequences come about, and give some of their interesting mathematical properties.     

The index of a semifree periodic mapping

We consider semi-free periodic mappings of homologic spheres and study the structure of the set of fixed points and the properties of the index of a periodic mapping; we establish relations for the degree of an equivariant mapping.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 79–86, January, 1973.The author wishes to thank Yu. G. Borisovich for his attention to the paper and for useful discussions.     

Numerical mapping of exterior domains

This paper describes an integral equation method for computing the conformal mapping of the exterior of a given simply-connected region onto the exterior of the unit circle. The method includes evaluation of the transfinite, diameter of the given region. Results are presented for a number of trial problems.     

On the conformal mapping of a circular disc with a curvilinear polygonal hole

A method of finding a mapping function which maps the doubly-connected region between a circle and a curvilinear polygon onto an annulus is described and numerical results for the ratio of the radii of the annulus given.     

The decomposition of a linear mapping

We give a practical algorithm for obtaining the unique decomposition of a linear mapping into a sum of commuting semisimple and nilpotent mappings.     

Unique continuation of semi-conformality for a harmonic mapping onto a surface

We show that a harmonic mapping ϕ from either a three-manifold (with a condition on its Ricci curvature) or from a surface with values in a surface which has rank 2 somewhere, satisfies the following unique continuation property: if ϕ is semi-conformal on an open set, then it is semi-conformal everywhere.     

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.     

Fast conformal mapping of multiply connected regions

An iterative method is presented which constructs for an unbounded region G with m holes and sufficiently smooth boundary a circular region H and a conformal mapping Φ from H to G. With the usual normalization both H and Φ are uniquely determined by G. With a few modifications the method can also be applied to a bounded region G with m holes. The canonical region H is then the unit disc with m circular holes. The proposed method also determines the centers and radii of the boundary circles of H and requires, at each iterative step, the solution of a Riemann–Hilbert (RH) problem, which has a unique solution. Numerically, the RH problem can be treated efficiently by the method of successive conjugation using the fast Fourier transform (FFT). The iteration for the solution of the RH problem converges linearly. The conformal mapping method converges quadratically. The results of some test calculations exemplify the performance of the method.     

Hybrid bootstrap for mapping quantitative trait loci

The hybrid bootstrap uses resampling ideas to extend the duality approach to interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the parametric bootstrap region in situations where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the location of quantitative trait loci (QTL) in interval mapping model. The conditional distribution of quantitative traits, given flanked genetic marker genotypes is often assumed to be the mixture model of two phenotype distributions. The mixing proportions in the model represent the recombination rate between a genetic marker and quantitative trait loci and provides information about the unknown location of the QTL. Since recombination events are unlikely, we will have less information about the location of the QTL than other parameters. This observation makes a hybrid approach to interval estimation for QTL appealing, especially since the necessary distribution theory, which is often a challenge for mixture models, can be handled by bootstrap simulation.     

On the period spectrum of a symplectic mapping

Let M be a compact Riemannian manifold. It has been known for a long time that the singularities of the wave trace, trace(cos √Δt), are located at the periods of the closed geodesics. Do these singularities also contain information about the geometry of M in the neighborhood of a closed geodesic? We prove that the Birkhoff canonical form of the Poincaré map can be determined from the singularities of the wave trace.     

On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

Approximation to a set-valued mapping,I: A proposal

Given a multivalued mappingF, we address the problem of finding another multivalued mappingS that agrees locally withF. We limit ourselves to the case whenF(x) is convex compact for eachx. We propose anS obtained by linearizing the support function ofF(x) and give conditions under which thisS approximatesF in the sense of Hausdorff distance. We show equality betweenS and other proposals obtained from tangent cones to the graph ofF. Finally, we apply these results to the approximate subdifferential of a convex function.     

A conformal mapping for a rectilinear congruence

In this paper we deal with oriented rectilinear congruences in a three-dimensional Euclidean space E³ establishing a conformal mapping between their middle surface and their middle envelope. We give some properties and determine a special class of them, which have a minimal middle envelope.     

Cones with a mapping cone symmetry in the finite-dimensional case

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio?kowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.     

The relation between majorants and conformal mapping. Part 2

We make use here of the definitions and theorems from our paper: Majorants in classes of subharmonic functions, from the previous issue of this journal.     

A three-dimensional quantum integrable mapping

We derive a three-dimensional integrable mapping that gives the R-matrix of the Zamolodchikov model in the cyclic representation limit. We construct the discrete-time evolution generated by this mapping and derive a generating function for the integrals of motion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 479–487, March, 1999.     

A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping

The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in the framework of a Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under appropriate conditions. Additionally, the idea of our results are applied to find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space.