The classification of the closed sets on R and the representation of the majorants. Part 3

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 52, pp. 21–33, 1989.     

On conformal mapping and linear measure

Journal d'Analyse Mathématique -     

Zolotarev's conformal mapping and Chebotarev's problem

Consider the Chebotarev problem of finding a continuum S in the complex plane including some given points such that the logarithmic capacity of S is minimal. In this paper, we give a complete solution of this problem for the case of three given points with the help of Zolotarev's conformal mapping using Jacobian elliptic and theta functions. Moreover, for four given points, some special cases can be treated.     

Keldysh-Lavrentiev counterexample and means of powers of a conformal mapping. 1. construction of a mapping

We construct a region with infinitely many cusps on its boundary such that it is possible to estimate the derivative of a conformal mapping of the units disk onto this region from below. We use some ideas taken from the construction of a famous Keldysh-Lavrentiev example. Bibliography: 2 titles.     

Geodesic and contour optimization using conformal mapping

The art gallery problem (AGP) is one of the classical problems in computational geometry. It asks for the minimum number of guards required to achieve visibility coverage of a given polygon. The AGP is well-known to be NP-hard even in restricted cases. In this paper, we consider the AGP with fading (AGPF): A polygonal region is to be illuminated with light sources such that every point is illuminated with at least a global threshold, light intensity decreases over distance, and we seek to minimize the total energy consumption. Choosing fading exponents of zero, one, and two are equivalent to the AGP, laser scanner applications, and natural light, respectively. We present complexity results as well as a negative solvability result. Still, we propose two practical algorithms for AGPF with fixed light positions (e.g. vertex guards) independent of the fading exponent, which we demonstrate to work well in practice. One is based on a discrete approximation, the other on non-linear programming by means of simplex-partitioning strategies. The former approach yields a fully polynomial-time approximation scheme for the AGPF with fixed light positions. The latter approach obtains better results in our experimental evaluation.     

The application of the Lichtenstein-Gershgorin integral equation in conformal mapping

Conclusion Provided that the domain which should be mapped onto the unit circle is nearly circular the method seems to be very good. It is fast and accurate. Furthermore the results can be obtained by numerical differentiation if the boundary values of r=r(v) are specified by a table.In order to find the values of in situations when the domain D is not nearly circular, it will often be possible to find a transformation by means of which D can be mapped onto a domain D₁ which is nearly circular, and the method can be used to get the final results.This problem and other questions in connection with the method will be examined in the nearest future.     

The determination of the poles of the mapping function and their use in numerical conformal mapping

Let ? be the function which maps conformally a simply-connected domain Ω onto the unit disc. This paper is concerned with the problem of determining the dominant poles of ? in compl(Ω ∪ ? Ω), and of using this information in order to obtain accurate numerical approximations to ? by means of the Bergman kernel method.     

Point systems with extremal properties and conformal mapping

Summary LetD be the unit disk. It is a well-known fact that by use of simply connected domain methods the general conformal mapping problem of doubly connected domains can be reduced to the special case of a regionD bounded by the unit circle and a Jordan curve inD, where . Here we treat this special case and assume to be piecewise analytic without cusps. Let be the conformal mapping of {<|w|<1} onto the doubly connected domainD with (1)=1. We approximate by interpolation with finite Laurent series using point systems with extremal properties. Numerical results for four examples are given.     

$L^2$-Approximations of power and logarithmic functions with applications to numerical conformal mapping

Summary. For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best polynomial approximation to functions of the form , and , where . Furthermore, Andrievskii's lemma that provides an upper bound for the norm of a polynomial in terms of the norm of is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to . For the case when the boundary of G is piecewise analytic without cusps, the results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained. Received September 1, 1999 / Published online August 17, 2001     

Analysis and design of polygonal resistors by conformal mapping

To compute the electrical resistance ( conformal modulus) of a polygonally shaped resistor cut from a sheet of uniform resistivity, it suffices to find a conformal map of the polygon onto a rectangle. Constructing such a map requires the solution of a Schwarz-Christoffel parameter problem. First we show by examples that this is practical numerically. Then we consider an inverse resistor trimming problem in which the aim is to cut a slit in a given polygon just long enough to increase its resistance to a prescribed value. We show that here the solution can be obtained by solving a generalized parameter problem. The idea of a generalized parameter problem is applicable also in many other Schwarz-Christoffel computations.

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Zusammenfassung Um den elektrischen Widerstand eines polygonalen Resistors aus einem Material homogener Leitfähigkeit zu berechnen, genügt es, eine konforme Abbildung des Polygons auf ein Rechteck zu finden. Die Konstruktion einer solchen Abbildung erfordert die Lösung eines Schwarz-Christof-felschen Parameterproblems. Wir zeigen zunächst anhand von Beispielen, daß dies numerisch durchführbar ist. Dann betrachten wir ein inverses Problem: Die Aufgabe besteht hier darin, einen Schlitz in ein gegebenes Polygon zu schneiden, dessen Länge gerade so gewählt ist, daß der Widerstand auf einen vorgegebenen Wert erhöht wird. Wir zeigen, daß dieses Problem auf ein verallgemeinertes Parameterproblem zurückgeführt werden kann. Die Idee des verallgemeinerten Parameterproblems ist auch auf viele weitere Schwarz-Christoffel-Probleme anwendbar.

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Supported by NSF Mathematical Sciences Postdoctoral Fellowship, and by the U.S. Dept. of Energy under contract DE-AC02-76-ER03077-V. This work was performed at the Courant Institute of Mathematical Sciences, New York University.     

Universal central extensions of Lie conformal algebras,Part 2: Supercase

In the previous part of this study we considered universal central extensions of Lie conformal algebras and conditions for existence of such extensions. The aim of this paper is some refinement of previous results and extension of these results to Lie conformal superalgebras.