Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new hp-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the hp-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds.for which numerical results were reported, too. Again, it is also possible to use the Schwarz-Christoffel method for doubly connected domains [25].We next consider the case of simply connected plane domains. For such a domain D and for a quadruple {z 1 , z 2 , z 3 , z 4 } of its boundary points we call (D; z 1 , z 2 , z 3 , z 4 ) a quadrilateral if z 1 , z 2 , z 3 , z 4 occur in this order when the boundary curve is traversed in the positive direction. The points z k , k = 1, .., 4 , are called the vertices and the part of the oriented boundary between two successive vertices such as z 1 and z 2 is called a boundary arc (z 1 , z 2 ) . The modulus M(D; z 1 , z 2 , z 3 , z 4 ) of the quadrilateral (D; z 1 , z 2 , z 3 , z 4 ) is defined to be the unique h > 0 for which there exists a conformal mapping of D onto the rectangle with vertices 1 + ih, ih, 0, 1 such that the points z 1 , z 2 , z 3 , z 4 correspond to the vertices in this order. This conformal mapping is called the canonical conformal mapping associated with the quadrilateral. As in the case of doubly connected domains discussed above, it is well-known that the computation of the modulus h of the quadrilateral may be reduced to solving the Dirichlét-Neumann boundary value problem in the original domain D with the Dirichlét boundary values 1 on the boundary arc (z 1 , z 2 ) and 0 on the arc (z 3 , z 4 ) and Neumann boundary values 0 on the arcs (z 3 , z 4 ) and (z 4 , z 1 ) .Conformal moduli of rings and quadrilaterals have independent theoretical interest because of their crucial role in the theory of quasiconformal mappings [30]. These quantities are closely related to certain physical constants, e.g. they play an important role in applications involving the measurement of resistance values of integrated circuit networks. But the problem of computing the moduli is also interesting in the wider engineering context. The reciprocal identities (4.1) and (6.1) can be used to generate test cases for curvilinear Lipschitz domains and thus should be standard tools in the FEM-software development community. Unfortunately these identities are missing from the introductory FEM textbooks. Although the experimental results in this paper show that the reciprocal identities provide error estimates similar to the true error (in cases where the exact analytic result is known) more in...
Abstract. We study the convexity and starlikeness of metric balls on Banach spaces when the metric is the quasihyperbolic metric or the distance ratio metric. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.
Abstract. We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain Ω in a Banach space such that there is no geodesic between any given pair of points x, y ∈ Ω. In addition, we prove that if X is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of X, is smooth.
We prove several improved versions of Bohr's inequality for the harmonic mappings of the form f = h + g, where h is bounded by 1 and |g ′ (z)| ≤ |h ′ (z)|. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. [9], for example a term related to the area of the image of the disk D(0, r) under the mapping f is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.2000 Mathematics Subject Classification. Primary: 30A10, 30H05, 30C35; Secondary: 30C45 .
In this paper, we investigate the relationship between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric spaces. Our main objectives are to (1) generalize the main result in [14] together with other related results, and (2) give a complete answer to the open problem given in [14]. As an application, we prove that the composition of two locally weakly quasisymmetric mappings is a locally weakly quasisymmetric mapping and that it is quasiconformal.2000 Mathematics Subject Classification. Primary: 30C65, 30F45; Secondary: 30C20.
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