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On some convolution theorems

100%

Piejko K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2002

tom [Z] 42(1)

103-112

Let ℌ be the class of functions regular in the unit disc. For two functions ƒ(z) = [formula], g (z) = [formula] of the class ℌ and for all complex m we define the convolution (ƒ★m g) (z) [formula]. For given complex numbers a, A, B; aB ≠ A, |B| ≤ 1 we define Pa (A, B) := [formula]. The object of this paper is to give the solution of the problem of finding all m ϵ C, for which convolution ƒ★m g belongs to the Pab (X. Y) if ƒ ϵ Pa (A, B) and g ϵ Pb (C, D).

On generalization of close-to-convexity for complex holomorphic functions in Cn

75%

Marchlewska A.

Demonstratio Mathematica

2005

tom Vol. 38, nr 4

847--856

Sveral authors (I. I. Bawrin [1], K. Dobrowolska, I. Dziubinski, P. Liczberski, R. Sitarski [3], [4], [5], [13], S. Gong, S. S. Miller [6], Z. J. Jakubowski and J. Kaminski [8], J. Janiec [10] and others) studied various families of complex holomorphic functions in Cn and in Banach space, corresponding with famous subclasses of univalent functions. In this paper we study a class of holomorphic functions of n complex variables analogous to the class of close-to-convex functions of one variable considered by M. Biernacki, W. Kaplan and Z. Lewandowski (see [2], [11], [12]).

Transformation of the Vistula Lagoon onto a canonical domain

51%

Prosnak W. J. , Cześnik P. P.

Oceanologia

2001

tom No. 43 (2)

169-200

The paper deals with the conformal mapping of finite, plane, simply connected domains, representing oceans, lakes, estuaries, bays, lagoons, and other natural water bodies of this kind. As a rule, they are bounded by geometrically complex shorelines. The partial differential problems investigated in Oceanology and posed in such domains have turned out to be difficult to solve for at least three reasons. They follow on from the mathematical properties of the differential equations governing such problems, from the just-mentioned geometrical complexity of the domains of solution, and from the sensitivity of the solutions to boundary conditions. In view of the last reason the contours admitted as boundaries of the domains of the solution ought to be as close to the real shorelines as possible. The obviously inaccurate approximation of the shorelines by "staircases", which appears rather often (cf. Catewicz & Jankowski 1983, Lin & Chandler-Wilde 1996) as a consequence of applying finite difference methods to the solution of the partial differential problems, raises serious doubts from the point of view of Numerical Fluid Mechanics. It is recalled in the paper that such inaccuracies are not unavoidable: that complicated plane domains can be transformed accurately by means of properly applied conformal mapping onto regular, canonical domains - in particular, onto discs or squares. Such a transformation is demonstrated on the rather difficult example of the Vistula Lagoon. The transformation begins with the decomposition of the domain into five plane subdomains, each one of which is eventually transformed onto a disc. Every such result is arrived at quite independently of the remaining subdomains, by means of a set of properly selected consecutive mappings. Hence, the final canonical domain consists in this case of a system of five discs which, however, within the framework of this differential problem, have to be treated as interconnected. The interconnections involve images of four segments of straight lines, separating the original subdomains. The transformations and the resulting canonical domain presented in the paper are intended to be applied to the solution of certain hydrodynamical problems concerning the Vistula Lagoon, which will be published elsewhere.