A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.

Generation of advanced Escher-like spiral tessellations

In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give a way to achieve simple spiral tilings of cyclic symmetry. Then, we introduce several conformal mappings to generate three derived spiral tilings. To obtain Escher-like tessellations on the generated tilings, given pre-designed wallpaper motifs, we specify the tessellations’ implementation details. Finally, we exhibit a rich gallery of the generated Escher-like tessellations. According to the proposed method, one can produce a great variety of exotic Escher-like tessellations that have both good aesthetic value and commercial potential.

Correction to: Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

A Correction to this paper has been published: https://doi.org/10.1007/s10711-015-0119-z

GENERALIZ GENERALIZATION OF THE H TION OF THE HARDY CLASS FOR AN ASS FOR ANALYTIC FUNCTIONS

Introduction. Quoting from a well-known American mathematician Lipman Bers [1] “It would be tempting to rewrite history and to claim that quasiconformal transformations have been discovered in connection with gas-dynamical problems. As a matter of fact, however, the concept of quasiconformality was arrived at by Grotzsch [2] and Ahlfors [3] from the point of view of function theory”. The present work is devoted to the theory of analytic solutions of the Beltrami equation which directly related to the quasi-conformal mappings. The function is, in general, assumed to be measurable with almost everywhere in the domain under consideration. Solutions of equation (1) are often referred to as analytic functions in the literature. Research methods.

On one extremal problem for nonlinear Cauchy-Riemann-Beltrami systems

The study of nonlinear Cauchy--Riemann--Beltrami systems is conditioned study of certain problems of hydrodynamics and gas dynamics, in which there is an inhomogeneity of media and a certain singularity. The paper considers a nonlinear Cauchy--Riemann--Beltrami type system in the polar coordinate system in which the radial derivative is expressed through the complex coefficient, the angular derivative and its m-degree module. In particular, if m is equal to zero, then this system of equations is reduced to the ordinary linear system of Beltrami equations. Note that general first-order systems were used by M.А. Lavrentyev to define quasiconformal mappings on the plane, see \cite{L}. The problem of area distortion under quasi-conformal mappings is due to the work of B. Boyarsky, see \cite{Bo}. For the first time, the upper estimate of the area of the disk image under quasi-conformal mappings was obtained by M.А. Lavrentyev, see \cite{L}. A refinement of the Lavrentyev inequality in terms of the angular dilatation was obtained in the monograph \cite{BGMR}, see Proposition 3.7. In the present paper, it is found an exact upper estimate of the area of the image of the disk, which is analogous to the known result by Lavrentyev. Also, we find here a mapping on which the estimate is achieved. Thus, the work solves the extreme problem for the area functional of the image of disks under a certain class of regular homeomorphic solutions of nonlinear systems of the Cauchy--Riemann--Beltrami type with generalized derivatives integrated with a square. The work uses p-angular dilatation. In the conformal case, angular dilatation is important in the theory of quasi-conformal mappings and nondegenerate Beltrami equations. Proof of the main result of the article is based on the differential relation for the area function of the image of disks of arbitrary radii, which was established in the previous work of the authors for regular homeomorphisms with Luzin's N-property.