scholarly journals Approximate Conformal Mappings and Elasticity Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Pyotr N. Ivanshin ◽  
Elena A. Shirokova
Keyword(s):  
Conformal Mapping ◽  
Elasticity Theory ◽  
Unit Disk ◽  
Connected Domain ◽  
Smooth Boundary ◽  
Mapping Function ◽  
Conformal Mappings ◽  
New Method ◽  
Taylor Polynomial ◽  
Elasticity Problems

Here, we present the new method of approximate conformal mapping of the unit disk to a one-connected domain with smooth boundary without auxiliary constructions and iterations. The mapping function is a Taylor polynomial. The method is applicable to elasticity problems solution.

scholarly journals Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains

2020 ◽  
Author(s):  
Haiqing Xu
Keyword(s):  
Conformal Mapping ◽  
Unit Disk ◽  
Conformal Mappings ◽  
Optimal Regularity ◽  
Finite Distortion ◽  
Homeomorphic Extension

AbstractThe conformal mapping $$f(z)=(z+1)^2 $$f(z)=(z+1)2 from $${\mathbb {D}}$$D onto the standard cardioid has a homeomorphic extension of finite distortion to entire $${\mathbb {R}}^2 .$$R2. We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $${\mathbb {D}}$$D onto cardioid-type domains.

Uniform stresses inside both a non-parabolic inhomogeneity and a non-elliptical inhomogeneity located in the vicinity of a circular Eshelby inclusion

2021 ◽  
pp. 108128652110134
Author(s):  
Ping Yang ◽  
Xu Wang ◽  
Peter Schiavone
Keyword(s):  
Conformal Mapping ◽  
Connected Domain ◽  
Mapping Function ◽  
Elliptical Inhomogeneity ◽  
Surrounding Matrix ◽  
Eshelby Inclusion ◽  
Doubly Connected Domain ◽  
The Matrix ◽  
Uniformity Property ◽  
Complex Cases

We establish the uniformity of stresses inside both a non-parabolic open inhomogeneity and a non-elliptical closed inhomogeneity interacting with a nearby circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Our procedure involves the introduction of a conformal mapping function for the doubly connected domain occupied by the matrix and the circular Eshelby inclusion. Two conditions are established in order to achieve the uniformity property inside each of the two inhomogeneities. Our results indicate that: (a) the internal uniform stresses are independent of the specific shapes of the two inhomogeneities and the existence of the nearby circular Eshelby inclusion; (b) the open and closed shapes of the respective inhomogeneities are significantly affected by the presence of the circular Eshelby inclusion. We also consider the two more complex cases involving: (a) an arbitrary number of circular Eshelby inclusions undergoing uniform eigenstrains; (b) a circular Eshelby inclusion undergoing linear eigenstrains. Detailed numerical results demonstrate the feasibility and effectiveness of the proposed theory.

scholarly journals Relative Distance and Quasi-Conformal Mappings

1960 ◽  
Vol 16 ◽  
pp. 111-117
Author(s):  
D. A. Storvick
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Distance Function ◽  
Connected Domain ◽  
General Class ◽  
Conformal Mappings ◽  
Relative Distance ◽  
Simply Connected Domain ◽  
Simply Connected

1. Introduction. M. A. Lavrentiev made use of a relative distance function to establish some important results concerning the correspondence between the frontiers under a conformal mapping of a simply connected domain onto the unit circle. The purpose of this note is to show that some of these results are valid for the boundary correspondences induced by the more general class of quasi-conformal mappings.

scholarly journals Conformal mapping of the Misner–Sharp mass from gravitational collapse

2016 ◽  
Vol 25 (07) ◽  
pp. 1650081 ◽  
Author(s):  
Fayçal Hammad
Keyword(s):  
Conformal Mapping ◽  
Gravitational Collapse ◽  
Conformal Transformation ◽  
Conformal Mappings ◽  
Conformal Transformations ◽  
Theories Of Gravity ◽  
Definition Of ◽  
Geometric Definition

The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.

scholarly journals Dirichlet Problems on Riemann Surfaces and Conformal Mappings

1951 ◽  
Vol 3 ◽  
pp. 91-137 ◽  
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Riemann Surface ◽  
Conformal Mapping ◽  
Riemann Surfaces ◽  
Conformal Mappings ◽  
Dirichlet Problems ◽  
Boundary Correspondence ◽  
Abstract Riemann Surface

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.

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