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Conformal mappings
Published in Martin Vermeer, Antti Rasila, Map of the World, 2019
An important result concerning conformal mappings is the Riemann mapping theorem. This result says that there are many conformal mappings: in fact, every simply connected domain2D, which is not the whole complex plane, can be mapped conformally into an open unit disc D={z:|z|<1}. If it is additionally assumed that z0 ∈ D, f(z0) = 0 and |f′(z0)| > 0, then, based on Poincaré3's result, the mapping is also unique.
Thermal stress around a smooth cavity in a plate subjected to uniform heat flux
Published in Journal of Thermal Stresses, 2021
Zhaohang Lee, Yu Tang, Wennan Zou
Consider an infinite body Ω in two-dimensional space consisting of a homogeneous and isotropic medium whose thermal conduction behavior is governed by Fourier’s law, and elastic behavior by Hooke’s law. We are concerned with the perturbance effect due to a free cavity with a traction-free, thermally insulated boundary while the matrix is subjected to uniform heat flux at infinity, as shown in Figure 1. By the Riemann mapping theorem in complex analysis, there is a unique function in the form of Laurent series (see, e.g., Zou et al. [16]) mapping the exterior of the cavity onto the exterior of the unit circle with the origin as its center. In the above expression, h is a point inside the cavity, R is a positive real parameter indicating the size of the cavity, and are the complex variable parameters representing the shape. It can be seen that the point t on the boundary of any simply connected shape can be accurately described by In general, we only need to take a limited number of terms N to meet the accuracy requirement. To continue to improve the accuracy, we only need to increase the number of terms. So, one might as well use to describe the cavity and for the problem of a sole cavity, is usually taken.
Uniformly perfect and hereditarily non uniformly perfect analytic and conformal non-autonomous attractor sets
Published in Dynamical Systems, 2021
Mark Comerford, Kurt Falk, Rich Stankewitz, Hiroki Sumi
We call a doubly connected domain A in that can be conformally mapped onto a true (round) annulus , for some 0<r<R, a conformal annulus with the modulus of A given by , noting that R/r is uniquely determined by A (see, e.g. the version of the Riemann mapping theorem for multiply connected domains in [1]).