Thesis--Bonn. / Includes bibliographical references (p. 58-59).

Thesis--Bonn. / Includes bibliographical references (p. 58-59).

It is well known that, as a consequence of the Identity Theorem, we cannot "glue" together two analytic functions to create a new globally analytic function. In this paper we will both introduce and investigate special homeomorphisms, called quasiconformal maps, that are generalizations of the well known conformal maps. We will show that quasiconformal maps make this "gluing," up to conjugation, possible. Quasiconformal maps are a valuable tool in the field of complex dynamics. We will see how quasiconformal maps of infinitesimal circles have an image of an infinitesimal ellipse. Although quasiconformal maps are nice homeomorphisms, they might only be differentiable in the real sense almost everywhere and, surprisingly, complex differentiable nowhere. We shall rely on the work of Lehto and Virtanen as well as Shishikura in exploring these interesting complex valued functions. / Department of Mathematical Sciences

Lam, Ka Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 64-68). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (leaves 108-111).

In 2014, Astala, Päivärinta, Reyes, and Siltanen conducted numerical experiments reconstructing a piecewise continuous conductivity. The algorithm of the shortcut method is based on the reconstruction algorithm due to Nachman, which assumes a priori that the conductivity is Hölder continuous. In this dissertation, we prove that, in the presence of infinite-precision data, this shortcut procedure accurately recovers the scattering transform of an essentially bounded conductivity, provided it is constant in a neighborhood of the boundary. In this setting, Nachman’s integral equations have a meaning and are still uniquely solvable. To regularize the reconstruction, Astala et al. employ a high frequency cutoff of the scattering transform. We show that such scattering transforms correspond to Beltrami coefficients that are not compactly supported, but exhibit certain decay at infinity. For this class of Beltrami coefficients, we establish that the complex geometric optics solutions to the Beltrami equation exist and exhibit the same subexponential decay as described in the 2006 work of Astala and Päivärinta. This is a first step toward extending the inverse scattering map of Astala and Päivärinta to non-compactly supported conductivities.

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