Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

We wish to study those domains in Cn,for n ≥ 2, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We demonstrate that this study is radically different from that of domains in C by discussing some examples of special types of domains in Cn , n ≥2, such that every function holomorphic on them extends to strictly larger domains. Given a domain in Cn , n ≥ 2, we wish to construct the maximal domain of existence for the holomorphic functions defined on the given domain. This leads to Thullen’s construction of a domain (not necessarily in Cn)spread overCn, the so-called envelope of holomorphy, which fulfills our criteria. Unfortunately this turns out to beavery abstract space, far from giving us sense in general howa domain sitting in Cn can be constructed which is strictly larger than the given domain and such that all the holomorphic functions defined on the given domain extend to it. But with the help of this abstract approach we can give a characterization of the domains of holomorphyin Cn , n ≥ 2. The aforementioned characterization is as follows: adomain in Cn is a domain of holomorphy if and only if it is holomorphically convex. However, holomorphic convexity is a very difficult property to check. This calls for other (equivalent) criteria for a domain in Cn , n ≥ 2, to be a domain of holomorphy. We survey these criteria. The proof of the equivalence of several of these criteria are very technical – requiring methods coming from partial differential equations. We provide those proofs that rely on the first part of our survey: namely, on analytic continuation theorems. If a domain Ω Cn , n ≥ 2, is not a domain of holomorphy, we would still like to explicitly describe a domain strictly larger than Ω to which all functions holomorphic on Ω continue analytically. Aspects of Thullen’s approach are also useful in the quest to construct an explicit strictly larger domain in Cn with the property stated above. The tool used most often in such constructions s called “Kontinuitatssatz”. It has been invoked, without a clear statement, in many works on analytic continuation. The basic (unstated) principle that seems to be in use in these works appears to be a folk theorem. We provide a precise statement of this folk Kontinuitatssatz and give a proof of it.

The object of this paper is to define the elementary transcendental functions of a complex variable by means of integrals, and to discuss their properties.

In this paper we show how to give an intrinsic definition of a constructible circle on the sphere. The classical definition of constructible circle in the plane, using straight edge and compass is there by translated in ters of so called Lenart tools. The process by which we achieve our goal involves concepts from the algebra of Hermitian matrices, complex variables, and Sterographic projection. However, the discussion is entirely elementary throughout and hopefully can serve as a guide for teachers in advanced geometry.

Orientador: Suzete Maria Silva Afonso / Banca: Eliris Cristina Rizziolli / Banca: Luciane Parron Gimenes Arantes / Resumo: O objetivo principal deste trabalho é desenvolver um estudo introdutório, porém detalhado, sobre Análise Complexa e algumas de suas aplicações. Apresentamos o corpo dos números complexos, exploramos as funções complexas de uma variável complexa, exibimos parte da teoria das funções analíticas e parte da teoria de integração complexa. Provamos importantes resultados, tais como o Teorema de Cauchy, o Teorema de Taylor, o Teorema dos Resíduos, entre outros igualmente relevantes. Como aplicação da teoria, destacamos a utilização do Teorema dos Resíduos para determinar a transformada inversa de Laplace de uma função F(s) / Abstract: The main objective of this work is to develop an introductory but detailed study on Complex Analysis and some of its applications. We present the field of the complex numbers, explore the complex functions of a complex variable, exhibit part of the theory of analytic functions, and part of the complex integration theory. We prove important results, such as Cauchy's Theorem, Taylor's Theorem, Residue Theorem, among others equally relevant. As an application of the theory, we highlight the use of the Residue Theorem to determine the inverse Laplace transform of a function F(s) / Mestre

This paper demonstrates the properties of a Hilbert structure. In order to have a Hilbert structure it is necessary to satisfy certain properties or axioms. The main body of the paper is centered on six questions that develop these ideas.

Indiana University-Purdue University Indianapolis (IUPUI) / The main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.

Indiana University-Purdue University Indianapolis (IUPUI) / Let f:X\rightarrow X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of \mathbb P^1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z\rightarrow z^b, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W^s_loc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by resenting two examples with a < b for which W^s_loc(S) is not real analytic in the neighborhood of any point.

Indiana University-Purdue University Indianapolis (IUPUI) / Invariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.

<p dir="ltr">This thesis presents applications of one-point quadrature domains to encryption and decryption as well as a method for estimating average temperature. In addition, it investigates methods for finding explicit formulas for certain functions and introduces results regarding quadrature domains for harmonic functions and for double quadrature domains. We use properties of quadrature domains to encrypt and decrypt locations in two dimensions. Results by Bell, Gustafsson, and Sylvan are used to encrypt a planar location as a point in a quadrature domain. A decryption method using properties of quadrature domains is then presented to uncover the location. We further demonstrate how to use data from the decryption algorithm to find an explicit formula for the Schwarz function for a one-point area quadrature domain. Given a double quadrature domain, we show that the fixed points within the area and arc length quadrature identities must be the same, but that the orders at each point may differ between these identities. In the realm of harmonic functions, we demonstrate how to uncover a one-point quadrature identity for harmonic functions from the quadrature identity for a simply-connected one-point quadrature domain for holomorphic functions. We use this result to state theorems for the density of one-point quadrature domains for harmonic functions in the realm of smooth domains with $C^{\infty}$-smooth boundary. These density theorems then lead us to discuss applications of quadrature domains for harmonic functions to estimating average temperature. We end by illustrating examples of the encryption process and discussing the building blocks for future work.</p>