by Yu Wai-kuen. / Thesis (M. Phil.)--Chinese University of Hong Kong, 1981. / Bibliography: l. 52-55.

<p>This thesis consists of three parts. The first part is a introduction to the theory of domains of holomorphy through holomorphic convexity. The second part gives a introduction to Stein neighborhood bases in <strong>C</strong>ⁿ<strong> </strong>and presents some minor results on the Nebenhülle of a compact set in <strong>C</strong>ⁿ. The third and final part reviews some results on the existence of Stein neighborhood bases.</p> / <p>Denna uppsats består av tre delar. Den första delen introducerar holomorfiområden genom teorin för konvexitet med avseende på holomorfa funktioner. Den andra delen är en introduktion till Steinomgivningsbaser i <strong>Cⁿ</strong> och presenterar några smärre resultat rörande Nebenhüllet till en kompakt mängd i <strong>Cⁿ</strong>. Den tredje och sista delen ger en överblick över en del resultat om existensen av Steinomgivningsbaser.</p>

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1997. / Includes bibliographical references. Also available on the Internet.

This thesis consists of three parts. The first part is a introduction to the theory of domains of holomorphy through holomorphic convexity. The second part gives a introduction to Stein neighborhood bases in Cn and presents some minor results on the Nebenhülle of a compact set in Cn. The third and final part reviews some results on the existence of Stein neighborhood bases. / Denna uppsats består av tre delar. Den första delen introducerar holomorfiområden genom teorin för konvexitet med avseende på holomorfa funktioner. Den andra delen är en introduktion till Steinomgivningsbaser i Cn och presenterar några smärre resultat rörande Nebenhüllet till en kompakt mängd i Cn. Den tredje och sista delen ger en överblick över en del resultat om existensen av Steinomgivningsbaser.

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This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing. We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal. We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions $\mathcal{R}$ we associate a sheaf $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ of analytic germs which is closed under application of the implicit function theorem. We prove that $\mathcal{O}^{\scriptscriptstyle{\mathcal{R}}}$ is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call $\mathcal{R}$-analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in $\mathcal{R}$. If this condition is satisfied then the $\mathcal{R}$-analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture.

Thesis (Ph. D.)--Brandeis University, 2008. / "UMI:3306459." MICROFILM COPY ALSO AVAILABLE IN THE UNIVERSITY ARCHIVES. Includes bibliographical references.

In this thesis, we study integral representations of Herglotz-Nevanlinna functions, that is to say holomorphic functions defined on a product of several copies of the complex upper half-plane having non-negative imaginary part. The manuscript is divided into three parts, beginning with a general introduction followed by two papers. In the general introduction, we familiarize ourselves with the concept of a Herglotz-Nevanlinna function as well as providing a comprehensive introduction into the theory of integral representations for this particular class of functions. Paper I treats exclusively the two-variable case and presents an integral representation of Herglotz-Nevanlinna functions in two complex variables in terms of a real number, two non-negative numbers and a positive Borel measure satisfying two properties. Three properties that hold for the class of measures appearing in such integral representations are also proven. In Paper II, we provide an integral representation for the class of Herglotz-Nevanlinna functions in arbitrarily many complex variables in terms of a real number, a linear term and a positive Borel measure satisfying two properties. Properties of the class of measures appearing in this representation are then discussed in detail as well as alternative descriptions of said class. Finally, a symmetry formula satisfied by Herglotz-Nevanlinna functions is proved at the end.