Uniform Approximation on Riemann Surfaces

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is usually not possible to approximate f uniformly by functions holomorphic on all of R. Firstly, we show, however, that for every open Riemann surface R and every closed subset E of R; there is closed subset F of E, which approximates E extremely well, such that every function holomorphic on F can be approximated much better than uniformly by functions holomorphic on R. Secondly, given a function f from a closed subset of a Riemann surface R to the Riemann sphere C; we seek to approximate f in the spherical distance by functions meromorphic on R. As a consequence we generalize a recent extension of Mergelyan\u27s theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis. The problem of approximating by meromorphic functions pole-free on E is equivalent to that of approximating by meromorphic functions zero-free on E, which in turn is related to Voronin\u27s spectacular universality theorem for the Riemann zeta-function. The reection principles of Schwarz and Caratheodory give conditions under which holomorphic functions extend holomorphically to the boundary and the theorem of Osgood-Caratheodory states that a one-to-one conformal mapping from the unit disc to a Jordan domain extends to a homeomorphism of the closed disc onto the closed Jordan domain. Finally, in the last Chapter, we study similar questions on Riemann surfaces for holomorphic mappings

Flow of fluid through a curved tube

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Real root isolation for exact and approximate polynomials using descartes' rule of signs

Collins und Akritas (1976) have described the Descartes method for isolating the real roots of an integer polynomial in one variable. This method recursively subdivides an initial interval until Descartes' Rule of Signs indicates that all roots have been isolated. The partial converse of Descartes' Rule by Obreshkoff (1952) in conjunction with the bound of Mahler (1964) and Davenport (1985) leads us to an asymptotically almost tight bound for the resulting subdivision tree. It implies directly the best known complexity bounds for the equivalent forms of the Descartes method in the power basis (Collins/Akritas, 1976), the Bernstein basis (Lane/Riesenfeld, 1981) and the scaled Bernstein basis (Johnson, 1991), which are presented here in a unified fashion. Without losing correctness of the output, we modify the Descartes method such that it can handle bitstream coefficients, which can be approximated arbitrarily well but cannot be determined exactly. We analyze the computing time and precision requirements. The method described elsewhere by the author together with Kerber/Wolpert (2007) and Kerber (2008) to determine the arrangement of plane algebraic curves rests in an essential way on variants of the bitstream Descartes algorithm; we analyze a central part of it.Collins und Akritas (1976) haben das Descartes-Verfahren zur Einschließung der reellen Nullstellen eines ganzzahligen Polynoms in einer Veränderlichen angegeben. Das Verfahren unterteilt rekursiv ein Ausgangsintervall, bis die Descartes'sche Vorzeichenregel anzeigt, dass alle Nullstellen getrennt worden sind. Die partielle Umkehrung der Descartes'schen Regel nach Obreschkoff (1952) in Verbindung mit der Schranke von Mahler (1964) und Davenport (1985) führt uns auf eine asymptotisch fast scharfe Schranke für den sich ergebenden Unterteilungsbaum. Daraus folgen direkt die besten bekannten Komplexitätsschranken für die äquivalenten Formen des Descartes-Verfahrens in der Monom-Basis (Collins/Akritas, 1976), der Bernstein-Basis (Lane/Riesenfeld, 1981) und der skalierten Bernstein-Basis (Johnson, 1991), die hier vereinheitlicht dargestellt werden. Ohne dass die Korrektheit der Ausgabe verloren geht, modifizieren wir das Descartes-Verfahren so, dass es mit "Bitstream"-Koeffizienten umgehen kann, die beliebig genau angenähert, aber nicht exakt bestimmt werden können. Wir analysieren die erforderliche Rechenzeit und Präzision. Das vom Verfasser mit Kerber/Wolpert (2007) und Kerber (2008) an anderer Stelle beschriebene Verfahren zur Bestimmung des Arrangements (der Schnittfigur) ebener algebraischer Kurven fußt wesentlich auf Varianten des Bitstream-Descartes-Verfahrens; wir analysieren einen zentralen Teil davon

Fluctuations of eigenvalues of random normal matrices

In this note, we prove Gaussian field convergence of fluctuations of eigenvalues of random normal matrices in the interior of a quantum droplet

Difference Equations Everywhere : Some Motivating Examples

Publicació amb motiu de la International Conference on Difference Equations and Applications (July 24-28, 2017, Timişoara, Romania) amb el títol Difference Equations, Discrete Dynamical Systems and ApplicationsThis work collects several situations where discrete dynamical systems or difference equations appear. Most of them are different from the examples used in textbooks and from the usual mathematical models appearing in Biology or Economy. The examples are presented in detail, including some appropriate references. Although most of them are known, the fact of collecting all together aims to be a source of motivation for studying DDS and difference equations and to facilitate teaching these subjects

Study of the Equilibria of Parabolic Differential Equations with Interfaces Intersecting the Boundary

Existence of steady state solutions for the Allen-Cahn and Cahn-Hilliard equations in two dimensional domains is discussed. We are in particular interested in establishing existence of single layered equilibria with the property that their transition layer intersects the boundary. In the case of the Allen-Cahn equation we consider bone-like domains and seek solutions intersecting the flat part of the boundary. We establish conditions for the domain which ensure existence of such equilibria. Their stability is also analyzed. For the Cahn-Hilliard equations we show that there exist equilibria near every point of a local maximum of the curvature of the boundary