On the numerical performance of a domain decomposition method for conformal mapping

This paper is a sequel to a recent paper [14], concerning a domain decomposition method (hereafter referred to as DDM ) for the conformal mapping of a certain class of quadrilaterals. For the description of the DDM we proceed exactly as in [14:§1], by introducing the following terminology and notations

A domain decomposition method for conformal mapping onto a rectangle

Let g be the function which maps conformally a simply-connected domain G onto a rectangle R so that four specified points z1, z2, z3, z4,o n ∂G are mapped respectively onto the four vertices of R. This paper is concerned with the study of a domain decomposition method for computing approximations to g and to an associated domain functional in cases where: (i) G is bounded by two parallel straight lines and two Jordan arcs. (ii) The four points z1, z2, z3, z4, are the corners where the two straight lines meet the two arcs

A domain decomposition method for approximating the conformal modules of long quadrilaterals

This paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals. The method has been studied already by us and also by D Gaier and W K Hayman, but only in connection with a special class of quadrilaterals, viz. quadrilaterals where: (a) the defining domain is bounded by two parallel straight lines and two Jordan arcs, and (b) the four specified boundary points are the four corners where the arcs meet the straight lines. Our main purpose here is to explain how the method may be extended to a wider class of qua-drilaterals than that indicated above

A numerical method for the computation of faber polynomials for starlike domains

We describe a simple numerical process (based on the Theodorsen method for conformal mapping ) for computing approximations to Faber polynomials for starlike domains

Orthogonal polynomials for area-type measures and image recovery

Let $G$ be a finite union of disjoint and bounded Jordan domains in the complex plane, let $\mathcal{K}$ be a compact subset of $G$ and consider the set $G^\star$ obtained from $G$ by removing $\mathcal{K}$; i.e., $G^\star:=G\setminus \mathcal{K}$. We refer to $G$ as an archipelago and $G^\star$ as an archipelago with lakes. Denote by $\{p_n(G,z)\}_{n=0}^\infty$ and $\{p_n(G^\star,z)\}_{n=0}^\infty$, the sequences of the Bergman polynomials associated with $G$ and $G^\star$, respectively; that is, the orthonormal polynomials with respect to the area measure on $G$ and $G^\star$. The purpose of the paper is to show that $p_n(G,z)$ and $p_n(G^\star,z)$ have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for $G^\star$ are determined by the boundary of $G$. As a consequence we can analyze certain asymptotic properties of $p_n(G^\star,z)$ by using the corresponding results for $p_n(G,z)$, which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.Comment: 24 pages, 9 figure

Approximating the conformal map of elongated quadrilaterals by domain decomposition

Let $Q:=\{ \Omega;z_1,z_2,z_3,z_4\}$ be a quadrilateral consisting of a Jordan domain $\Omega$ and four points $z_1$, $z_2$, $z_3$, $z_4$ in counterclockwise order on $\partial \Omega$ and let $m(Q)$ be the conformal module of $Q$. Then, $Q$ is conformally equivalent to the rectangular quadrilateral $\{R_{m(Q)};0,1,1+im(Q),im(Q)\},$, where $R_{m(Q)}:=\{(\xi,\eta):0<\xi<1, \ 0 <\eta<m(Q)\},$ in the sense that there exists a unique conformal map $f: \Omega \rightarrow R_{m(Q)}$ that takes the four points $z_1$, $z_2$, $z_3$, $z_4$, respectively onto the four vertices $0$, $1$, $1+im(Q)$, $im(Q)$ of $R_{m(Q)}$. In this paper we consider the use of a domain decomposition method (DDM) for computing approximations to the conformal map $f$, in cases where the quadrilateral $Q$ is "long". The method has been studied already but, mainly, in connection with the computation of $m(Q)$. Here we consider certain recent results of Laugesen, for the DDM approximation of the conformal map $f: \Omega \rightarrow R_{m(Q)}$ associated with a special class of quadrilaterals (viz. quadrilaterals whose two non-adjacent boundary segments $(z_2,z_3)$ and $(z_4,z_1)$ are parallel straight lines) and seek to extend these results to more general quadrilaterals. By making use of the available DDM theory for conformal modules, we show that the corresponding theory for $f$ can, indeed, be extended to a much wider class of quadrilaterals than those considered by Laugesen

Curvilinear crosscuts of subdivision for a domain decomposition method in numerical conformal mapping

Let $Q:=\{\Omega;z_1,z_2,z_3,z_4\}$ be a quadrilateral consisting of a Jordan domain $\Omega$ and four distinct points $z_1$, $z_2$, $z_3$ and $z_4$ in counterclockwise order on $\partial \Omega$. We consider a domain decomposition method for computing approximations to the conformal module $m(Q)$ of $Q$ in cases where $Q$ is "long'' or, equivalently, $m(Q)$ is "large''. This method is based on decomposing the original quadrilateral $Q$ into two or more component quadrilaterals $Q_1$, $Q_2,\ldots$ and then approximating $m(Q)$ by the sum of the the modules of the component quadrilaterals. The purpose of this paper is to consider ways for determining appropriate crosscuts of subdivision and, in particular, to show that there are cases where the use of curved crosscuts is much more appropriate than the straight line crosscuts that have been used so far

Finite Number and Finite Size Effects in Relativistic Bose-Einstein Condensation

Bose-Einstein condensation of a relativistic ideal Bose gas in a rectangular cavity is studied. Finite size corrections to the critical temperature are obtained by the heat kernel method. Using zeta-function regularization of one-loop effective potential, lower dimensional critical temperatures are calculated. In the presence of strong anisotropy, the condensation is shown to occur in multisteps. The criteria of this behavior is that critical temperatures corresponding to lower dimensional systems are smaller than the three dimensional critical temperature.Comment: 18 pages, 9 figures, Fig.3 replaced, to appear in Physical Review

O(N) Quantum fields in curved spacetime

For the O(N) field theory with lambda Phi^4 self-coupling, we construct the two-particle-irreducible (2PI), closed-time-path (CTP) effective action in a general curved spacetime. From this we derive a set of coupled equations for the mean field and the variance. They are useful for studying the nonperturbative, nonequilibrium dynamics of a quantum field when full back reactions of the quantum field on the curved spacetime, as well as the fluctuations on the mean field, are required. Applications to phase transitions in the early Universe such as the Planck scale or in the reheating phase of chaotic inflation are under investigation.Comment: 31 pages, 2 figures, uses RevTeX 3.1, LaTeX 2e, AMSfonts 2.2, graphics 0.6; To appear in Phys. Rev. D (7/15/97