Green's functions for multiply connected domains via conformal mapping

A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iteration. By making the end of the strip jagged, the method can be generalised to weighted Green's functions and weighted approximations

Path Planning with Divergence-Based Distance Functions

Distance functions between points in a domain are sometimes used to automatically plan a gradient-descent path towards a given target point in the domain, avoiding obstacles that may be present. A key requirement from such distance functions is the absence of spurious local minima, which may foil such an approach, and this has led to the common use of harmonic potential functions. Based on the planar Laplace operator, the potential function guarantees the absence of spurious minima, but is well known to be slow to numerically compute and prone to numerical precision issues. To alleviate the first of these problems, we propose a family of novel divergence distances. These are based on f-divergence of the Poisson kernel of the domain. We define the divergence distances and compare them to the harmonic potential function and other related distance functions. Our first result is theoretical: We show that the family of divergence distances are equivalent to the harmonic potential function on simply-connected domains, namely generate paths which are identical to those generated by the potential function. The proof is based on the concept of conformal invariance. Our other results are more practical and relate to two special cases of divergence distances, one based on the Kullback-Leibler divergence and one based on the total variation divergence. We show that using divergence distances instead of the potential function and other distances has a significant computational advantage, as, following a pre-processing stage, they may be computed up to an order of magnitude faster than the others when taking advantage of certain sparsity properties of the Poisson kernel. Furthermore, the computation is "embarrassingly parallel", so may be implemented on a GPU with up to three orders of magnitude speedup

Volume-preserving deformation using generalized barycentric coordinates

The cage-based deformation of a 3D object through generalized barycentric coordinates is a simple, e fficient, effective and hence widely used shape manipulation scheme. Editing vertices of the polyhedral cage induces a smooth space deformation of its interior; the vertices thus become control handles of the final deformation. However, in some application fi elds, as medicine, constrained volume preserving deformations are required. In this paper, we present a solution that takes advantage of the potential of the deformations based on generalized barycentric coordinates while adding the constraint of keeping a volume constant. An implementation of the proposed scheme is presented and discussed. A measure of local stress of the deformed volume is also proposed.Peer ReviewedPostprint (author’s final draft

Asymptotics of wavelets and filters

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 126-131).by Jianhong (Jackie) Shen.Ph.D

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Free boundary problems in a Hele-Shaw cell

The motion of a free boundary separating two immiscible fluids in an unbounded Hele-Shaw cell is considered. In the one-phase problem, a viscous fluid is separated from an inviscid fluid by a simple closed boundary. Preliminaries for a complex variable technique are presented by which the one-phase problem can be solved explicitly via conformal mappings. The Schwarz function of the boundary plays a major role giving rise to the so called Schwarz function equation which governs the evolution of exact solutions. The Schwarz function approach is used to study the stability of a translating elliptical bubble due to a uniform background flow, and the stability of a blob (or bubble) subject to an external electric field. The one-phase problem of a translating free boundary and of a free boundary subject to an external field are studied numerically. A boundary integral method is formulated in the complex plane by considering the Cauchy integral formula and the complex velocity of a fluid particle on the free boundary. In the case of a free boundary subject to an external electric field due to a point charge, it is demonstrated that a stable steady state is achieved for appropriate charge strength. The method is also employed to study breakup of a single translating bubble in which the Schwarz function singularities (shown to be stationary) of the initial boundary play an important role. The two-phase problem is also considered, where the free boundary now separates two viscous fluids, and the construction of exact solutions is studied. The one-phase numerical model is enhanced, where a boundary integral method is formulated to accommodate the variable pressure in both viscous phases. Some numerical experiments are presented with a comparison to analytical results, in particular for the case where the free boundary is driven by a uniform background flow

Orthogonal polynomials, equilibrium measures and quadrature domains associated with random matrix models

Motivated by asymptotic questions related to the spectral theory of complex random matrices, this work focuses on the asymptotic analysis of orthogonal polynomials with respect to quasi-harmonic potentials in the complex plane. The ultimate goal is to develop new techniques to obtain strong asymptotics (asymptotic expansions valid uniformly on compact subsets) for planar orthogonal polynomials and use these results to understand the limiting behavior of spectral statistics of matrix models as their size goes to infinity. For orthogonal polynomials on the real line the powerful Riemann-Hilbert approach is the main analytic tool to derive asymptotics for the eigenvalue correlations in Hermitian matrix models. As yet, no such method is available to obtain asymptotic information about planar orthogonal polynomials, but some steps in this direction have been taken. The results of this thesis concern the connection between the asymptotic behavior of orthogonal polynomials and the corresponding equilibrium measure. It is conjectured that this connection is established via a quadrature identity: under certain conditions the weak-star limit of the normalized zero counting measure of the orthogonal polynomials is a quadrature measure for the support of the equilibrium measure of the corresponding two-dimensional electrostatic variational problem of the underlying potential. Several results are presented on equilibrium measures, quadrature domains, orthogonal polynomials and their relation to matrix models. In particular, complete strong asymptotics are obtained for the simplest nontrivial quasi-harmonic potential by a contour integral reduction method and the Riemann-Hilbert approach, which confirms the above conjecture for this special cas