1,798 research outputs found
Real Separated Algebraic Curves, Quadrature Domains, Ahlfors Type Functions and Operator Theory
The aim of this paper is to inter-relate several algebraic and analytic
objects, such as real-type algebraic curves, quadrature domains, functions on
them and rational matrix functions with special properties, and some objects
from Operator Theory, such as vector Toeplitz operators and subnormal
operators. Our tools come from operator theory, but some of our results have
purely algebraic formulation. We make use of Xia's theory of subnormal
operators and of the previous results by the author in this direction. We also
correct (in Section 5) some inaccuracies in two papers by the author in Revista
Matematica Iberoamericana (1998).Comment: 43 pages, 2 figures; zip archiv
The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc
Given a bounded n-connected domain in the plane bounded by non-intersecting
Jordan curves, and given one point on each boundary curve, L. Bieberbach proved
that there exists a proper holomorphic mapping of the domain onto the unit disc
that is an n-to-one branched covering with the properties that it extends
continuously to the boundary and maps each boundary curve one-to-one onto the
unit circle, and it maps each given point on the boundary to the point 1 in the
unit circle. We modify a proof by H. Grunsky of Bieberbach's result to show
that there is a rational function of 2n+2 complex variables that generates all
of these maps. We also show how to generate all the proper holomorphic mappings
to the unit disc via the rational function.Comment: 17 page
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
Robust and efficient solution of the drum problem via Nystrom approximation of the Fredholm determinant
The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian
with Dirichlet boundary condition-has many applications, yet remains
challenging for general domains when high accuracy or high frequency is needed.
Boundary integral equations are appealing for large-scale problems, yet certain
difficulties have limited their use. We introduce two ideas to remedy this: 1)
We solve the resulting nonlinear eigenvalue problem using Boyd's method for
analytic root-finding applied to the Fredholm determinant. We show that this is
many times faster than the usual iterative minimization of a singular value. 2)
We fix the problem of spurious exterior resonances via a combined field
representation. This also provides the first robust boundary integral
eigenvalue method for non-simply-connected domains. We implement the new method
in two dimensions using spectrally accurate Nystrom product quadrature. We
prove exponential convergence of the determinant at roots for domains with
analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency,
in a variety of domain shapes including ones with strong exterior resonances.Comment: 21 pages, 7 figures, submitted to SIAM Journal of Numerical Analysis.
Updated a duplicated picture. All results unchange
Topology of quadrature domains
We address the problem of topology of quadrature domains, namely we give
upper bounds on the connectivity of the domain in terms of the number of nodes
and their multiplicities in the quadrature identity.Comment: 37 pages, 11 figures in J. Amer. Math. Soc., Published
electronically: May 11, 201
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An orthonormalization method for the approximate conformal mapping of multiply-connected domains
We consider the use of an orthonormalization method for constructing approximations to one of the standard conformal maps for multiply-connected domains. The method has been used successfully in [12], but only for the mapping of doubly-connected domains. Our purpose here is to consider its application to the mapping of domains whose connectivity is greater than two
Complexity in complex analysis
We show that the classical kernel and domain functions associated to an
n-connected domain in the plane are all given by rational combinations of three
or fewer holomorphic functions of one complex variable. We characterize those
domains for which the classical functions are given by rational combinations of
only two or fewer functions of one complex variable. Such domains turn out to
have the property that their classical domain functions all extend to be
meromorphic functions on a compact Riemann surface, and this condition will be
shown to be equivalent to the condition that an Ahlfors map and its derivative
are algebraically dependent. We also show how many of these results can be
generalized to finite Riemann surfaces.Comment: 30 pages, to appear in Advances in Mat
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