Quadrature Identities and Deformation of Quadrature Domains

Sjödin, Tomas

doi:10.1007/3-7643-7316-4_12

Cited by 8 publications

(7 citation statements)

References 7 publications

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“…It follows that multiply connected quadrature domains for analytic functions for a given µ occur in continuous families. It even turns out [48], [149] that any two algebraic domains for the same µ can be deformed into each other through families as above. Thus there is a kind of uniqueness at a higher level: given any µ there is at most one connected family of algebraic domains belonging to it.…”

Section: Signed Measures Instability Uniquenessmentioning

confidence: 99%

Random matrices in 2D, Laplacian growth and operator theory

Mineev-Weinstein

Putinar

Teodorescu³

2008

J. Phys. A: Math. Theor.

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Abstract. Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.

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Section: Signed Measures Instability Uniquenessmentioning

confidence: 99%

Random matrices in 2D, Laplacian growth and operator theory

Mineev-Weinstein

Putinar

Teodorescu³

2008

J. Phys. A: Math. Theor.

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“…Now by Theorem 4.2 in [6] and the fact that any harmonic function on has a harmonic conjugate it follows that there is a real Radon measure with support in ! such that…”

Section: Another Existence Results For the Ellipsementioning

confidence: 94%

Mother bodies of algebraic domains in the complex plane

Sjödin

2006

Complex Variables and Elliptic Equations

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We give a definition of a mother body of a domain in the complex plane, and prove some continuity properties of its potential in terms of the Schwarz function (which is explicitly assumed to exist). We end the article by studying the case of the ellipse, and use the previous results to prove existence and uniqueness of a mother body in this case, as well as a related existence result about graviequivalent measures for the ellipse.

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“…, 0) = Ω and It follows that multiply connected quadrature domains for analytic functions for a given µ occur in continuous families. It even turns out [19], [73] that any two algebraic domains for the same µ can be deformed into each other through families as above. Thus there is a kind of uniqueness at a higher level: given any µ there is at most one connected family of algebraic domains belonging to it.…”

Section: Remark 42mentioning

confidence: 99%

Selected topics on quadrature domains

Gustafsson

Putinar

2007

Physica D: Nonlinear Phenomena

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This is a selection of facts, old and new, about quadrature domains. The text, written in the form of a survey, is addressed to non-experts and covers a variety of phenomena related to quadrature domains. Such as: the difference between quadrature domains for subharmonic, harmonic and respectively complex analytic functions, geometric properties of the boundary, instability in the reverse Hele-Shaw flow, dependence and non-uniqueness on the quadrature data, interpretation in terms of function theory on Riemann surfaces, a matrix model and a reconstruction algorithm. Plus some low degree/order examples where computations can be carried out in detail.

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Quadrature Identities and Deformation of Quadrature Domains

Cited by 8 publications

References 7 publications

Random matrices in 2D, Laplacian growth and operator theory

Random matrices in 2D, Laplacian growth and operator theory

Mother bodies of algebraic domains in the complex plane

Selected topics on quadrature domains

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