Selected topics on quadrature domains

Gustafsson, Björn; Putinar, Mihai

doi:10.1016/j.physd.2007.04.015

“…Conformal images of the unit disc by rational functions are also called quadrature domains. We refer to [23,31] for details and ramifications of the theory of quadrature domains. In particular, up to the inversion z → 1/z, the complements of the domains in class Q are quadrature domains.…”

Section: Algebraic Domains Of Class Qmentioning

confidence: 99%

Shape reconstruction of nanoparticles from their associated plasmonic resonances

Ammari

¹

,

Putinar

²

,

Ruiz

³

et al. 2019

Journal de Mathématiques Pures et Appliquées

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We prove by means of a couple of examples that plasmonic resonances can be used on one hand to classify shapes of nanoparticles with real algebraic boundaries and on the other hand to reconstruct the separation distance between two nanoparticles from measurements of their first collective plasmonic resonances. To this end, we explicitly compute the spectral decompositions of the Neumann-Poincaré operators associated with a class of quadrature domains and two nearly touching disks. Numerical results are included in support of our main findings.Mathematics Subject Classification (MSC2000): 35R30, 35C20.

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“…For reconstruction of polygonal domains see [17,20]. In general, an important class of quadrature domains (see [2,21] and references therein) provides a natural framework for a study of the "finite dimensional" reconstruction problem, as well as its possible non-uniqueness, in particular, the invisibility phenomenon.…”

Section: Complex Moments Of Planar Domainsmentioning

confidence: 99%

Reconstruction of Planar Domains from Partial Integral Measurements

Batenkov¹,

Golubyatnikov²,

Yomdin³

2013

Complex Analysis and Dynamical Systems V

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We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewisealgebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [5] and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given.Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects.----------------

show abstract

“…In general, an important class of quadrature domains (see [2,21] and references therein) provides a natural framework for a study of the "finite dimensional" reconstruction problem, as well as its possible non-uniqueness, in particular, the invisibility phenomenon.…”

Section: Complex Moments Of Planar Domainsmentioning

confidence: 99%

Complex Analysis and Dynamical Systems V

Batenkov¹,

Golubyatnikov²,

Yomdin³

2013

Contemporary Mathematics

3

0

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We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewisealgebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [5] and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given.Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects.----------------

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Selected topics on quadrature domains

Cited by 27 publications

References 72 publications

Shape reconstruction of nanoparticles from their associated plasmonic resonances

Shape reconstruction of nanoparticles from their associated plasmonic resonances

Reconstruction of Planar Domains from Partial Integral Measurements

Complex Analysis and Dynamical Systems V

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