Ullemar's formula for the jacobian of the complex moment mapping

Kuznetsova, O. S.; Tkachev, Vladimir G.

doi:10.1080/02781070310001634610

Cited by 10 publications

(12 citation statements)

References 12 publications

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“…. , c n ) has been established by O. Kouznetsova and V. Tkachev [142], [143], who proved an explicit formula for the (nonzero) Jacobi determinant of the map from the coefficients of f to the moments (c 1 , . .…”

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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“…when f restricted to the class of polynomials of a fixed degree. This formula, which was proved by to O. Kuznetsova and V. Tkachev [12], [19] after an initial conjecture of C. Ullemar [20], will be discussed in some depth below.…”

Section: The String Equation For Univalent Conformal Mapsmentioning

confidence: 93%

The string equation for polynomials

Gustafsson

2018

Anal.Math.Phys.

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For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we investigate to what extent the string equation makes sense and holds for non-univalent analytic functions.We give positive answers in two cases: for polynomials and for a special class of rational functions.

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“…An explicit expression for this determinant has been found by O. Kuznetsova and V. Tkachev [338], [556], based on a conjecture of C. Ullemar [558]. .…”

Section: Moment Coordinatesmentioning

confidence: 99%

Classical and Stochastic Laplacian Growth

Gustafsson

Teodorescu

Васильев

2014

Advances in Mathematical Fluid Mechanics

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More information about this series at http://www.springer.com/series/5032 Advances in Mathematical Fluid Mechanics is a forum for the publication of high quality monographs, or collections of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its mathematical aims and scope are similar to those of the Journal of Mathematical Fluid Mechanics. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on fluid mechanics.The monographs and collections of works published here may be written in a more expository style than is usual for research journals, with the intention of reaching a wide audience. Collections of review articles will also be sought from time to time.

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Ullemar's formula for the jacobian of the complex moment mapping

Cited by 10 publications

References 12 publications

Random matrices in 2D, Laplacian growth and operator theory

Random matrices in 2D, Laplacian growth and operator theory

The string equation for polynomials

Classical and Stochastic Laplacian Growth

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