Green's Functions and Infinite Products

Melnikov, Yuri A.

doi:10.1007/978-0-8176-8280-4

Cited by 26 publications

(14 citation statements)

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“…In this section we give the detailed computation of the Green's function. Similar computations can be found in [19]. We will work on the annulus of radii a and b with a < b.…”

Section: Green's Function On the Annulusmentioning

confidence: 76%

Liouville quantum gravity on the annulus

Rémy

2018

Journal of Mathematical Physics

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In this work we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists initiated by Polyakov in 1981 [22]. It is also a very important example of a conformal field theory (CFT). Results have already been obtained on the Riemann sphere [4] and on the unit disk [13] so this paper will follow the same approach. The case of the annulus contains two difficulties: it is a surface with two boundaries and it has a non-trivial moduli space. We recover the Weyl anomaly -a formula verified by all CFT -and deduce from it the KPZ formula. We also show that the full partition function of Liouville quantum gravity integrated over the moduli space is finite. This allows us to give the joint law of the Liouville measures and of the random modulus and to write the conjectured link with random planar maps.

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“…In this section we give the detailed computation of the Green's function. Similar computations can be found in [19]. We will work on the annulus of radii a and b with a < b.…”

Section: Green's Function On the Annulusmentioning

confidence: 76%

Liouville quantum gravity on the annulus

Rémy

2018

Journal of Mathematical Physics

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“…Defining Green's function using Brownian motion allows for a number of infinite product identities to be deduced, including Euler's celebrated infinite product representation for the sine function. This is shown in [53], while the corresponding analytic derivations of some of these identities can be found in [59] and [60].…”

Section: Stochastic Loewner Evolutionmentioning

confidence: 88%

Planar Brownian Motion and Complex Analysis

Markowsky

2020

Preprint

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“…Acoustic fields in the concave part of a corner or acute-angled wedges can be analysed. However, there are certain restrictions in realising a wedgeshaped acoustic domain through the method of images (17,26). Having a finite number of image scatterers requires a wedge angle of β = π/Q where Q is the number of total wedges in a given halfplane; hence, 2Q wedges in total.…”

Section: Methods Of Imagesmentioning

confidence: 99%

Image conditions for polar and cylindrical coordinate separation-of-variables acoustic multiple scattering models with perfectly reflecting flat boundaries

Shin

2018

The Quarterly Journal of Mechanics and Applied Mathematics

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This article addresses efficient implementation of the method of images for acoustic multiple scattering models (MSM) with perfectly reflecting flat boundaries. Time-harmonic problems are first solved in the polar coordinate system for circular scatterers; then the models are extended to the cylindrical coordinate system with (semi-)infinitely long circular cylinders. The MSM in this article is based on the method of separation of variables and Graf's addition theorem. Derivations are provided for 'image conditions' which relate the unknown coefficients of outgoing waves from image scatterers with those of real counterparts. The method of images is applied to wedgeshaped domains with apex angles of π/n rad for a positive integer n. Image conditions make the MSM numerically more efficient: the system of linear equations for unknown coefficients is formulated 2n times faster; its memory requirements are reduced by 4n 2 times for direct solvers. The proposed model is applied to a benchmark wedge in ocean environment with n = 64. Good agreement is observed between the MSM with image conditions and the boundary element method. Furthermore, half-and quarter-space measurements in an anechoic chamber are in accordance with the correct use of image conditions. Incorrect image conditions reported elsewhere for polar coordinates are also discussed.

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Green's Functions and Infinite Products

Cited by 26 publications

References 0 publications

Liouville quantum gravity on the annulus

Liouville quantum gravity on the annulus

Planar Brownian Motion and Complex Analysis

Image conditions for polar and cylindrical coordinate separation-of-variables acoustic multiple scattering models with perfectly reflecting flat boundaries

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