Towards an Effectivisation of the Riemann Theorem

Natanzon, Sergei

doi:10.1007/s10455-005-7276-5

Cited by 7 publications

(8 citation statements)

References 13 publications

(11 reference statements)

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“…with universal coefficients which will be found. For a particular function f (t 0 ) they were found in [22]. In the general case the arguments are similar.…”

Section: Taylor Expansion Of Symmetric Solutionsmentioning

confidence: 58%

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Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

Natanzon

Zabrodin

2013

Preprint

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We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 double Hurwitz numbers. They can also serve as a starting point for a constructive approach to the Riemann mapping problem and the inverse potential problem in 2D.

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“…with universal coefficients which will be found. For a particular function f (t 0 ) they were found in [22]. In the general case the arguments are similar.…”

Section: Taylor Expansion Of Symmetric Solutionsmentioning

confidence: 58%

“…r m . Our method is an extension of the method developed in [22,23] for the formal c = 1 string solution. Convergence of the Taylor series for this case was investigated in [11].…”

Section: Introductionmentioning

confidence: 99%

Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics

Natanzon

Zabrodin

2013

Preprint

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“…Substituting r = 1 into series (5) and comparing the result with WF (3) it is easy to find that exact solution of IDP (1-3) is equal to:…”

Section: The Internal Dirichlet Problem and Fractality Of Boundary Conditionmentioning

confidence: 99%

“…Generally speaking one may derive this mapping f (z, ε) explicitly in the framework of formalism of the harmonic moments of exterior domain C\ ε [5] but this way is too hard. On the other hand due to representation (23) of ε we can restrict ourselves by construction of approximate conformal mapping of the nearly circular domain on unit disk.…”

Section: The Internal Dirichlet Problem and Fractality Of Boundary Conditionmentioning

confidence: 99%

Fractality and the Internal Dirichlet Problem

Azamova¹,

Alekseeva²,

Потапов

et al. 2021

13th Chaotic Modeling and Simulation International Conference

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In this chapter influence of fractality on solution of the two-dimensional internal Dirichlet problem is analyzed. Two different situations are considered namely the first of them deals with fractal boundary condition on the unit disk. In this case exact solution of the Laplace equation proves to obey to some analog of the de Rham functional equation. Also norm and the Dirichlet integral for this solution has been estimated. In the second situation boundary condition is supposed to be regular but boundary of the domain is fractally perturbed. For clarification of this case both approximate conformal mapping technique and the Potapov concept of physical fractals has been applied.

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“…Following [15], we define the combinatorial constantsP ij Proof. Since ∆(z) = k≥1 z −k k ∂h k , the Hirota equation (26) yields…”

Section: Theh-kp Hierarchy In Terms Of ∂H Imentioning

confidence: 99%