Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

Antipov, Y. A.; Crowdy, Darren

doi:10.1007/s11785-007-0020-3

Cited by 6 publications

(10 citation statements)

References 13 publications

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“…Its solution was derived [6] for an (N + 1)-connected circular domain in terms of an automorphic analogue of the Cauchy kernel. The kernel was expressed through the Schottky-Klein prime function of the associated Schottky double.…”

Section: ) With Discontinuous Functions A(t) B(t) and C(t)mentioning

confidence: 99%

“…In general, the kernel can be expressed through the Schottky-Klein prime function ω(z, τ ) associated with the group G [6] by the formula to indicate the starting and the terminal points of the arc t νj t νj+1 , respectively (it is assumed that t νm ν +1 = t ν1 ). On the arc t ν1 t ν2 , a branch of the function arg p(τ ) can be fixed arbitrarily.…”

Section: ) With Discontinuous Functions A(t) B(t) and C(t)mentioning

confidence: 99%

“…The Maxwell equations written for a source-free 2-d medium in the steady-state case imply the harmonicity of the current in the domain D. On the electrodes, the tangential component E τ of the electric field intensity vanishes, whilst on the dielectrics (insulated walls), the normal component J n of the current intensity vanishes: 5) where θ is the polar angle in the parametrization of the circle L ν , t − δ ν = ρ ν e iθ , e ν and d ν are the unions of the electrodes and the dielectrics on the circle L ν , respectively, 6) and t ν2n ν +1 = t ν1 . By employing Ohm's law (8.4), we rewrite the first boundary condition in (8.5) in the form…”

Section: Piecewise Constant Coefficients A(t) and B(t)mentioning

confidence: 99%

“…One of the steps of the solution to the Hilbert problem with discontinuous coefficients is the factorization problem. Its solution was derived [6] for an (N + 1)connected circular domain in terms of an automorphic analogue of the Cauchy kernel. The kernel was expressed through the Schottky-Klein prime function of the associated Schottky double.…”

Section: Introductionmentioning

confidence: 99%

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Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

Antipov

Silvestrov

2010

Quart. Appl. Math.

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Abstract.In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an (N + 1)-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetric Schottky group. We prove the existence theorem for the second (quasimultiplicative) kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem, which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with N + 1 circular holes with electrodes and dielectrics on the walls when the conductor is placed at a right angle to the applied magnetic field.

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Section: ) With Discontinuous Functions A(t) B(t) and C(t)mentioning

confidence: 99%

Section: ) With Discontinuous Functions A(t) B(t) and C(t)mentioning

confidence: 99%

Section: Piecewise Constant Coefficients A(t) and B(t)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

Antipov

Silvestrov

2010

Quart. Appl. Math.

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“…For their solution we employ a qusiautomorphic analogue of the Cauchy kernel (19), (21). Note that the Cauchy kernel analogue (29) in terms of the Schottky-Klein prime function of the Schottky group could also be employed. In Section 4, for the first class Schottky groups (30), we write down a series representation of a family of conformal maps solving the problem.…”

Section: Introductionmentioning

confidence: 99%

Method of automorphic functions for an inverse problem of antiplane elasticity

Antipov

2019

The Quarterly Journal of Mechanics and Applied Mathematics

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A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of n uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a (3n − 4)-parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.

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Geometric function theory: a modern view of a classical subject

Crowdy

2008

Nonlinearity

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Geometric function theory is a classical subject. Yet it continues to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, nonlinear integrable systems theory and the theory of partial differential equations. This paper surveys, with a view to modern applications, open problems and challenges in this subject. Here we advocate an approach based on the use of the Schottky-Klein prime function within a Schottky model of compact Riemann surfaces.

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Riemann–Hilbert Problem for Automorphic Functions and the Schottky–Klein Prime Function

Cited by 6 publications

References 13 publications

Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

Method of automorphic functions for an inverse problem of antiplane elasticity

Geometric function theory: a modern view of a classical subject

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