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

Abstract

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.

Riemann–Hilbert problem on a hyperelliptic surface and uniformly stressed inclusions embedded into a half-plane subjected to antiplane strain

“…For the general case, another technique may be developed. It includes employing a circular false(n+1false)-connected domain as a parametric domain and solving the associated Riemann–Hilbert problems of the theory of automorphic functions [23,24]. This is planned as a part of future research.…”

Circular map for supercavitating flow in a multiply connected domain

“…The existence of the G-quasimultiplicative kernel M(ζ, ξ ) was recently proved by Antipov and Silvestrov (24). For the particular case of the group G, a first class group, such a kernel was constructed by Silvestrov (22) in terms of series similar to the Poincaré theta series.…”

The Ultimate Noise Limit for Hall Plates in Voltage, Current, and Hybrid Operating Modes

“…In (17) we note that the N-th weighing coefficient d N is subtracted from all other coefficients. Therefore d N is superfluous: suppose we search for optimum coefficients with some arbitrary value for d N , say 3.…”

“…However, there the theory is limited to thin, plane, single connected Hall effect regions with peripheral contacts. Although these are no severe restrictions for practical use, it rules out some "exotic" variants like [15] [16] [17] [18]. The formulae of [14] apply for circular Hall plates.…”