A numerical method for quasiconformal mapping of doubly connected domains onto annuli is presented. The ratio R of the radii of the annulus is not known a priori and is determined as part of the solution procedure. The numerical method presented in this paper requires solving iteratively a sequence of inhomogeneous Beltrami equations, each for a diierent R. R is updated using a procedure based ...

We study magnetic field evolution in flows with fluctuating in time governing parameters in electrically conducting fluid. We use a standard mean-field approach to derive equations for large-scale magnetic field for the fluctuating Arnold-Beltrami-Childress (ABC) flow as well as for the fluctuating Roberts flow. The derived mean-field dynamo equations have growing solutions with growth rate of ...

Harmonic morphisms are mappings between Riemannian manifolds which preserve Laplace’s equation. They can be characterized as harmonic maps which enjoy an extra property called horizontal weak conformality or semiconformality. We shall give a brief survey of the theory concentrating on (i) twistor methods, (ii) harmonic morphisms with one-dimensional fibres; in particular we shall outline the co...

We consider first-order bosonic string theory, perturbed by the primary operator, corresponding to deformation of the target-space complex structure. We compute the effective action in this theory and find that its consistency with the worldsheet conformal invariance requires necessarily the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. We discuss the symmetr...

We consider the stochastic heat equations on Lie groups, that is, equations of the form ∂tu = ∆xu + b(u) + F (u)Ẇ on R+ × G, where G is a compact Lie group, ∆ is the Laplace-Beltrami operator on G, b and F are Lipschitz coefficients, and where Ẇ is a Gaussian space-correlated noise, which is white-noise in time. We find necessary and sufficient conditions on the space correlation of Ẇ such that...

We study Beltrami-type equations with two given complex characteristics. Under certain conditions imposed on the coefficients, we prove theorems existence of homeomorphic ACL-solutions this equation. In addition, under some relatively weak conditions, establish corresponding continuous equation that are logarithmic Hölder in a domain.

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of di...

The Laplace Beltrami operator, known as an elliptic operator for functions defined on surfaces, appears in some applications in sciences and engineerings. In this paper we consider the Laplace Beltrami operator ∆Γ on surfaces Γ defined as graphs of C2 functions on a flat domain Ω ⊂ Rd−1 (d ≥ 2), ∆Γu = f on Γ, u = 0 on ∂Γ. Based on some properties of differential geometry, we transformed the Lap...