arXiv subjects

Dmitry Bryukhov

Publications and source records attributed to Dmitry Bryukhov.

3 recordsLinked to original sources

arXiv2024

Potential Vector Fields in $\mathbb R^3$ and $\alpha$-Meridional Mappings of the Second Kind $(\alpha \in \mathbb R)$

This paper extends approach developed in a recent author's paper on analytic models of potential fields in inhomogeneous media. New three-dimensional analytic models of potential vector fields in some layered media are constructed. Properties of various analytic models in Cartesian and cylindrical coordinates in $\mathbb R^3$ are compared. The original properties of the Jacobian matrix $\mathbf{J}(\vec V)$ of potential meridional fields $\vec V$ in cylindrically layered media, where $\phi( \rho) = \rho^{-\alpha}$ $(\alpha \in \mathbb R)$, lead to the concept of \emph{$\alpha$-meridional mappings of the first and second kind}. The concept of \emph{$\alpha$-Meridional functions of the first and second kind} naturally arises in this way. When $\alpha =1$, the special concept of \emph{Radially holomorphic functions in $\mathbb R^3$}, introduced by G\"{u}rlebeck, Habetha and Spr\"{o}ssig in 2008, is developed in more detail. Certain key properties of the radially holomorphic functions $G$ and functions reversed with respect to $G$ are first characterized. Surprising properties of the radially holomorphic potentials represented by superposition of the radially holomorphic exponential function $e^{\breve{\beta} x}$ $(\breve{\beta} \in \mathbb R)$ and function reversed with respect to $e^{\breve{\beta} x}$ are demonstrated explicitly. The basic properties of the radially holomorphic potential represented by the radially holomorphic extension of the Joukowski transformation in $\mathbb R^3$ are studied.

math.CV↗

arXiv2021

Potential Vector Fields in $\mathbb R^4$ and New Generalizations of the Cauchy-Riemann System

This paper extends approach of recent author's paper devoted to special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media and new generalizations of the Cauchy-Riemann system in $\mathbb R^3$. Two families of generalizations of the Cauchy-Riemann system with variable coefficients in $\mathbb R^4$ are presented in the context of non-Euclidean geometry. Analytic models of a wide range of potential meridional vector fields in $\mathbb R^4$ are characterized using a family of Vekua type systems in cylindrical coordinates. The specifics of meridional fields allows us to introduce the concept of four-dimensional $α$-meridional mappings of the first and second kind depending on the values of a real parameter $α$. In case $α=2$ tools of the radially holomorphic potential in $\mathbb R^4$ are developed in the context of generalized axially symmetric potential theory (GASPT). Analytic models of potential meridional fields in $\mathbb R^4$ generated by Bessel functions of the first kind of integer order and quaternionic argument are described in case $α=2$. In case $α=0$ the geometric specifics of four-dimensional harmonic meridional mappings of the second kind is demonstrated explicitly in the context of the theory of gradient dynamical systems with harmonic potential.

math.CV↗

arXiv2021

The Static Maxwell System in Three Dimensional Inhomogeneous Isotropic Media, Generalized Non-Euclidean Modification of the System $(R)$ and Fueter's Construction

This paper extends approach of our joint paper with Kähler and recent paper of the author, published in 2021, on problems of the static Maxwell system in three dimensional inhomogeneous media. Applied pseudoanalytic function theory developed by Kravchenko et al. allows to characterize, in particular, meridional and transverse fields in cylindrically layered media. Geometric properties of the electric field gradient ($EFG$) tensor within a wide range of meridional fields allows us to introduce the concept of $α$-meridional mappings of the first and second kind depending on the values of a real parameter $α$. In case $α=1$ tools of the radially holomorphic potential provide essentially new meridional models in the context of generalized axially symmetric potential theory (GASPT). Integral representations of Bessel functions of the first kind of integer order and the reduced quaternionic argument are first established. In case $α=0$ geometric properties of harmonic meridional mappings of the second kind are described. Some open problems in three dimensional inhomogeneous anisotropic media are discussed using a generalized Riemannian modification of the system $(R)$.

math.AP↗