Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I

  • S. V. Gryshchuk

Abstract

Among all two-dimensional algebras of the second rank with unity $e$ over the field of complex numbers $C$, we find a semisimple algebra $B_0 = \{ c_1e + c_2\omega: c_k \in C, k = 1, 2\} , \omega^2 = e$, containing bases $(e_1, e_2)$, such that $e^4_1 + 2pe^2_1e^2_2 + e^4_2 = 0$ for every fixed $p > 1$. A domain $\{ (e1, e2)\}$ is described in the explicit form. We construct $B_0$ -valued “analytic” functions $\Phi$ such that their real-valued components satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl(\frac{\partial^4}{\partial x^4} + 2p \frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr) u(x, y) = 0,$$ where $x, y$ are real variables.
Published
25.08.2018
How to Cite
GryshchukS. V. “Сommutative сomplex Algebras of the Second Rank With Unity and Some Cases of Plane Orthotropy. I”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, no. 8, Aug. 2018, pp. 1058-71, http://umj.imath.kiev.ua/index.php/umj/article/view/1617.
Section
Research articles