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Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy. I

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Ukrainian Mathematical Journal Aims and scope

A Correction to this article was published on 15 March 2019

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Among all two-dimensional algebras of the second rank with unity e over the field of complex numbers ℂ, we find a semisimple algebra 𝔹0: ={c1e + c2ω : ck ∈ ℂ, k = 1, 2}, ω2 = e, containing bases (e1, e2) such that \( {e}_1^4+2p{e}_1^2{e}_2^2+{e}_2^4=0 \) for any fixed p > 1. A domain {(e1, e2)} is described in the explicit form. We construct 𝔹0-valued “analytic” functions Φ such that their real-valued components satisfy the equation for the stress function u in the case of orthotropic plane deformations \( \left(\frac{\partial^4}{\partial {x}^4}+2p\frac{\partial^4}{\partial {x}^2\partial {y}^2}+\frac{\partial^4}{\partial {y}^4}\right)u\left(x,y\right)=0 \), where x and y are real variables.

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Change history

  • 20 May 2019

    The author���s name should read S. V. Gryshchuk.

  • 20 May 2019

    The author���s name should read S. V. Gryshchuk.

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  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    S. V. Hryshchuk

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 8, pp. 1058–1071, August, 2018.

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Hryshchuk, S.V. Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy. I. Ukr Math J 70, 1221–1236 (2019). https://doi.org/10.1007/s11253-018-1564-2

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