Abstract
Among all two-dimensional commutative associative algebras of the second rank with unity over the field of complex numbers we want to find all pairs \((\mathbb {B}_{\ast }, \{e_1,e_2\})\), where \(\mathbb {B}_{\ast }\) is an algebra and {e 1, e 2} are its bases such that \(e_1^4+ 2p e_1^2 e_2^2 + e_2^4 = 0\) for every fixed p, − 1 < p < 1. This problem is solved in an explicit form. An approach of \(\mathbb {B}_{\ast }\)-valued “analytic” functions Φ(xe 1 + ye 2) ({e 1, e 2} is fixed, x and y are real variables), such that their real-valued functions-components satisfy the equation on finding the stress function in certain cases of orthotropic plane deformations, is developing.
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Acknowledgements
The research is partially supported by the Fundamental Research Programme funded by the Ministry of Education and Science of Ukraine (project No. 0116U001528) and a common grant of the National Academy of Sciences of Ukraine and the Poland Academy of Sciences (No. 32/2018).
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V. Gryshchuk, S. (2022). Bases in Commutative Algebras of the Second Rank and Monogenic Functions Related to Some Cases of Plane Orthotropy. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-87502-2_16
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