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An Algebraic Algorithm of Pseudotensors Weights Eliminating and Recovering

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Abstract

The paper deals with an algebraic algorithm for the pseudotensors weights eliminating and recovering. The proposed algorithm allows us to reduce pseudotensors of arbitrary ranks and weights to absolute tensors of higher ranks. The weight of a pseudotensor is assumed to be an integer (positive or negative). The algorithm is based on the transformation of a pseudotensor of arbitrary rank and integer weight by using tensor product of permutation symbols. The requisite equations from algebra and analysis of pseudotensors are given and discussed. Based on the proposed algebraic algorithm, covariant constancy of permutation symbols and fundamental orienting pseudoscalar powers, a realisation of covariant differentiation of a pseudotensor field of arbitrary rank and integer weight is developed. The definition of the pseudotensor field gradient is then introduced.

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Notes

  1. As some authors pointed out [3, p. 159], weights of rational pseudoinvariants can be sometimes fractional.

  2. In micropolar elasticity microrotation vector can be represented by the covariant pseudovector \({{\mathop \phi \limits^{[ - 1]} }_{{s}}}\), absolute vectors \({{\phi }^{s}}\) (or \({{\phi }_{s}}\)), and contravariant pseudovector \({{\mathop \phi \limits^{[ + 1]} }^{{s}}}\). Pseudovector components are known insensitive to the mirror reflections of local coordinate frame.

  3. Note that this realisation can be used not only in Euclidean spaces, but also in Riemannian ones [2].

  4. The equation (15) also makes sense for fractional weights, but with one limitation: in left-handed coordinate systems powers ew must be real.

  5. The Hamilton nabla operator is defined according to: \(\nabla = \mathop {\boldsymbol\imath} \limits^k {\kern 1pt} {{\partial }_{k}}\).

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Funding

This study was in part financially supported by the Ministry of Science and Higher Education of the Russian Federation (State Registration Number AAAA-A20-120011690132-4) and by the Russian Foundation for Basic Research project no. 20-01-00666.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics RAS, 119526, Moscow, Russia

    E. V. Murashkin & Yu. N. Radayev

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  1. E. V. Murashkin
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  2. Yu. N. Radayev
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Correspondence to E. V. Murashkin or Yu. N. Radayev.

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Translated by M. Katuev

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Murashkin, E.V., Radayev, Y.N. An Algebraic Algorithm of Pseudotensors Weights Eliminating and Recovering. Mech. Solids 57, 1416–1423 (2022). https://doi.org/10.3103/S0025654422060085

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