Development of mathematical methods

The theory of symmetry for tensor quantities is developed. The new definition for tensor invariants is given (2003) [1 - 3]. This definition coincides with the traditional one only for the Euclidean tensors. It is shown that any invariant can be obtained as a solution of a differential equation of the first order. The number of independent solutions of this equation determines the minimum number of invariants necessary to fix the system of tensors as a solid unit.
  1. Zhilin P.A. Modified theory of symmetry for tensors and their invariants // Izvestiya VUZov. Severo-Kavkazskii region. Estestvennye nauki (Transactions of Universities. South of Russia. Natural sciences). Special issue “Nonlinear Problems of Continuum Mechanics”. 2003. P. 176-195. (In Russian.)
  2. Zhilin P.A. Symmetries and Orthogonal Invariants in Oriented Space> // Proceedings of XXXII Summer School - Conference “Advanced Problems in Mechanics”. St. Petersburg, Russia. 2005. P. 470-483.
  3. Altenbach H., Naumenko K., Zhilin P.A. A note on transversely-isotropic invariants // ZAMM. Z. Angew. Math. Mech. 86, N 2. P. 162–168. (2006)