By using the notion of a quantum double we introduce analogs partial derivatives on Reflection Equation algebra, associated with Hecke symmetry GL(N) type. We construct matrix L=MD, where M is generating algebra and D composed prove that matrices M, L satisfy identity, called Capelli one. Upon applying trace, it becomes scalar relation, which far-reaching generalization classical identity. Also...

Let $J_q^z(n)$ be the quadratic matrix algebra associated to quantized enveloping $U_q\left(A_{2 n-1}\right)$. It is shown explicitly that defining relations of form a Gröbner-Shirshov basis. Consequently, several structural properties are derived. Received: June 13, 2023Accepted: July 21, 2023

We develop a matrix algebra approach for optimizing condition-based maintenance with planning time gradually deteriorating, continuously monitored equipment. This avoids the calculation times of simulation and approximations more direct approaches. consider model in which after failure should be planned as well, resulting downtime, emergency repairs upon failure. Numerical examples are provided...

Building upon the concepts of the matrix Lie group and the matrix Lie algebra, we explore the natural connections between the Lie groups and Lie algebras via the exponential map. We later introduce the matrix commutator as a Lie bracket operation to aid our investigation of Lie algebra representations, which we illustrate with the example of the adjoint representation on the special unitary group.