A theorem on generalized nonions and their properties for the applied structures in physics

The central part of the paper consists of a theorem on generalized nonions governing dynamical systems modelling of special ternary, quaternary, quinary, senary, etc. structures, due to the third named author. Let M n (C), n ≥ 2, be the set of n × n-matrices with complex entries. The theorem states that in M n (C) there exists a basis such that PQ− λ s QP = 0, s = 0, 1, 2,.., n − 1, where {P,Q,}, u, v are specified in Section 1, formulae (1) and (2).The particular cases n = 2, 3, 4 with other choices of u, v were discussed by James Joseph Sylvester (1883, 1884) and by Charles Sanders Peirce (1882).In particular, λ = j, j3 = 1, j ≠ 1, generates nonions. Before the section on the above theorem and its visualization on a two-sheeted Riemann surface, we give three physical motivations for the topic: controlled noncommutativity: Sylvester–Peirce approach vs. Max Planck approach (1900), supersonic flow of a ternary alloy in gas, and changing hexagonal to pentagonal structure in pentacene.