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

• Published in: Lobachevskii journal of mathematics 2017-03, Vol.38 (2), p.255-261
• Main Authors: Frątczak, E. Z., Ławrynowicz, J., Nowak-Kępczyk, M., Polatoglou, H. M., Wojtczak, L.
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Geometry; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Ternary alloys
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S199508021702007X

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, j 3 = 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.