The article continues a presentation of modern quantum mathematics backgrounds started in [Quantum Mathematics and its Applications. Part 1. Automatyka, vol. 6, AGH Publisher, Krakow, 2002, No. 1, pp. 234–2412; Quantum Mathematics: Holonomic Computing Algorithms and Their Applications. Part 2....

We present a number of alternative sufficient conditions for the existence of pre-symplectic or pre-implectic (Poisson) structures, for both ordinary differential (ODE) and ordinary difference (OΔE) equations. Four alternative sets of conditions are obtained for ODEs and OΔEs in n dependent...