This opening lecture is intended to serve as a propaedeutic for the papers to be presented at this conference whose nonhomogeneous audience includes scientists, mathematicians, engineers and educators. This expository and developmental lecture, a case study of mathematical growth, surveys the origin and development of a mathematical idea from its birth in intellectual curiosity to applications....

q-analogue non-homogeneous wave equations are solved by a Duhamel solution strategy using constructions with q-analogue Fourier multipliers to compensate for the dependence of the analogue differential Leibnitz rule on the parity of the functions involved.

We construct field theory on noncommutative κ-Minkowski spacetime. Having the Lorentz action on the noncommutative space-time coordinates we show that the field lagrangian is invariant. We show that noncommutativity requires replacing the Leibnitz rule with the coproduct one.

In view of the recent interest in a short proof of the Jacobi identity for the Poisson-brackets, we provide an alternative simple proof for the same. Our derivation is based on the validity of the Leibnitz rule in the context of dynamical evolution. PACS number(s): 45.20.-d