Descartes’ Dream: Cartesian Products
Keiichi Takahashi · Pure and Applied Mathematics Journal · 2015
In the last century, especially in the last half of the century, there was the paradigm of sectionalism prevailing and sciences and engineering were divided into very small parts which are mutually independent. It was like in Babel where there was no common language to communicate. The purpose of this paper is to present one of the possible glues—the notion of Cartesian product—to stick some remotely separated parts of science and engineering together. This concept appears in various places and it will turn out that it can unify the scattered notions quite well. Our two main objectives are the interpretation of cyclic codes as polynomials and nested PSO. We make clear the meaning of polynomials through Cartesian product or rather as terminating formal power series. The latter, formal power series, is not touched in engineering disciplines but is quite useful in unifying and interpreting various notions. In particular, it will make clear the meaning of addition of polynomials. This reminds us of topologization of adéles. PSO (Particle Swarm Optimization), a developed form of genetic algorithm, has come to our attention through the papers [4], [23] and [24]. In [4], the PSO is used to find optimal choice of parameters in the FOPID. In other two papers, PSO algorithm is used in cell balancing in the Lithium-ion battery pack for EV’s. Motivated by the passage on [3] that the stability is preserved by the Cartesian product of many copies of the attractor, we may conceive of the nested PSO.