Fundamental notions for the second generation Fukui project and a prototypal problem of the normed repeat space and its super spaces

Abstract

Fukui’s DNA problem is a long-range target of the First and Second Generation Fukui Project, whose underlying motive has been to cultivate a new interdisciplinary region between chemistry and mathematics. The normed repeat space \({\fancyscript{X}_r ({q,d,p})}\) and its super space \({\fancyscript{X}_B ({q,d,p})}\) are special Banach algebras which are fundamental in the second generation Fukui project. In the present article, keeping Fukui’s DNA problem in mind, we formulate and solve a prototypal problem (problem I) of the normed repeat space \({\fancyscript{X}_r ({q,d,p})}\) and its super space \({\fancyscript{X}_B ({q,d,p})}\). The present article provides also three “Challenging Problems” II–IV. These are called parallel problems in view of the structural parallelism observed when compared with the original problem I. The challenging problems, which would allow multiple approaches, are specially designed for those who are interested in interdisciplinary investigations.