Normed repeat space and its super spaces: fundamental notions for the second generation Fukui project

Abstract

Fukui’s DNA problem is a long-range target of the international and interdisciplinary joint project initiated by Kenichi Fukui in 1992, whose underlying motive has been to cultivate a new interdisciplinary region between chemistry and mathematics for a future development of theoretical chemistry. “Can the conductivity and other properties of a single-walled carbon nanotube be analyzed in the setting of a *-algebra equipped with a complete metric?” This metric problem is fundamental to proceed towards a solution of Fukui’s DNA problem. To affirmatively solve this metric problem, we establish, here in this paper, the new notion of normed repeat space ${\fancyscript{X}_{r}(q, d, p)}$. The normed repeat space ${\fancyscript{X}_{r}(q, d, p)}$ is an intermediate theoretical device to shift from periodic polymers to aperiodic polymers like DNA and RNA in the above-mentioned Fukui Project. The space ${\fancyscript{X}_{r}(q, d, p)}$ is a Banach algebra for all 1 ≤ p ≤ ∞, and ${\fancyscript{X}_{r}(q, d, p)}$ forms a C*-algebra for p = 2. Here, polymer moiety size number q and dimension number d are arbitrarily given positive integers. The generalized repeat space ${\fancyscript{X}_{r}(q, d)}$ is included in the normed repeat space ${\fancyscript{X}_{r}(q, d, p)}$, which in turn is included in one of its super spaces ${\fancyscript{X}_{B}(q, d, p)}$ so that aperiodic polymers can be represented and investigated in the setting of this super space ${\fancyscript{X}_{B}(q, d, p)}$.