Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. VI*

Abstract

ABSTRACT:The present Part VI of this series of articles provides a mathematical and methodical link between (i) fundamental methodology in the repeat space theory (RST), which is referred to as the approach via the aspect of form and general topology and which has universal unifying power to handle additivity problems of molecules that have many identical moieties, and (ii) frontier electron theory of reactivity indices. Using theoretical tools required to link (i) and (ii), we establish a theorem from which the Generalized Alpha Existence Theorem (a theorem essential in the RST and proved in the previous Part V) directly follows.

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution VII

“…Recalling the definition of the generalized repeat space 𝒳 r (1, 1) (cf. 1, 7), one immediately notes that { M N } ∈ 𝒳 r (1, 1).…”

“…In this section, after fixing some notation, we first recall Problem 2 in Part VI 1 of this series. We then formulate a new illustrative problem (Problem 3) that is formulated in the context of the generalized repeat space 𝒳 r (1, 1).…”

“…The reader is referred to Part VI 1 for the motivation of the above Problem 2 and also for the preceding Problem 1, which was formulated in Part V 6. We remark that M N given by (2.2) can be expressed in terms of matrices P N and S N : M N = aP N + bP + cP + ( d − b ) P S N P + ( e − c ) P S N P + ( f − a ) P N S N P + ( g − b ) P N S N P .…”

“…This article is a direct extension of Part VI 1 of this series of articles 1–6. In Part VI, in the setting of the generalized repeat space, the definition of which was first constructed in Ref.…”

Multidimensional Magic Mountains and Matrix Art for the generalized repeat space theory

“…Among them is the following which has been playing a significant role in the development of the repeat space theory (RST) (cf. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]), which is the central unifying theory in the First (cf. [1,2]) and the Second (cf.…”