Shigeru Arimoto

The Asymptotic Linearity Theorem for the study of additivity problems of the zero-point vibrational energy of hydrocarbons and the total pi-electron energy of alternant hydrocarbons

By using the approach via the aspect of form and general topology, as well as basic notions of abstract algebra, a theoretical framework has been developed which elucidates the mechanism of the additivity relationships between structure and properties in molecules having many identical moieties. The main theorem, the Asymptotic Linearity Theorem (ALT), together with an auxiliary theorem, the a Independence Theorem, implies that the zero-point vibrational energy (or total pi-electron energy for the case of alternant hydrocarbons)En of a linearly extended system B-An-B′ having n repeating identical moieties has the asymptotic expansionEn = αn +β +o(1) as n → ∞, where α ∈ ℝ is independent of the choice of the end moieties B and B′. The theorem being formulated in a general context, the actual implication of the ALT is much broader than the above two applications would indicate.

A Finer Aspect of Eigenvalue Distribution of Selfadjoint Band Toeplitz Matrices

The asymptotics of eigenvalues of Toeplitz operators has received a lot of attention in the mathematical literature and has been applied in several disciplines. This paper describes two such application disciplines and provides refinements of existing asymptotic results using new methods of proof. The following result is typical: Let

Repeat space theory applied to carbon nanotubes and related molecular networks. III

T(\varphi)

Practical version of the asymptotic linearity theorem with applications to the additivity problems of thermodynamic quantities

be a selfadjoint band limited Toeplitz operator with a (real valued) symbol

Aspects of form and general topology: Alpha Space Asymptotic Linearity Theorem and the spectral symmetry of alternants

\varphi

Preliminary considerations towards a theoretical foundation of the zero point energy additivity rules

, which is a nonconstant trigonometric polynomial. Consider finite truncations

Magic Mountain and Devil's Staircase swapping problems

T_n(\varphi)

The courant-fischer theorem and the spectrum of selfadjoint block band Toeplitz operators

of

Symmorphy transformations and operators in the repeat spaceX r(q) for additivity problems

T(\varphi)

A unifying approach to the additivity problems of the zero point vibrational energy of hydrocarbons and the total pi electron energy of alternant hydrocarbons

, and a finite union of finite intervals of real numbers