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

The present article is part III of a series devoted to extending the Repeat Space Theory (RST) to apply to carbon nanotubes and related molecular networks. In this part III, four problems concerning the above-mentioned extension of the RST have been formulated. Affirmative solutions of these problems imply (i) asymptotic analysis of carbon nanotubes (CNTs) via the new techniques of normed repeat space, Banach algebra, and C*-algebra becomes possible; (ii) a new linkage is formed between the investigations of CNTs and those of ‘spectral symmetry’. In the present paper, we give affirmative solutions to all of the four problems, together with (a) estimates of the norms of matrix sequences representing CNTs, (b) Challenging Problem A^#, which complements Problems A, (c) several pictures of ‘CNT Matrix Art’ which has heuristic power to lead one to get the affirmative answers to the problems formulated in an abstract algebraic manner.     

Chemical Processing of Potassium Substituted (Pb0.6Ba0.4)Nb2O6 Powders and Thin Films Through Metallo-Organics

Highly oriented tungsten bronze K_0.4(Pb_0.6Ba_0.4)_0.8Nb₂O₆ ferroelectric thin films have been prepared from metal alkoxides and metal acetate by chemical process. The formation of solid solution with potassium was found to be very effective to form the tungsten bronze phase at lower temperatures compared with (Pb_0.6Ba_0.4)Nb₂O₆. The amount of potassium in the composition of K_x(Pb_0.6Ba_0.4)_{1 – x/2}Nb₂O₆ [0 x 0.4] is also important to crystallize in the tungsten bronze phase. K_0.4(Pb_0.6Ba_0.4)_0.8Nb₂O₆ films with c-axis preferred orientation were successfully synthesized on MgO(1 0 0) and Pt(1 0 0)/MgO(1 0 0) substrates above 700°C. KPBN60 thin film on Pt(1 0 0)/MgO(1 0 0) showed a remnant polarization of 20 C/cm² and a coercive field of 140 kV/cm at –150°C.     

Determination of the angular separation and the spectra of two-point sources from spectral coherence measurements

We present a novel method not only for determining the angular separation of unknown two-point spectral sources but also for obtaining each of their spectral profiles from measurements of both their complex degree of spectral coherence and their spectra across an observation area in the space-frequency domain. The theoretical predictions are fully proved by an experimental demonstration with a pair of uncorrelated two-point spectral sources.     

Governing equations of envelopes created by nearly bichromatic waves and relation to the nonlinear Schrödinger equation

In this paper, we derive the governing equations of the envelope created by nearly bichromatic waves defined as the waves with wavenumber spectrum has two closely approaching peaks and spreads over around two peaks. The obtained governing equation has higher-order terms related to the distinctive quantities of nearly bichromatic waves. We show a nonlinear Schrödinger type equation governs the envelope created by nearly bichromatic waves, up to third order, under some conditions.     

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

We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set (F) of the eigenvalues of the truncationsT _n (F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound.     

Determination of lidocaine in human serum by capillary gas chromatography with surface ionization detection

A novel analytical method has been developed for the determination of lidocaine (2-(diethylamino)-N-(2,6-dimethyl)phenylacetamide) in human serum. After solvent extraction from serum the extract was analyzed by gas chromatography employing a van den Berg type solventless injector, a new sensitive and selective detector, the surface ionization detector, and a bonded phase flexible fused silica capillary column. The detection limit was ca. 30–50 ng/ml.     

Note on the Repeat Space Theory

This note provides a chronological sketch of the development from the early 1990s of the Repeat Space Theory (RST), which had originated in the study of the zero-point energy additivity problems of hydrocarbons in 1985. Interacting with the theories of dynamical systems, operator algebra, and so forth, the RST has developed into a comprehensive theoretical framework of axiomatic nature, which unites and solves, in particular, what we call globally-pertaining-type problems, or, for short, g-type problems; these constitute physico-chemical problems for whose solutions global mathematical contextualization is essential. In conjunction with the author's communications with Prof. Kenichi Fukui, the genesis of the notion of g-type problems has also been presented in this note. Through the vision the RST provides, it is foreseeable that investigations of the peripheral research domains of g-type problems in chemistry will play a significant role for future investigations, especially for those related to macromolecules, physico-chemical network systems, and biochemical network systems, in the vast uncharted interdisciplinary regions between chemistry and modern mathematics.