Proof of the Fukui conjecture via resolution of singularities and related methods. IV

The present article is a direct continuation of the previous part III of this series of articles, which have been devoted to cultivating a new interdisciplinary region between chemistry and mathematics. In the present part IV, we develop two sets of fundamental theoretical tools, using methods from the field of resolution of singularities and analytic curves. These two sets of tools are essential in structurally elucidating the assertion of the Fukui conjecture (concerning the additivity problems) and the crux of the functional asymptotic linearity theorem (functional ALT) that proves the conjecture in a broad context. This conjecture is a vital guideline for a future development of the repeat theory (RST)—the central unifying theory in the First and the Second Generation Fukui Project.     

New Proof of the Fukui Conjecture by the Functional Asymptotic Linearity Theorem

The present article provides a new proof of the Fukui conjecture concerning the additivity problem of the zero-point vibrational energies of hydrocarbons. This conjecture played a prominent role in the initial development of the repeat space theory (RST), and continues to be of vital significance in the recent development of the theory of the generalized repeat space X _r (q,d). The new proof of the Fukui conjecture has been given here by establishing the functional version of the Asymptotic Linearity Theorem (ALT), the Functional ALT. This enhanced version of the ALT directly implies the validity of the Fukui conjecture; it easily unifies, in a broad perspective, a variety of additivity phenomena in physico-chemical network systems having many identical moieties, and efficiently solves some interpretational problems of the empirical additivity formulae from experimental chemistry. The proof of the functional version of the ALT is based on a new method transferable to the extended theoretical framework of the generalized repeat space X _r (q,d).     

Proof of the Fukui conjecture via resolution of singularities and related methods. V

The present article is a direct continuation of part IV of this series. The Local Analyticity Proposition (LAP1), which admits a proof via resolution of singularities is a major key to proving the Fukui conjecture via resolution of singularities and related methods. By LAP1, the essential part of the mechanism of the “asymptotic linearity phenomena” is extracted and is elucidated by using tools from the theory of algebraic and analytic curves. Here in the present article, we complete the proof of the LAP1 by using fundamental tools developed in parts III and IV of this series, thus completing the proof of the Fukui conjecture via resolution of singularities and related methods. This series of articles I-V establishes, for the first time, a new linkage between (i) the mathematical field of resolution of singularities and (ii) the chemical field of additivity problems tackled and solved in a unifying manner via the repeat space theory (RST), which is the central theory in the First and Second Generation Fukui Project. A new development called the Matrix Art Program in the Second Generation Fukui Project has also been expounded with a graphical representation of energy band curves of a carbon nanotube.     

The Functional Delta Existence Theorem and the Reduction of a Proof of the Fukui Conjecture to That of the Special Functional Asymptotic Linearity Theorem

This article establishes a fundamental existence theorem, called the Functional Delta Existence Theorem (DET), which is significant for a new development in the repeat space theory (RST) and also for elucidating an empirical asymptotic principle from experimental chemistry. By using the Functional DET, we reduce a proof of the Fukui conjecture to that of a special and simpler version of the Asymptotic Linearity Theorem (ALT). This reduction provides a basis for the forthcoming series of articles entitled “Proof of the Fukui conjecture via resolution of singularities and related methods”. A proof of the Functional DET is given here in a unifying manner so that an investigative link is formed among: (i) fundamental methodology in the RST, which is referred to as the approach via the aspect of form and general topology, (ii) frontier electron theory of reactivity indices, and (iii) the Shingu–Fujimoto empirical asymptotic principle for long chain molecules.     

Proof of the Fukui conjecture via resolution of singularities and related methods. I

The present article is the preliminary part of a series devoted to extending the foundation of the Asymptotic Linearity Theorems (ALTs), which prove the Fukui conjecture concerning the additivity problem of the zero-point vibrational energies of hydrocarbons. In this article, we establish a theorem, referred to as the Boundedness Theorem, through which one can easily form a chain of logical implications that reduces a proof of the Fukui conjecture to that of the Piecewise Monotone Lemma (PML). This chain of logical implications serves as a basis throughout this series of articles. The PML, which has been indispensable for demonstrating any version of the ALTs and has required for its proof a mathematical language not generally known to chemists, is directly related to the theory of algebraic curves. Proofs of the original and enhanced versions of the PML are obtainable via resolution of singularities and related methods.Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Proof of the Fukui conjecture via resolution of singularities and related methods. II $$^{\star}$$

The present article is a direct continuation of the first part of this series. We reduce a proof of the Fukui conjecture (concerning the additivity problem of the zero-point vibrational energies of hydrocarbons) to that of a proposition related to the theory of algebraic curves, so that we can focus on the key mechanism of the additivity phenomena. Namely, by establishing what is called the Basic Piecewise Monotone Theorem (BPMT), we reduce a proof of the Fukui conjecture to that of a proposition, called the Local Analyticity Proposition, Version 1 (LAP1), which admits a proof via resolution of singularities. By LAP1, the essential part of the mechanism of the asymptotic linearity phenomena is extracted and is elucidated by using tools from the mathematical theory of algebraic curves, whose language is of vital importance in analyzing the crux of the additivity mechanism. Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Proof of the Fukui conjecture via resolution of singularities and related methods: III

The present article is a direct continuation of the second part of this series. In conjunction with the analysis of the energy band curves of carbon nanotubes, we develop here fundamental theoretical tools, which are essential to prove the Local Analyticity Proposition (LAP). The LAP enables one to prove the Fukui conjecture (the guiding conjecture for developing the repeat space theory) in a new and powerful context of the theory of algebraic curves and resolution of singularities. The present fundamental tools also serve as modular tools for the repeat space theory, by which one can solve a variety of additivity and molecular network problems in a unifying manner.     

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.     

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

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. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 389–400, 2001     

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

The Asymptotic Linearity Theorem (ALT), which proves the Fukui conjecture in a broader context, plays a significant role in the repeat space theory (RST), which is the central unifying theory in the First and the Second Generation Fukui Project. Proving the Asymptotic Linearity Theorem Extension Conjecture (ALTEC) is a fundamental problem in the repeat space theory. The present paper constructs a class of functions MagicMtθ, which serves as a powerful tool for proving the Asymptotic Linearity Theorem Extension Conjecture and related propositions. The d-dimensional generalization?μ _d,n,θ of MagicMt _θ , which is given in the present paper and is called a ‘d-dimensional Magic Mountain’, provides inwardly repeating fractals in multidimensional spaces useful for interdisciplinary research that uses the generalized repeat space theory.     

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.     

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

The present Part VII of this series of articles is a direct extension of Part VI, where (1) fundamental methodology in the repeat space theory (RST) and (2) frontier electron theory of reactivity indices were theoretically linked. This part presents an estimate of the size of the regular index set, which was a central notion in Part VI, and two new theorems that are simpler and more powerful than the main theorem in Part VI. The main theorem in this part enables one to globally contextualize the Generalized Alpha Existence Theorem (a theorem essential in the RST and proved in Part V) and the μ Existence Theorem (derived from the main theorem in Part VI) into the star algebra structure of the generalized repeat space ??_r(q, d). © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Substituent effect on the molecular stability, group interaction, detonation performance, and thermolysis mechanism of nitroamino-substituted cyclopentanes and cyclohexanes

Density functional theory (DFT) method has been employed to study the effect of nitroamino group as a substituent in cyclopentane and cyclohexane, which usually construct the polycyclic or caged nitra-mines. Molecular structures were investigated at the B3LYP/6-31G** level, and isodesmic reactions were designed for calculating the group interactions. The results show that the group interactions ac-cord with the group additivity, increasing with the increasing number of nitroamino groups. The dis-tance between substituents influences the interactions. Detonation performances were evaluated by the Kamlet-Jacobs equations based on the predicted densities and heats of formation, while thermal stability and pyrolysis mechanism were studied by the computations of bond dissociation energy (BDE). It is found that the contributions of nitroamino groups to the detonation heat, detonation velocity, detonation pressure, and stability all deviate from the group additivity. Only 3a, 3b, and 9a-9c may be novel potential candidates of high energy density materials (HEDMs) according to the quantitative cri-teria of HEDM (ρ≈ 1.9 g/cm3, D ≈ 9.0 km/s, P ≈ 40.0 GPa). Stability decreases with the increasing number of N-NO2 groups, and homolysis of N-NO2 bond is the initial step in the thermolysis of the title com-pounds. Coupled with the demand of thermal stability (BDE > 20 kcal/mol), only 1,2,4-trinitrotriazacy-clohexane and 1,2,4,5-tetranitrotetraazacyclohexane are suggested as feasible energetic materials. These results may provide basic information for the molecular design of HEDMs.     

Epoxy-amine composites with ultralow concentrations of single-layer carbon nanotubes

The physical and mechanical properties of composite materials based on epoxy-amine systems—diglycidyl ether of diphenylolpropane-eutectic mixture of aromatic amines (40 wt % m-phenylenediamine-60 wt % 4,4′-diaminodiphenylmethane) and epoxy-diane resin ED-20-eutectic mixture of aromatic diamines with ultralow contents (≤0.1 wt %) of single-layer carbon nanotubes—are studied. It is shown that, for the concentration dependence of the modulus of elasticity during tension, the additivity rule is obeyed only at the lowest concentrations of carbon nanotubes. In this case, the presence of near-surface layers of a matrix with an increased elastic modulus should be considered.     

Donor-acceptor interaction and intramolecular proton transfer in aminopolyenes

The influence of donor and acceptor substituents at chain termini on the geometry of the chain and charge distribution on atoms was studied for the ground and lower triplet electronically excited state of model ω-dimethylaminopolyene molecules (CH₃)₂N(CH=CH) _n CH=C(CN)₂, n = 1–3. Calculations were performed by the B3LYP/6-31+G** method. The influence of substituents on bond lengths and the amplitude of deviations from the equilibrium carbon-carbon bond length in unsubstituted polyenes increased as the conjugation chain grew longer. The deviations of the effects of both donor and acceptor groups from additivity, however, decreased. In the lower triplet electronically excited state of the molecule, the effect of substituents on changes in C-C bond lengths along the chain was not damped. The section of the potential energy surface for intramolecular proton shift from the donor amino to the acceptor nitrile group in “cyclic” (cis) conformers of the H₂N-CH=CH-CN and H₂N-CH=CH-CH=CH-CN molecules was analyzed. The structure of the reaction transition state and the height of the barrier to proton transfer were calculated.     

Investigation of solvatochromism in the low-lying singlet states of dithienyl polyenes

To understand the low-lying singlet states of dithienyl polyenes, we investigated the solvatochromism of a series of α,ω-di(2-dithienyl 3,4-butyl) polyenes having n=1–5 double bonds. Absorption and emission spectra were collected in a series of aprotic solvents. The absorption energy dispersion effect sensitivity increased smoothly with n, reaching asymptotic behavior as n approached 5. The emission energy had less solvent sensitivity. The trends gave evidence for the existence of a ¹B^*_u absorbing state and a ¹A^*_g emitting state. We observed sensitivity of the absorbing and emitting states to solute–solvent electrostatic interactions, suggesting the dithienyl polyenes had a polar ground state conformation.     

A Complete Solution to a Conjecture on the β-Polynomials of Graphs

In 1990, Gutman and Mizoguchi conjectured that all roots of the -polynomial (G,C,x) of a graph G are real. Since then, there has been some literature intending to solve this conjecture. However, in all existing literature, only classes of graphs were found to show that the conjecture is true; for example, monocyclic graphs, bicyclic graphs, graphs such that no two circuits share a common edge, graphs without 3-matchings, etc, supporting the conjecture in some sense. Yet, no complete solution has been given. In this paper, we show that the conjecture is true for all graphs, and therefore completely solve this conjecture.     

Fluorometric enzyme immunosensing system based on natural product resveratrol for horseradish peroxidase and Ag/SiO<Subscript>2</Subscript> nanoparticles

The properties of resveratrol (3′, 4′, 5-trihydroxystlbene, RST) were for the first time evaluated as a potential substrate for horseradish peroxidase (HRP)-catalyzed fluorogenic reaction. The properties of RST for use as fluorogenic substrates for HRP and its application in immunoassays were compared with commercially available substrates such as p-hydroxyphenylpropionic acid (pHPPA), chavicol and Amplex red by a fluoroimmunosensing method in the use of Schistosomia japonicum antibody (SjAb) as a model analyte. The fluoroimmunosensing device was constructed by dispersing Schistosomia japonicum antigen (SjAg), nano-Ag/SiO₂ particles and sol-gel at low temperature. In pH 5.8 Britton-Robinson buffer (B-R), HRP-SjAb conjugates can catalyze the polymerization reaction of RST by H₂O₂ forming fluorescent dimmers. The increase of the fluorescence intensity of the dimmers product at emission of 462 nm (excitation: 315 nm) is proportional to the concentration of HRP-SjAb binding to the SjAg entrapped in the nano-Ag/SiO₂ particles-sol-gel matrix. A competitive binding assay has been used to determine SjAb in rabbit serum with the aid of SjAb labeled with HRP. Substrate RST showed comparable ability for HRP detection and its enzyme-linked immunosensing reaction system, in a linear detection ranging of 1.5×10⁻⁶–7.3×10⁻⁴ g/L and with a detection limit of 1.5×10⁻⁶ g/L. The immobilized biocomposites surface could be regenerated by simply polishing with an alumina paper, with an excellent reproducibility (RSD = 4.7%). The proposed method has been successfully used for analysis of the rabbit serum sample with satisfactory results. Supported by the Projects of Scientific Research Fund of Hunan Provincial Education Department of China (Grant Nos. 05B020 and 06C098)