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.     

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.     

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).     

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.     

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.     

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).     

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     

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).     

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     

On the fundamental conjecture of HDMR: a Fourier analysis approach

Although the HDMR decomposition has become an important tool for the understanding of high dimensional functions, the fundamental conjecture underlying its practical utility is still open for theoretical analysis. In this paper, we introduce the HDMR decomposition in conjunction with the Fourier-HDMR approximation leading to the following conclusions: (1) we suggest a type of Fourier-HDMR approximation for certain classes of differentiable functions; (2) utilizing the Fourier-HDMR method, we prove the fundamental conjecture about the dominance of low order terms in the HDMR expansion under relevant conditions, and we also obtain error estimates of the truncated HDMR expansion up to order u; (3) we prove the domain decomposition approximation theorem which shows that the global Fourier-HDMR approximation is not always optimal for a given accuracy order; (4) and finally, a piecewise Fourier-HDMR approach is discussed for high dimensional modeling. These results help to further understand how to efficiently represent the high dimensional functions.     

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

A theoretical framework has been presented, which links two diverse molecular problems: the study of symmorphy transformations of molecular shape analysis (further developed in the present paper) and that of additivity of the zero-point vibrational energy of hydrocarbons and the total pi-electron energy of alternant hydrocarbons. The linkage, using fundamental tools of (general) topology and algebra, makes it possible to mutually introduce the methodologies used in fields hitherto separately investigated. By establishing this linkage, topological patterns described by symmorphy groups can be treated by the algebraic methods developed for the above additivity problems. The linkage also brings forth new techniques of topologizing the repeat spaceX _r(q) for the additivity problems. Moreover, this connection paves the way to analyzing molecular homologous series and their properties by means of associating sequences of molecular structures with elements of a repeat space equipped with a topology.On leave from: Institute for Fundamental Chemistry, 34-4 Nishihiraki-cho, Takano, Sakyo-ku, Kyoto 606, Japan.     

Repeat Space Theory Applied to Carbon Nanotubes and Related Molecular Networks. I

The present article is the first part of a series devoted to extending the Repeat Space Theory (RST) to apply to carbon nanotubes and related molecular networks. Four key problems are formulated whose affirmative solutions imply the formation of the initial investigative bridge between the research field of nanotubes and that of the additivity and other network problems studied and solved by using the RST. All of these four problems are solved affirmatively by using tools from the RST. The Piecewise Monotone Lemmas (PMLs) are cornerstones of the proof of the Fukui conjecture concerning the additivity problems of hydrocarbons. The solution of the fourth problem gives a generalized analytical formula of the pi-electron energy band curves of nanotube (a, b), with two new complex parameters c and d. These two parameters bring forth a broad class of analytic curves to which the PMLs and associated theoretical devices apply. Based on the above affirmative solutions of the problems, a central theorem in the RST, called the asymptotic linearity theorem (ALT) has been applied to nanotubes and monocyclic polyenes. Analytical formulae derived in this application of the ALT illuminate in a new global context (i) the conductivity of nanotubes and (ii) the aromaticity of monocyclic polyenes; moreover an analytical formula obtained by using the ALT provides a fresh insight into Hückel’s (4n+2) rule. The present article forms a foundation of the forthcoming articles in this series. The present series of articles is closely associated with the series of articles entitled ‘Proof of the Fukui conjecture via resolution of singularities and related methods’ published in the JOMC.     

Microchip-based cellular biochemical systems for practical applications and fundamental research: from microfluidics to nanofluidics

By combining cell technology and microchip technology, innovative cellular biochemical tools can be created from the microscale to the nanoscale for both practical applications and fundamental research. On the microscale level, novel practical applications taking advantage of the unique capabilities of microfluidics have been accelerated in clinical diagnosis, food safety, environmental monitoring, and drug discovery. On the other hand, one important trend of this field is further downscaling of feature size to the 10¹–10³ nm scale, which we call extended-nano space. Extended-nano space technology is leading to the creation of innovative nanofluidic cellular and biochemical tools for analysis of single cells at the single-molecule level. As a pioneering group in this field, we focus not only on the development of practical applications of cellular microchip devices but also on fundamental research to initiate new possibilities in the field. In this paper, we review our recent progress on tissue reconstruction, routine cell-based assays on microchip systems, and preliminary fundamental method for single-cell analysis at the single-molecule level with integration of the burgeoning technologies of extended-nano space.     

C60: Buckminsterfullerene,The Celestial Sphere that Fell to Earth

In 1975–1978 the long-chained polyynylcyanides, HC₅N, HC₇N, and HC₉N were surprisingly discovered in the cold dark clouds of interstellar space by radioastronomy. The subsequent quest for their source indicated that they were being blown out of red giant, carbon stars. In 1985 carbon-cluster experiments aimed at simulating the chemistry in such stars confirmed these objects as likely sources. During these cluster studies a serendipitous discovery was made; a stable pure-carbon species, C₆₀, formed spontaneously in a chaotic plasma produced by a laser focused on a graphite target. A closed spheroidal cage structure was proposed for this molecule, which was to become the third well-characterized allotrope of carbon and was named buckminsterfullerene. It has taken five years to produce sufficient material to prove the correctness of this conjecture. There may be a timely object lesson in the fact that exciting new and strategically important fields of chemistry and materials science have been discovered overnight due to fundamental research, much of which was unable to attract financial support, and all of which was stimulated by a fascination with the role of carbon in space and stars. In this account, interesting aspects of this discovery, its origins, and its sequel are presented. The story has many facets, some of which relate to the way scientific discoveries are made.     

Fluorescent probes for detection and bioimaging of leucine aminopeptidase

Leucine aminopeptidase (LAP) is one of important proteolytic enzymes and closely related with pathogenesis of cancer and liver injury. Determination of LAP activity in serum is used clinically for liver disorder diagnosis. The level of expressed LAP is very low in normal cells, but overexpressed in tumors and liver diseases, especially drug-induced hepatitis. LAP has become a predictive biomarker for many cancers and diverse physiological processes. Therefore, in situ dynamic monitoring and identifying intracellular LAP is imperative for LAP-related disease diagnosis. This review focuses on LAP-specific fluorescence imaging probes for the detection and tracking of intracellular LAP actively in vitro and in vivo. The progress suggests that fluorescence imaging is a vital and rapidly growing technology for early diagnosis of tumors.     

Substituent effects in parallel-displaced pi-pi interactions

High-quality quantum-mechanical methods are used to examine how substituents tune pi-pi interactions between monosubstituted benzene dimers in parallel-displaced geometries. The present study focuses on the effect of the substituent across entire potential energy curves. Substituent effects are examined in terms of the fundamental components of the interaction (electrostatics, exchange-repulsion, dispersion and induction) through the use of symmetry-adapted perturbation theory. Both second-order M?ller-Plesset perturbation theory (MP2) with a truncated aug-cc-pVDZ' basis and spin-component-scaled MP2 (SCS-MP2) with the aug-cc-pVTZ basis are found to mimic closely estimates of coupled-cluster with perturbative triples [CCSD(T)] in an aug-cc-pVTZ basis. Substituents can have a significant effect on the electronic structure of the pi cloud of an aromatic ring, leading to marked changes in the pi-pi interaction. Moreover, there can also be significant direct interactions between a substituent on one ring and the pi-cloud of the other ring.     

Adsorption d'eau sur la kaolinite: Influence de la nature des sites actifs

Water adsorption on kaolinite is a specific cooperative adsorption which does not satisfy the fundamental hypothesis of the BET theory.The adsorption isotherms on different homoionic samples show the effect of the hydration energy of the active sites (exchangeable cations) on quantitative adsorption data.The corresponding calorimetric curves present a maximum which characterizes interactions in the adsorbed phase. A relationship is apparent between these interactions and the electric field or the polarizability of the fixed cation, these factors determining the nature of the bond between the surface and cation.From the experimental data, we may propose an approximative value for the number of molecules which compose the primary hydration sheath of the active sites.     

On the steady-state solutions of the kinetics of homogeneous nucleation

A general solution to the steady-state equations for the kinetics of homogeneous nucleation is determined, in agreement with irreversible thermodynamics. Most of the steady-state distributions of clusters present a minimum. Our theory provides a conjecture for understanding the apparent contradictions existing in the present experimental data.