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. 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.     

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.     

Doubly Bridged Dinuclear Cadmium(II)azido Schiff Base Complexes: Synthesis, X-Ray Structure and Luminescence Properties of [Cd(L)(N3)]2(Y)2 [L = Schiff Bases, Y = ClO4 −, PF6 −]

Three new Schiff bases, -(bis(pyridin-2-yl)alkylidene)butane-1,4-diamine [(py)(R)C=N-(CH)N=C(R)(py), L: py = pyridine; R = H, -(bis(pyridin-2-yl)formylidene)butane-1,4-diamine (bpfd); R = Me, -(bis(pyridin-2-yl)methylidene)butane-1,4-diamine (bpmd); R = C, -(bis(pyridin-2-yl)bezylidene)butane-1,4-diamine (bpbd)] were prepared and used to synthesize six dinuclear cadmium(II)azido complexes of type [Cd(L)(N)](Y) [L = bpfd, Y = ClO (1a); L = bpfd, Y = PF (1b); L = bpmd, Y = ClO (2a); L = bpmd, Y = PF (2b); L = bpbd, Y = ClO (3a); and L = bpbd, Y = PF (3b)]. Two representative members of the series, 3a and 3b, have been characterized by single crystal X-ray diffraction measurements. Structural study reveals that each cadmium center in both dinuclear compounds is located in a distorted octahedral coordination environment surrounded by six nitrogen atoms—four (N1, N2, N3, N4) from tetradentate ligand, and the fifth and sixth positions occupied by nitrogen atoms (N5, N5^*) of doubly end-on bridging azides. The complexes display intraligand ¹ fluorescence and intraligand ³ phosphorescence in glassy solutions (MeOH at 77 K).     

Synthesis and characterization of poly(lactic acid) and glucosamine-formaldehyde/dodecylbenzenesulfonate composite films

The composite films of poly(lactic acid) (PLA) doped with glucosamine(Gluc)-formaldehyde(FA) polymer/sodium dodecylbenzenesulfonate (SDBS) complexes at 1–5 wt% were synthesized to demonstrate striking improvement of their structural and mechanical properties. The polymer complexes were obtained by the hydrothermal polymerization of Gluc and FA at a molar ratio of 1:2 in the presence of SDBS. The atomic ratios of S in to N in (=S/N) in the polymer complexes limitedly range from 0.52 to 0.69, indicating that the complexation develops through the nonstoichiometric reaction between groups of (Gluc-FA) polymer and ones of SDBS and 31–48% of the groups remain unbound. The PLA composite film doped with 1 wt% (Gluc-FA)/SDBS showed the elongation-at-break of as large as 194% compared with 37% for PLA film, together with an appreciable increase of the crystallites size (D ₂₀₀) of PLA from 21.8 to 33.3 nm.     

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).     

Synthesis and magneto-structural correlation of a new maleato bridged copper(II) coordination polymer

A one-dimensional (1-D) copper(II) coordination polymer [Cu(maleate)(2,2′-bipyridyl)] _n ·2H₂O has been synthesised. Single crystal X-ray diffraction study reveals that maleate ion bridges two adjacent copper(II) centres along the chain in a syn–anti fashion. A complete cryomagnetic investigation of the title complex correlates well with the distorted square pyramidal geometry of the central copper(II) ion and bridging nature of the maleate. A τ value of 0.26 indicates the distortion towards tbp coordination allowing the magnetic orbital to acquire some character leading to a weak antiferromagnetic interaction having J = −0.26 cm⁻¹. The complex has also been firmly established from several other instrumental techniques like Fourier transform infrared (FT-IR) and EPR spectroscopies.     

Understanding the chemical reactivity of phenylhalocarbene systems: an analysis based on the spin-polarized density functional theory

Global and local indices based on the spin-polarized density functional theory (SP-DFT) have been used to rationalize the philicity power and spin polarization pattern of a family of singlet substituted phenylhalocarbenes, (pYPhXC, Y = –NO₂ , –CN, –CHO, –F, –H, –CH₃ , –OH, –OCH₃ , –NH₂ ; X = –F, –Cl, –Br). The local reactivity may be traced out by the simple condensed-to-atoms model for the SP-DFT Fukui functions, namely and . For the addition of some singlet phenylhalocarbenes on tetramethylethylene a linear correlation among the global and local electrophilicity index , and the observed rate constants were found. This result supports a mechanistic model where the carbene adds to the olefin in a single step that is controlled by the carbene electrophilicity. These results emphasize the usefulness of general SP-DFT philicities in the rationalization of chemical reactivity at initial stages of reactions that could involve both charge transfer and spin polarization processes.     

Interaction of (hydroxypropyl) cellulose with anionic surfactants in dilute regime

(Hydroxypropyl)cellulose (HPC) dilute aqueous solutions in the presence of sodium cholate (CS), sodium deoxycholate (DC), and sodium dodecylsulphate (SDS) were investigated. The hydrophobicity parameter (I ₁/I ₃) from fluorescence has shown a critical aggregation concentration (CAC) lower than the critical micellar concentration (CMC). One or two breakpoints were observed in the curve conductivity vs surfactant concentration. The thermodynamic parameters of aggregation (, and ) and the degree of counterion dissociation were calculated. Evidences for the secondary aggregation of CS/water system were found. The relative viscosity increases for HPC/bile salt solutions only at high surfactant concentrations, whereas for HPC/SDS, it passes through a maximum. The cloud points of both HPC/bile salt solutions at higher surfactant concentrations reach a temperature plateau value around 324 K, while for HPC/SDS, it exceeds 373 K at low SDS concentrations. Dynamic light scattering has demonstrated that the surfactants bind to HPC already at concentrations lower than CAC.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

Luminescence Behavior of Polynuclear Alkynylcopper(I) Phosphines

A series of soluble trinuclear and tetranuclear copper(I) complexes containing ₃-^l acetylides , and have been synthesized and shown to exhibit rich photoluminescent behavior at room temperature. The electrochemistry of the trinuclear Cu(I) acetylide complexes and the excited-state redox properties of have been investigated. The X-ray crystal structures of and have been determined.     

A bound for the reversal distance of genome rearrangements

This paper answers the conjecture posed by Jianxiu Hao regarding an estimate of sorting by reversals. More precisely, for every permutation π the right-hand inequality of will be proved.     

Emulsion polymerization of styrene by horseradish peroxidase-mediated initiation

A new horseradish peroxidase (HRP)-catalyzed redox initiating system was successfully applied in the emulsion polymerization of styrene, producing stable polymer colloids and nanospheres. The particle size was about 30–50 nm with a diameter distribution (CV) 14–20% obtained by SEM. The molecular weight, was around 10⁵, and was around 10⁶. All the results were well consistent with the control, potassium persulfate (KPS)-initiating polymerization. The obtained experimental results supported a micelle mechanism similar to that of conventional emulsion polymerization of styrene.     

Solvent effects on the dielectric dispersion of poly(vinyl pyrrolidone)-poly(ethylene glycol) blends

The complex relative dielectric function , loss tangent , complex electric modulus and alternating current electrical conductivity dispersion behaviour of liquid poly(vinyl pyrrolidone)-poly(ethylene glycol) (PVP–PEG) blends in water, ethyl alcohol and 1,4–dioxane solvent over the entire volume mixture concentration range has been investigated in the frequency range of 20 Hz to 1 MHz at 25 °C. The PVP–PEG blends show the dielectric dispersion corresponding to the micro-Brownian motion of the PVP chain in the upper frequency region, whereas in the lower frequency region, dielectric dispersion is due to ionic conduction and electrode polarization phenomena. Results show that the conductivity values of these blends in water and dioxane can be monitored with the change in the solvent concentration, whereas it has a small variation with ethyl alcohol concentration. The comparative dielectric dispersion shape study confirms that the chain dynamics of polymers blend is influenced by heterogeneous interactions and solvent polar strength.     

A class of orthogonal polynomials suggested by a trigonometric Hamiltonian: Antisymmetric states

This is the second in a series of papers dealing with the sets of orthogonal polynomials generated by a trigonometric Hamiltonian. In the first of this series, a subclass of the Jacobi polynomials denoted by and referred to as the -polynomial of the first kind, which arises in the investigation of the symmetric state eigenfunctions of the Hamiltonian under consideration, was examined. Another subclass of the Jacobi polynomials denoted by is introduced here representing the antisymmetric states, and is called in accordance the -polynomial of the second kind. Moreover, by the derivation of the ultraspherical polynomial wavefunctions, interrelations between the -polynomials of the first and second kinds as well as the other orthogonal polynomial systems are also emphasized.AMS subject classification: 33C45, 81Q05, 33C05     

Kinetics and Equilibria for Complex Formation between Palladium(II) and Chloroacetate

Kinetics and equilibria for the formation of a 1:1 complex between palladium(II) and chloroacetate were studied by spectrophotometric measurements in 1.00 mol HClO₄ at 298.2 K. The equilibrium constant, K, of the reaction

On minimal energies of trees of a prescribed diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let denote the set of trees on n vertices and diameter d, . Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in with minimal energy. In this article, the trees in with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35     

Mathematical relations between codon bias indexes and their applications

Codon Adaptation Index (CAI), Effective Number of Codons as well as its modifications , can be used to measure gene codon bias. In this article, we prove is more efficient and unbiased than and by revisiting correlations of them with CAI in the level of individual amino acid’s codon bias. Correlations are studied by mathematical expressions rather than statistic methods, because the latter unavoidably depend on the data set used. Additionally, the immediate cause of correlations of with CAI (as well as those of RSCU with CAI) are also described in mathematical language. Perhaps, mathematics provides us a new way to study correlations between biological indexes.