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).     

Practical version of the Asymptotic Linearity Theorem with applications to the additivity problems of thermodynamic quantities

By using fundamental notions and theorems of the Lesbesgue integral, a practical version of the Asymptotic Linearity Theorem has been derived. From this new version of the theorem, it easily follows that the theorem is also applicable to the additivity problems of thermodynamic quantities of molecules having many identical moieties, as well as to those of the zero-point vibrational energy of hydrocarbons and the total pi electron energies of altemant hydrocarbons.On leave from: Institute for Fundamental Chemistry, 34-4 Nishihiraki-cho, Takano, Sakyo-ku, Kyoto 606, Japan.     

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.     

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)E _n of a linearly extended system B-A _n -B having n repeating identical moieties has the asymptotic expansionE _n = n + +o(1) as n , where 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.On leave from: Institute for Fundamental Chemistry, 34-4 Nishihiraki-cho, Takano, Sakyo-ku, Kyoto 606, Japan.     

Some aspects of the symmetry and topology of possible carbon allotrope structures

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C₆₀ of I _h symmetry and ellipsoidal C₇₀ of D _5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C₇₆, C₇₈, C₈₂ and C₈₄ can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n ² carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C₆₀ fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C₁₆₈ schwarzite unit cell has local O _h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C₆₀ fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

Aspects of topology, genus and isomerism in closed 3-valent networks

Recent work on toroidal and Klein bottle fullerenes is reported, including, in particular, attempts to generate useful visual models with the NiceGraph program (part of the Vega package). Some strengths and weaknesses of this modelling software are mentioned. The so‐far‐unanswered question of what ratio of small radius (r) to large radius (R) might be preferred by a geometric realisation of a toroidal network, and broad categories of isomerism are briefly discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Expressions of the general form of the acid-base titration curve

The titration of a multicomponent mixture can be described either with de Levies integral representation [1] or with Moisios and Heikonens derivative representation [2]. In the latter method an experimental titration curve is numerically differentiated and the computations are based on the buffer capacity. Numerical calculations demonstrate that the results of these two methods coincide very well.     

Dynamic Aspects and Controllability of the MELiSSA Project: A Bioregenerative System to Provide Life Support in Space

Manmade ecosystems differ from their prototype biosphere by the principle of control. The Earth Biosphere is sustainable by stochastic control and very large time constants. By contrast, in a closed ecosystem such as the micro-ecological life support system alternative (MELiSSA system) developed by the European Space Agency for space exploration, a deterministic control is a prerequisite of sustainable existence. MELiSSA is an integrated sum of interconnected biological subsystems. On one hand, all unit operations in charge of the elementary functions constitutive of the entire life support system are studied until a thorough understanding and mathematical modelling. On the other hand, the systemic approach of complex, highly branched systems with feedback loops is performed. This leads to study in the same perspective, with the same degree of accuracy and with the same language, waste degradation, water recycling, atmosphere revitalisation and food production systems prior to the integration of knowledge-based control models. This paper presents the mathematical modelling of the MELiSSA system and the interface between the control strategy of the entire system and the control of the bioreactors.     

Linearity of calibration curves: use and misuse of the correlation coefficient

The correlation coefficient is commonly used to evaluate the degree of linear association between two variables. However, it can be shown that a correlation coefficient very close to one might also be obtained for a clear curved relationship. Other statistical tests, like the Lack-of-fit and Mandel’s fitting test thus appear more suitable for the validation of the linear calibration model. A number of cadmium calibration curves from atomic absorption spectroscopy were assessed for their linearity. All the investigated calibration curves were characterized by a high correlation coefficient (r >0.997) and low quality coefficient (QC <5%), but the straight-line model was systematically rejected at the 95% confidence level on the basis of the Lack-of-fit and Mandel’s fitting test. Furthermore, significantly different results were achieved between a linear regression model (LRM) and a quadratic regression (QRM) model in forecasting values for mid-scale calibration standards. The results obtained with the QRM did not differ significantly from the theoretically expected value, while those obtained with the LRM were systematically biased. It was concluded that a straight-line model with a high correlation coefficient, but with a lack-of-fit, yields significantly less accurate results than its curvilinear alternative. Received: 15 January 2002 Accepted: 18 April 2002     

Adsorption integral equation via complex approximation with constraints: kernel of general form

For the monolayer adsorption on a homogeneous surface, including arbitrary range lateral interactions, the isotherm can be written as a power series of the Langmuir isotherm. If this isotherm is used as the kernel in the adsorption integral equation, this integral equation can be solved in an analytical form. Because the global isotherm is usually known as a set of experimental values, the use of a numerical method is inevitable. A new numerical method for solving the adsorption integral equation with a kernel of general form is developed. It is based on recent results concerning the structure of the local isotherm and on the ideas of complex approximation with constraints, and allows reduction of the problem under consideration to a linear‐quadratic programming problem. Results of numerical experiments are presented. The method can be useful for the evaluation of the adsorption energy distribution from experimental data. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1058–1066, 2001     

Continuity in symmetry and in distinguishability of states: the symmetry numbers of nonrigid molecules

We establish a connection between the concept of distinguishability of quantum states and the concept of continuity in geometric symmetry. For this purpose, we employ the continuous symmetry measure and the nuclear wave functions of a molecule, and evaluate the physical effects of deviation from geometric symmetry. We apply this tool in presenting a unified approach to assigning symmetry numbers to rigid and nonrigid molecules, and readdress, as a specific example, the evaluation of the temperature dependence of the entropy of cyclobutane and cyclohexane. We believe we show that the concept of continuity in geometric symmetry provides a more natural and practical approach to the inherent link between symmetry and entropy, compared with the classical approach.     

A general approach to the screening and confirmation of tryptamines and phenethylamines by mass spectral fragmentation

Certain characteristic fragmentations of tryptamines (indoleethylamine) and phenethylamines are described. Based on the GC-EI/MS, LC-ESI/MS and MALDI/TOFMS, the mass fragmentations of 13 standard compounds, including alpha-methyltryptamine (AMT), N,N-dimethyltryptamine (DMT), 5-methoxy-alpha-methyltryptamine (5-MeO-AMT), N,N-diethyltryptamine (DET), N,N-dipropyltryptamine (DPT), N,N-dibutyltryptamine (DBT), N,N-diisopropyltryptamine (DIPT), 5-methoxy-N,N-dimethyltryptamine (5-MeO-DMT), 5-methoxy-N,N-diisopropyltryptamine (5-MeO-DIPT), methamphetamine (MAMP), 3,4-methylenedioxyamphetamine (3,4-MDA), 3,4-methylenedioxymethamphetamine (3,4-MDMA) and 2-methylamino-1-(3,4-methylenedioxyphenyl)butane (MBDB), were compared. As a result, the parent ions of these analytes were hard to be obtained by GC/MS whereas the protonated molecular ions can be observed clearly by means of ESI/MS and MALDI/TOFMS. Furthermore, two major characteristic fragmentations, namely and alpha-cleavage ([M+H](+)-->[3-vinylindole](+)) and beta-cleavage ([M+H](+)-->[CH(2)N(+)R(N1)R(N2)]), are produced when the ESI and MALDI modes are used, respectively. In the case of ESI/MS, the fragment obtained from alpha-cleavage is the major process. In contrast to this, in the case of MALDI/TOFMS, the major fragment is produced via beta-cleavage. The ionization efficiency and fragments formed from either alpha- or beta-cleavages are closely related to the degree of alkylation of the side chain nitrogen in both cases.     

Structure rationalization and topology prediction of two-distinct-component organic crystals: the role of volume fraction and interface topology

We consider here small-length-scale crystal structures with two clearly different molecular components (e.g., hydrophobic and hydrophilic). Using a perspective developed by studies on large-length-scale block copolymers and liquid crystals, we focus on the crystalline interface between the two components. We examine four types of two-component crystals: aromatic ammonium carboxylates, aromatic oligo(ethylene oxides), cyclohexylammonium carboxylates, and ether-thioether compounds. Of the 111 crystal structures found in the Cambridge Structure Database (CSD), 108 adopt one of the five generic topologies found in diblock copolymers: spheres, columns, perforated layers, layers, and bicontinuous structures. As in diblock copolymers, a key factor controlling the interfacial topology is shown to be the volume ratio of the two components. When the volume fraction of one component is less than 30% of the whole, more than five-sixths of the examined crystal structures are of columnar or spherical type. For volume fractions between 40 and 50% more than three-quarters are of lamellar or bicontinuous type. We use this model to predict the topologies of small-length-scale two-component crystals. We predict the crystal topolgies of six new crystal structures: three are predicted to be columnar, and the other three, lamellar or bicontinuous. The crystal structures of these systems were then determined by single-crystal X-ray methods. Five of the structures form in topologies consistent with the predictions: three in columns and two in layers. The remaining one forms as a perforated layer instead of the predicted columnar structure. Such predictive accuracy is consistent with the statistics of the CSD investigation.     

Pyrromethene dyes (BODIPY) can form ground state homo and hetero dimers: photophysics and spectral properties

Homo and hetero dimerisation of two spectroscopically different BODIPY chromophores was studied, namely, 4,4-difluoro-4-bora-3a,4a-diaza-s-indacene and its 5-styryl-derivative. These exhibit very similar absorption and fluorescence spectral shape, but are mutually shifted by ca. 70 nm. For this reason the former and the latter are referred to as the green and red BODIPY, which here are denoted gB and rB, respectively. Various spectroscopic properties of the rB in different common solvents were determined. The calculated and experimental fluorescence quantum yield is found to be close to 100%, the fluorescence relaxation has a single exponential decay with a lifetime of about 4.5 ns, and the F?rster radius for donor-donor energy migration is 67+/-1A. The dimerisation in different solvents was examined by using custom synthesised; mono and bis BODIPY-labelled forms of 1,2-cis-diaminocyclohexane. It is shown that gB and rB can form ground state homo- as well as hetero dimers. The dimers are non-fluorescent, compatible with H-dimers and may act as excitation traps or as acceptors to the corresponding excited monomers.     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

Voltammetry of redox reactions using the general form of the oxidized or reduced species adsorption isotherm

The theory of linear-sweep voltammetry (LSV) applied to heterogeneous redox reactions accompanied by the nondissociative adsorption of the reactant or the product is developed. The basic criterial relationships of LSV in this case are invariant with respect to the type of adsorption isotherm and the number of adsorption sites occupied by one species. The degree of irreversibility of the discharge-ionization step can be evaluated from the effect of the potential scan rate on the peak potential. The nature of the adsorbate can be deduced from the effect of the reactant concentrations on the peak current.