The Clar number of a benzenoid hydrocarbon and linear programming

It is shown that the Clar number of a benzenoid hydrocarbon H (defined as the number of circles in a Clar formula, or equivalently as the maximum number of mutually resonant hexagons of H) can be determined by mixed-integer programming. Moreover, linear programming appears to suffice in practice to find in moderate computing time the Clar number of pericondensed hydrocarbons with more than a thousand hexagons.     

The system of harmonically interacting particles: An exact solution of the quantum-mechanical problem

An original procedure has been developed to determine the eigenvalues and the eigenfunctions of quantum-mechanical many-body systems with a harmonic oscillator interaction. Exact solutions in the explicit form have been obtained for the case of systems containing three groups of identical particles. For the system containing two groups of identical particles, a solution in the adiabatic approximation has been obtained and compared with the exact one.     

An Upper Bound for the Clar Number of Fullerene Graphs

A fullerene graph is a three-regular and three-connected plane graph exactly 12 faces of which are pentagons and the remaining faces are hexagons. Let F _n be a fullerene graph with n vertices. The Clar number c(F _n ) of F _n is the maximum size of sextet patterns, the sets of disjoint hexagons which are all M-alternating for a perfect matching (or Kekulé structure) M of F _n . A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: . Two famous members of fullerenes C₆₀ (Buckministerfullerene) and C₇₀ achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.     

Alternative Integer-Linear-Programming Formulations of the Clar Problem in Hexagonal Systems

We present two alternative objective functions for the integer-linear-programming formulation of the Clar problem in hexagonal systems proposed by Hansen and Zheng [1994, J. Math. Chem. 15, 93]. Also, we note that these formulations can be solved in polynomial-time with linear programming algorithms.     

Towards an exact theory of polyelectrolytes

Within the framework of the gas approximation an exact method is developed for calculating the electric potential distribution around a polyion. The polyion is modelled as an infinitely long impermeable cylinder and the mobile ions are assumed to be impermeable spheres. Equations are derived for two potentials ψ_min and ψ_max so that the genuine potential ψ lies in between. The equations are numerically solved for different parameter values. The results are compared with data obtained on the basis of the standard Poisson-Boltzmann equation.     

The iodine number and the unsaturation number of fats

Chemistry of Natural Compounds - In order to determine the degree of unsaturation of fats and to study the kinetics of their hydrogenation, it is proposed to use instead of the iodine number an...     

Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian

We report the implementation of analytic energy gradients for the evaluation of first-order electrical properties and nuclear forces within the framework of the spin-free (SF) exact two-component (X2c) theory. In the scheme presented here, referred to in the following as SFX2c-1e, the decoupling of electronic and positronic solutions is performed for the one-electron Dirac Hamiltonian in its matrix representation using a single unitary transformation. The resulting two-component one-electron matrix Hamiltonian is combined with untransformed two-electron interactions for subsequent self-consistent-field and electron-correlated calculations. The "picture-change" effect in the calculation of properties is taken into account by considering the full derivative of the two-component Hamiltonian matrix with respect to the external perturbation. The applicability of the analytic-gradient scheme presented here is demonstrated in benchmark calculations. SFX2c-1e results for the dipole moments and electric-field gradients of the hydrogen halides are compared with those obtained from nonrelativistic, SF high-order Douglas-Kroll-Hess, and SF Dirac-Coulomb calculations. It is shown that the use of untransformed two-electron interactions introduces rather small errors for these properties. As a first application of the analytic geometrical gradient, we report the equilibrium geometry of methylcopper (CuCH(3)) determined at various levels of theory.     

A DNA algorithm for the graph coloring problem

A DNA algorithm based on surfaces for the graph coloring problem is presented. First the whole combinatorial color assignments to the vertices of a graph are synthesized and immobilized on a surface; then a vertex is legally colored while those adjacent to it with illegal colors are deleted; and the cycle is repeated until finally the correct color assignments to the graph are reached. Compared with the other DNA algorithms, our algorithm is easy to implement and error-resistant.     

On the determination of the exact number of bound states of a given potential. I. General method

A general method is developed for determining the exact number of bound states for a given potential in a space of N dimensions (N ≧ 2). This is applied to a central potential of the form where λ, r₀, α are arbitrary positive numbers, and for which analytical solutions are obtained. Some properties of the degeneracy in the energy levels, with respect to the orbital angular momentum l, are also discussed.     

On the ordering of benzenoid chains and cyclo-polyphenacenes with respect to their numbers of Clar aromatic sextets

Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the concept of sextet polynomial introduced by Hosoya and Yamaguchi [Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)], we define a new ordering of benzenoid systems. For two isomeric benzenoid systems B₁ and B₂, we say B₁>B₂ if each coefficient of sextet polynomial of B₁ is no less than the corresponding coefficient of sextet polynomial of B₂. In this paper, we consider the ordering of the benzenoid chains. The maximal and second maximal benzenoid chains as well as the minimal, the second minimal up to the fourth minimal benzenoid chains are determined. Furthermore, under this ordering, we determine the maximal and second maximal cyclo-polyphenacenes as well as the minimal, the second minimal, and up to the seventh minimal cyclo-polyphenacenes.     

An exact analysis of low Peclet number heat transfer in laminar flow with axial conduction

An exact method of analysis is presented for solving the low Peclet number thermal entry region problem for laminar flow inside a circular tube, including the effects of axial conduction in both the upstream and the downstream directions. The local Nusselt number for uniform wall heat flux conditions is determined for Pe = 0.5, 1, 5, 20, 45 and ∞ at different locations along the thermal entry region. The method is simple and easy to apply, and provides a new approach for the analysis of thermal entry region problems with axial conduction.     

The enormity of the zinc deficiency problem and available solutions; an overview

The societal cost of micronutrient deficiency (MND) or the “hidden hunger” is in millions of dollars/year, reducing the GDP of some countries by as much as 11%. Zn is an important micronutrient for both plants and animals. An estimated 17% of the world population, or around 1.1 billion people, are at the risk of zinc (Zn) deficiency. The deficiency has been related to adverse pregnancy outcomes, stunted growth, premature deaths, immune system dysfunctions, neuro-behavioral disorders, and recently with the failure to recover from COVID-19. These health risks associated with Zn deficiency have compelled FAO and WHO to recommend Zn fortification of diet. Correcting Zn deficiency is a challenge due to several reasons. Close to half of the agricultural soils are Zn deficient, and chemical Zn fertilizers are costly and ineffective. Developing Zn-rich crops through plant breeding and genetic engineering is challenging. Zn-dense diet is costly and cannot be implemented in the low-income region most affected by Zn deficiency. Lack of consensus among regulatory bodies on defining and diagnosing Zn deficiency in plants and Humans. Awareness and other sociocultural issues. Among the most important available solutions are zinc biofortification of the cereal crops, use of zinc biofertilizers, development of Zn-efficient crops with reduced phytate content. The use of Zn supplements, dietary modification, and diversification, especially with fish, are proposed as the most accessible and affordable solutions. Awareness programs in areas suffering the most from Zn deficiency are required. Despite the suggestions from FAO and WHO, global efforts to combat Zn deficiency matching those for combating diseases like HIV are not in place. Coordinated efforts of the international community, especially policy-makers, agricultural scientists, dieticians, physicians, and others, are required to address the issue of hidden hunger.     

Nonlinear relaxation patterns in the Cahn-Hilliard equation: an exact solution

We consider the 1-D Cahn-Hilliard equation with the order parameter v and derive an equation for a modified order parameter g such that g'=v'. The new equation allows for separation of variables. This yields exact solutions for v expressed in terms of generalized hypergeometric functions. These solutions have an infinite gradient at their zeros and the first three derivatives of zero at their extrema. The amplitude of these patterns decreases as the inverse square root of time. It is suggested that the phenomenon of compartmentalization of evolving structures typically observed in evolutionary models of the Cahn-Hilliard type is a manifestation of relaxation patterns similar to those derived in this paper.     

The Clar covering polynomial of hexagonal systems Ⅱ.An application to resonance energy of condensed aromatic hydrocarbons

The Clar covering polynomial of hexagonal systems is a recently proposed1,2 concept which contains much more topological properties of condensed aromatic hydrocarbons, such as Kekule structure count, Clar number, first Herndon number, etc. It is shown that this polynomial can be used for calculating the resonance energy of condensed aromatic hydrocarbons with better accuracy.     

The Tn antigen-structural simplicity and biological complexity

Glycoproteins in animal cells contain a variety of glycan structures that are added co‐ and/or posttranslationally to proteins. Of over 20 different types of sugar–amino acid linkages known, the two major types are N‐glycans (Asn‐linked) and O‐glycans (Ser/Thr‐linked). An abnormal mucin‐type O‐glycan whose expression is associated with cancer and several human disorders is the Tn antigen. It has a relatively simple structure composed of N‐acetyl‐D ‐galactosamine with a glycosidic α linkage to serine/threonine residues in glycoproteins (GalNAcα1‐O‐Ser/Thr), and was one of the first glycoconjugates to be chemically synthesized. The Tn antigen is normally modified by a specific galactosyltransferase (T‐synthase) in the Golgi apparatus of cells. Expression of active T‐synthase is uniquely dependent on the molecular chaperone Cosmc, which is encoded by a gene on the X chromosome. Expression of the Tn antigen can arise as a consequence of mutations in the genes for T‐synthase or Cosmc, or genes affecting other steps of O‐glycosylation pathways. Because of the association of the Tn antigen with disease, there is much interest in the development of Tn‐based vaccines and other therapeutic approaches based on Tn expression.     

The complexity of Platonic and Archimedean polyhedral links

A mathematical methodology for understanding the complexity of Platonic and Archimedean polyhedral links has been developed based on some topological invariants from knot theory. Knot invariants discussed here include crossing number, unknotting number, genus and braid index, which are considered significant in view of DNA nanotechnology. Our results demonstrate that the braid index provides the most structural information; hence, it can be used, among four knot invariants, as the most useful complexity measure. Using such an invariant, it indicates that the complexity of polyhedral links is directed by the number of their building blocks. The research introduces a simple but important concept in the theoretical characterization and analysis of DNA polyhedral catenanes.