Showing papers in "Journal of Mathematical Chemistry in 2015"
TL;DR: In this article, it was shown that the structure-dependency of total electron energy depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton.
Abstract: In 1972, within a study of the structure-dependency of total $$\pi $$
-electron energy (
$${\mathcal {E}}$$
), it was shown that $${\mathcal {E}}$$
depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton. In the same paper, also the sum of cubes of degrees of vertices of the molecular graph was shown to influence $${\mathcal {E}}$$
, but this topological index was never again investigated and was left to oblivion. We now establish a few basic properties of this “forgotten topological index” and show that it can significantly enhance the physico-chemical applicability of the first Zagreb index.
536 citations
TL;DR: A mathematical analysis reveals that dual descriptor is a more accurate tool than nucleophilic and electrophilic Fukui functions and can be considered a more reliable descriptor to measure local reactivity than Fukui function.
Abstract: A mathematical analysis reveals that dual descriptor is a more accurate tool than nucleophilic and electrophilic Fukui functions. Although Fukui function has the capability of revealing nucleophilic and electrophilic regions on a molecule, the dual descriptor is able to unambiguously expose truly nucleophilic and electrophilic regions, but along with the latter, dual descriptor is less affected by the lack of relaxation terms than the Fukui function when the frontier molecular orbital approximation is applied. This implies that the dual descriptor can be considered a more reliable descriptor to measure local reactivity than Fukui function. This statement is demonstrated in the present work.
127 citations
TL;DR: In this paper, a Runge-Kutta type eight algebraic order two-step method with phase-lag and its first, second, third and fourth derivatives equal to zero is presented.
Abstract: A Runge–Kutta type (four stages) eighth algebraic order two-step method with phase-lag and its first, second, third and fourth derivatives equal to zero is produced in this paper. We also study the results of elimination of the phase-lag and its derivatives on the efficiency of the method. Our studies consist: (1) the construction of the method, (2) the determination of the local truncation error of the proposed method, (3) the investigation of the local truncation error analysis using the comparison with other similar methods of the literature, (4) the computation of the interval of periodicity (stability interval) of the developed method. For this calculation we use a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, (5) the definition of the error estimation based on methods with different algebraic orders and (6) the investigation of the effectiveness of the new obtained method studying the numerical solution of the coupled differential equations arising from the Schrodinger equation.
100 citations
TL;DR: In this article, a family of two stage low computational cost symmetric two-step methods with vanished phase-lag and its derivatives is developed, and the local truncation error, the interval of periodicity and the effect of the vanishing of the phaselag and their derivatives on the efficiency of the obtained method are also studied.
Abstract: A family of two stage low computational cost symmetric two-step methods with vanished phase-lag and its derivatives is developed in this paper. More specifically we produce:
The local truncation error, the interval of periodicity and the effect of the vanishing of the phase-lag and its derivatives on the efficiency of the obtained method are also studied in this paper.
92 citations
TL;DR: In this article, a low computational cost eight algebraic order hybrid two-step method with vanished phase-lag and its first, second, third and fourth derivatives is developed, which investigates the local truncation error, the stability and the result of the elimination of the phase lag and its derivatives on the effectiveness of the produced method.
Abstract: A low computational cost eighth algebraic order hybrid two-step method with vanished phase-lag and its first, second, third and fourth derivatives is developed in this paper. We also investigate the local truncation error, the stability and the result of the elimination of the phase-lag and its derivatives on the effectiveness of the produced method.
78 citations
TL;DR: In this article, the numerical solutions of one dimensional modified Burgers' equation with the help of Haar wavelet method are investigated, and the calculated numerical solutions are drawn graphically.
Abstract: In this paper, we investigate the numerical solutions of one dimensional modified Burgers’ equation with the help of Haar wavelet method. In the solution process, the time derivative is discretized by finite difference, the nonlinear term is linearized by a linearization technique and the spatial discretization is made by Haar wavelets. The proposed method has been tested by three test problems. The obtained numerical results are compared with the exact ones and those already exist in the literature. Also, the calculated numerical solutions are drawn graphically. Computer simulations show that the presented method is computationally cheap, fast, reliable and quite good even in the case of small number of grid points.
63 citations
TL;DR: Based on an optimized explicit four-step method, a new hybrid high algebraic order four step method is introduced in this paper, which investigates the procedure of vanishing of the phase-lag and its first, second, third and fourth derivatives.
Abstract: Based on an optimized explicit four-step method, a new hybrid high algebraic order four-step method is introduced in this paper. For this new hybrid method, we investigate the procedure of vanishing of the phase-lag and its first, second, third and fourth derivatives. More specifically, we investigate: (1) the construction of the new method, i.e. the computation of the coefficients of the method in order its phase-lag and first, second, third and fourth derivatives of the phase-lag to be eliminated, (2) the definition of the local truncation error, (3) the analysis of the local truncation error, (4) the stability (interval of periodicity) analysis (using scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis). Finally, we investigate computationally the new obtained method by applying it to the numerical solution of the resonance problem of the radial Schrodinger equation. The efficiency of the new developed method is tested comparing this method with well known methods of the literature but also using very recently developed methods.
59 citations
TL;DR: In this article, an algebraic order predictor-corrector explicit four-step method is studied and the consequences of the vanishing of the phase-lag and its first, second, third and fourth derivatives are investigated.
Abstract: In this paper an eighth algebraic order predictor–corrector explicit four-step method is studied. The main scope of this paper is to study the consequences of (1) the vanishing of the phase-lag and its first, second, third and fourth derivatives and (2) the high algebraic order on the efficiency of the new developed method. A theoretical and computational study of the obtained method is also presented. More specifically, the theoretical study of the new predictor–corrector method consists of:
Finally, the computational study of the new predictor–corrector method consists of the application of the new produced predictor–corrector explicit four-step method to the numerical solution of the resonance problem of the radial time independent Schrodinger equation.
52 citations
TL;DR: In this article, a new multistage high algebraic order four-step method is obtained, which has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives.
Abstract: A new Multistage high algebraic order four-step method is obtained in this paper. It is the first time in the literature that a method of this category is developed and has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives. We study this new method by investigating: (1) the development of the new method, i.e. the calculation of the coefficients of the method in order the phase-lag and its first, second, third, fourth and fifth derivatives of the phase-lag to be vanished, (2) the determination of the formula of the Local Truncation Error, (3) the comparative analysis of the Local Truncation Error (with this we mean the application of the new method and similar methods on a test problem and the analysis of their behavior), (4) the stability of the new method, by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the results of this application i.e. by investigating the interval of periodicity of the new obtained method. We finally study the computational behavior the new developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrodinger equation. We prove the effectiveness of the new obtained method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.
51 citations
TL;DR: In this article, a predictor-corrector explicit four-step method of sixth algebraic order is investigated, and the results of the elimination of the phase-lag and its first, second and third derivatives on the efficiency of the proposed method are investigated theoretically and computationally.
Abstract: A predictor–corrector explicit four-step method of sixth algebraic order is investigated in this paper. More specifically, we investigate the results of the elimination of the phase-lag and its first, second and third derivatives on the efficiency of the proposed method. The resultant method is studied theoretically and computationally. The theoretical investigation of the new hybrid method consists of: (1) the construction of the new method, (2) the definition (calculation) of the local truncation error, (3) the comparative local truncation error analysis (with other known methods of the same form), (4) the stability analysis using scalar test equation with frequency different than the frequency of the phase-lag analysis. Finally, we will study computationally the new obtained method. This study is based on the application of the new produced predictor–corrector explicit four-step method to the approximate solution of the resonance problem of the radial time independent Schrodinger equation.
50 citations
TL;DR: In this paper, the characterization of complex and detailed balancing for mass action kinetics chemical reaction networks is revisited from the perspective of algebraic graph theory, in particular Kirchhoff's Matrix Tree theorem for directed weighted graphs.
Abstract: The characterization of the notions of complex and detailed balancing for mass action kinetics chemical reaction networks is revisited from the perspective of algebraic graph theory, in particular Kirchhoff’s Matrix Tree theorem for directed weighted graphs. This yields an elucidation of previously obtained results, in particular with respect to the Wegscheider conditions, and a new necessary and sufficient condition for complex balancing, which can be verified constructively.
TL;DR: This work presents a new family of iterative methods for multiple roots whose multiplicity is known and is optimal in Kung–Traub’s sense, because only three functional values per iteration are computed.
Abstract: In this work we focus on the problem of approximating multiple roots of nonlinear equations. Multiple roots appear in some applications such as the compression of band-limited signals and the multipactor effect in electronic devices. We present a new family of iterative methods for multiple roots whose multiplicity is known. The methods are optimal in Kung–Traub’s sense (Kung and Traub in J Assoc Comput Mach 21:643–651, [1]), because only three functional values per iteration are computed. By adding just one more function evaluation we make this family derivative free while preserving the convergence order. To check the theoretical results, we codify the new algorithms and apply them to different numerical examples.
TL;DR: In this paper, the authors apply the Adomian decomposition method combined with the Duan-Rach modified recursion scheme to analyze a system of coupled nonlinear boundary value problems, which yields a rapidly convergent, easily computable, and readily verifiable sequence of analytic approximate solutions.
Abstract: In this paper, we examine a system of two coupled nonlinear differential equations that relates the concentrations of carbon dioxide CO\(_2\) and phenyl glycidyl ether in solution. This system is subject to a set of Dirichlet boundary conditions and a mixed set of Neumann and Dirichlet boundary conditions. We apply the Adomian decomposition method combined with the Duan–Rach modified recursion scheme to analytically treat this system of coupled nonlinear boundary value problems. The rapid convergence of our analytic approximate solutions is demonstrated by graphs of the objective error analysis instead of comparison to an alternate solution technique alone. The Adomian decomposition method yields a rapidly convergent, easily computable, and readily verifiable sequence of analytic approximate solutions that is suitable for numerical parametric simulations. Thus our sequence of approximate solutions are shown to identically satisfy the original set of model equations as closely as we please.
TL;DR: In this article, a model based on the spatially inhomogeneous, nonlinear Smoluchowski equations with random initial distribution density was proposed to describe the annihilation of spatially separate electrons and holes in a disordered semiconductor.
Abstract: To describe the annihilation of spatially separate electrons and holes in a disordered semiconductor, we suggest the use of a model based on the spatially inhomogeneous, nonlinear Smoluchowski equations with random initial distribution density. Furthermore, we present a Monte Carlo algorithm for solving this equation. Our approach provides a general method for the computation of the electron-hole kinetics in inhomogeneous media, taking into account both their radiative and nonradiative recombination by tunneling as well as their diffusion. To validate the simulation algorithm, we compare our model with a more general approach based on a statistical description and the Kirkwood closure of the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy equations. A comparison with recent experimental results is also discussed.
TL;DR: In this paper, two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented, one of them is extended to the multidimensional case.
Abstract: In this paper, by using a generalization of Ostrowski’ and Chun’s methods two bi-parametric families of predictor–corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton’s method, and the predictor of the second one is Steffensen’s scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.
TL;DR: In this article, a QP centroid and a variance are defined at two levels: functional and numerical, and the numerical QP variance can be associated to a collective QP squared distance involving the whole quantum mechanical density function set composing the QP vertices.
Abstract: Quantum polyhedra (QP) are geometrical constructs whose vertices are made by quantum mechanical density functions (DF). In this paper a QP centroid and a variance are defined at two levels: functional and numerical. The numerical QP variance can be shown associated to a collective QP squared distance involving the whole DF set composing the QP vertices. In this manner, a global dissimilarity index corresponding to the set of QP vertices can be defined. Extension of the mathematical and computational techniques developed on QP to shape functions polyhedra and to classical descriptor N-dimensional multimolecular polyhedra, are also discussed.
TL;DR: In this article, a parametric class of iterative methods for solving nonlinear systems of equations is proposed and its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability.
Abstract: In this manuscript, a new parametric class of iterative methods for solving nonlinear systems of equations is proposed. Its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability. These results are checked by solving the nonlinear system that arises from the partial differential equation of molecular interaction.
TL;DR: In this paper, a closed-form approach is presented to determine the degree of vulcanization of natural rubber (NR) vulcanized with sulphur in presence of different accelerators.
Abstract: The paper presents a novel efficient closed form approach to determine the degree of vulcanization of natural rubber (NR) vulcanized with sulphur in presence of different accelerators. The general reaction scheme proposed by Han and co-workers for vulcanized sulphur NR is re-adapted and suitably modified taking into account the single contributions of the different accelerators, focusing in particular on some experimental data, where NR was vulcanized at different temperatures (from 150 to $$180\, ^{\circ }\hbox {C}$$
) and concentrations of sulphur, using TBBS and DPG in the mixture as co-agents at variable concentrations. In the model, chain reactions initiated by the formation of macro-compounds responsible for the formation of the unmatured crosslinked polymer are accounted for. It is assumed that such reactions depend on the reciprocal concentrations of all components and their chemical nature. In presence of two accelerators, reactions are assumed to proceed in parallel, making the assumption that there is no interaction between the two accelerators. Despite there is experimental evidence that a weak process by which each accelerator affects the other, the reaction chemistry is still not well understood and therefore its effect cannot be translated into any mathematical model. In any case, even disregarding such interaction, good approximations of the rheometer curves are obtained. From the simplified kinetic scheme adopted, a closed form solution is found for the crosslink density, with the only limitation that the induction period is excluded from computations. The main capability of the model stands however in the closed form determination of kinetic constants representing the velocities of single reactions in the kinetic scheme adopted, which allows avoiding a numerically demanding least-squares best fitting on rheometer experimental data. Two series of experiments available, relying into rheometer curves at different temperatures and different concentrations of sulphur and accelerators, are utilized to evaluate the fitting capabilities of the mathematical model. Very good agreement between numerical output and experimental data is experienced in all cases analyzed.
TL;DR: In this paper, an efficient wavelet-based approximation method is established to nonlinear singular boundary value problems, where the shift second kind Chebyshev wavelet (S2KCWM) solution is addressed for the nonlinear differential equations in population biology.
Abstract: In this chapter, an efficient wavelet-based approximation method is established to nonlinear singular boundary value problems. To the best of our knowledge, until now there is no rigorous shifted second kind Chebyshev wavelet (S2KCWM) solution has been addressed for the nonlinear differential equations in population biology. With the help of shifted second kind Chebyshev wavelet operational matrices, the nonlinear differential equations are converted into a system of algebraic equations. The convergence of the proposed method is established. The power of the manageable method is confirmed. Finally, we have given some numerical examples to demonstrate the validity and applicability of the proposed wavelet method.
TL;DR: In this paper, the Kirchhoff index of a periodic linear chain is computed as non-trivial functions of the corresponding expressions for the path, which generalizes the previously known results for ladder-like and hexagonal chains.
Abstract: A periodic linear chain consists of a weighted $$2n$$
-path where new edges have been added following a certain periodicity. In this paper, we obtain the effective resistance and the Kirchhoff index of a periodic linear chain as non trivial functions of the corresponding expressions for the path. We compute the expression of the Kirchhoff index of any homogeneous and periodic linear chain which generalizes the previously known results for ladder-like and hexagonal chains, that correspond to periods one and two respectively.
TL;DR: This work describes a new construction algorithm for the recursive generation of all non-isomorphic IPR fullerenes and uses it to compute all I PR fullerene up to 400 vertices.
Abstract: We describe a new construction algorithm for the recursive generation of all non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm stays entirely within the class of IPR fullerenes, that is: every IPR fullerene is constructed by expanding a smaller IPR fullerene unless it belongs to a limited class of irreducible IPR fullerenes that can easily be made separately. The class of irreducible IPR fullerenes consists of 36 fullerenes with up to 112 vertices and 4 infinite families of nanotube fullerenes. Our implementation of this algorithm is faster than other generators for IPR fullerenes and we used it to compute all IPR fullerenes up to 400 vertices.
TL;DR: In this article, a simplified but efficient algorithm to compute triple, quadruple or higher order density similarity hypermatrices via an isometric decomposition of the pair similarity matrix is presented.
Abstract: Collective distances in quantum multimolecular polyhedra, which can be set as a scalar index associated to the variance vector, enhance the role of the pair density similarity matrix. This paper describes a simplified but efficient algorithm to compute triple, quadruple or higher order density similarity hypermatrices via an isometric decomposition of the pair similarity matrix. Such possibility opens the way to use these similarity elements in quantum QSAR and in the description of scalar condensed vector statistical like indices, for instance skewness and kurtosis. This might lead the way to describe the collective structure of quantum and classical multimolecular polyhedra.
TL;DR: In this article, the authors describe the general construction of the Hashin-Shtrikman bounds from first principles in the conductivity setting, and illustrate the implementation of the scheme with several examples.
Abstract: This paper is concerned with the estimation of the effective thermal conductivity of a transversely isotropic two phase composite. We describe the general construction of the Hashin–Shtrikman bounds from first principles in the conductivity setting. Of specific interest in composite design is the fact that the shape of the inclusions and their distribution can be specified independently. This case covers a multitude of composites used in applications by taking various limits of the spheroid aspect ratio, including both layered media and unidirectional composites. Furthermore the expressions derived are equally valid for a number of other effective properties due to the fact that Laplace’s equation governs a significant range of applications, e.g. electrical conductivity and permittivity, magnetic permeability and many more. We illustrate the implementation of the scheme with several examples.
TL;DR: This work discusses the chemical synthesis of topological links, in particular higher order links which have the Brunnian property (namely that removal of any one component unlinks the entire system).
Abstract: We discuss the chemical synthesis of topological links, in particular higher order links which have the Brunnian property (namely that removal of any one component unlinks the entire system). Furthermore, we suggest how to obtain both two dimensional and three dimensional objects (surfaces and solids, respectively) which also have this Brunnian property.
TL;DR: In this paper, analytical solutions for a family of two-stage reactions, in which the later process is first order with respect to the product of the previous, non-first order step, are derived.
Abstract: Analytical solutions are derived for a family of two-stage reactions, in which the later process is first order with respect to the product of the previous, non-first order step. A general strategy is shown that is suitable to handle typical cases. The strategy is demonstrated by considering, zeroth order, second order, mixed second order and third order initial reactions, analytical solutions for all of which can be obtained and advantageously used. The solutions can also be used as archetypes of intermediate formation and decay in chemical kinetics.
TL;DR: A frequently used representation of mass-action type reaction networks is extended to a more general system class where the reaction rates are in rational function form and it is shown that under some technical assumptions, the so-called dense realization containing the maximal number of reactions, forms a super-structure.
Abstract: In this paper, a frequently used representation of mass-action type reaction networks is extended to a more general system class where the reaction rates are in rational function form. An algorithm is given to compute a possible reaction graph from the kinetic differential equations. However, this structure is generally non-unique, as it is illustrated through the phenomenon of dynamical equivalence, when different reaction network structures correspond to exactly the same dynamics. It is shown that under some technical assumptions, the so-called dense realization containing the maximal number of reactions, forms a super-structure in the sense that the reaction graph of any dynamically equivalent reaction network is the sub-graph of the dense realization. Additionally, optimization based methods are given to find dynamically equivalent realizations with preferred properties, such as dense realizations or sparse realizations. The introduced concepts are illustrated by examples.
TL;DR: In this paper, a closed form procedure to determine reaction kinetic constants for NR vulcanized with sulphur is presented, which is much more stable and efficient from a computational standpoint.
Abstract: A closed form procedure to determine kinetic constants for NR vulcanized with sulphur is presented. The kinetic scheme originally proposed by Han et al. (Polymer (Korea) 22:223–230, 1998) and further modified by Milani et al. (Polym Test 32:1052–1063, 2013) is adopted as starting point to deduce a closed form expression for rubber curing degree. Rheometer experimental data collected at different temperatures are used to tune model parameters. After the normalization of the rheometer curves and the exclusion of induction period from calculations, the model requires the estimation of three kinetic constants, two of them describing incipient curing and stable crosslinks formations, the last reproducing reversion phenomenon. Whilst such constants are almost always determined by least-squares best fitting, here a numerical iterative procedure—much more stable and efficient from a computational standpoint—is proposed. The approach requires as input parameters only the degree of vulcanization at infinite (i.e. at the end of vulcanization), the instant where the maximum torque is reached and initial rate of vulcanization. The condition that the numerical curve reaches a maximum at a given time translates mathematically into a non-linear equation in two of the kinetic constants, which are determined iteratively in the paper. The numerical initial vulcanization rate is tuned in such a way to globally minimize the absolute error between numerical and experimental curves. The main capability of the procedure proposed stands in the very straightforward determination of reaction kinetic constants, avoiding demanding least-squares fittings on rheometer experimental data. A set of experimental data available, relying into rheometer curves of the same rubber blend conducted at five different temperatures are used to estimate the fitting capabilities of the mathematical model proposed. Very good agreement with experimental data is observed.
TL;DR: The resultant bond descriptors combining the classical and non-classical terms, due to the probability and current distributions, respectively, are proposed as generalized communication-noise and information-flow concepts in the quantum IT.
Abstract: The classical Information Theory (IT) deals with entropic descriptors of the probability distributions and probability-propagation (communication) systems, e.g., the electronic channels in molecules reflecting the information scattering via the system chemical bonds. The quantum IT additionally accounts for the non-classical (current/phase)-related contributions in the resultant information content of electronic states. The classical and non-classical terms in the quantum Shannon entropy and Fisher information are reexamined. The associated probability-propagation and current-scattering networks are introduced and their Fisher- and Shannon-type descriptors are identified. The non-additive and additive information descriptors of the probability channels in both the Atomic Orbital and local resolution levels are related to the network conditional-entropy and mutual-information, which represent the IT covalency and ionicity components in the classical communication theory of the chemical bond. A similar partition identifies the associated bond indices in the molecular current/phase channels. The resultant bond descriptors combining the classical and non-classical terms, due to the probability and current distributions, respectively, are proposed as generalized communication-noise (covalency) and information-flow (iconicity) concepts in the quantum IT.
TL;DR: In this paper, a direct method for computation of the energy effect of cycles in conjugated molecules is elaborated, based on numerical calculation of the (complex) zeros of certain graph polynomials.
Abstract: A direct method for computation of the energy-effect (ef) of cycles in conjugated molecules is elaborated, based on numerical calculation of the (complex) zeros of certain graph polynomials. Accordingly, the usage of the Coulson integral formula can be avoided, and thus the ef-values can be calculated for arbitrary cycles of arbitrary conjugated systems.
TL;DR: In this paper, the second order Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation and exact bound state solutions are obtained using convolution theorem.
Abstract: The second order \(N\)-dimensional Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution theorem. The Ladder operators are also constructed for the Mie-type potentials in \(N\)-dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.