Abstract: The inverse degree index, also called inverse index, first attracted attention through numerous conjectures generated by the computer programme Graffiti. Since then, its relationship with other graph invariants has been studied by several authors. In this paper, we obtain new inequalities involving the inverse degree index, and we characterize graphs which are extremal with respect to them. PubDate: 2019-03-22 DOI: 10.1007/s10910-019-01022-3

Abstract: A protein is commonly visualized as a discrete piecewise linear curve, conventionally characterized in terms of the extrinsically determined Ramachandran angles. However, in addition to the extrinsic geometry, the protein has also two independent intrinsic geometric structures, determined by the peptide planes and the side chains respectively. Here we develop a novel 3D visualization method that instead of the extrinsic geometry utilizes the intrinsic geometry of side chains. We base our approach on a series of orthonormal coordinate frames along the protein side chains in combination with a mapping of the atoms positions onto a unit sphere, for visualization purposes. We develop our methodology in terms of an example, by analyzing the acidity dependence of the presumed myoglobin ligand gate. In the literature, the ligand gate is often asserted to be highly localized at the distal histidine, its functioning being regulated by environmental changes. Thus, we investigate whether any \({\mathrm{pH}}\) dependence can be detected in the orientation of the distal and proximal histidine residues, using existing crystallographic data. We observe no \({\mathrm{pH}}\) dependence, in support of the alternative proposals that the ligand gate is more complex and might even be located elsewhere. Our methodology should help the planning of future myoglobin structure experiments, to identify the ligand gate position and its mechanism. More generally, our methodology is designed to visually depict the spatial orientation of side chain covalent bonds in a protein. As such, it can be eventually advanced into a general visual 3D tool for protein structure analysis for purposes of prediction, validation and refinement. It can serve as a complement to widely used visualization suites such as VMD, Jmol, PyMOL and others. PubDate: 2019-03-20 DOI: 10.1007/s10910-019-01021-4

Abstract: In this paper and for the first time in this research area we formulate a new multistage multistep full in phase method with meliorated properties. A theoretical, computational and numerical contemplation is also presented. The sufficiency of the new scheme is tried on using systems of coupled differential equations which represent quantum chemistry problems. PubDate: 2019-03-14 DOI: 10.1007/s10910-019-01018-z

Abstract: Recently, the problem about Pfaffian property of graphs has attracted much attention in the matching theory. Its significance stems from the fact that the number of perfect matchings in a graph G can be computed in polynomial time if G is Pfaffian. The Möbius strip with unique surface structure strip has potentially significant chirality properties in molecular chemistry. In this paper, we consider the Pfaffian property of graphs on the Möbius strip based on topological resolution. A sufficient condition for Pfaffian graphs on the Möbius strip is obtained. As its application, we characterize Pfaffian quadrilateral lattices on the Möbius strip. PubDate: 2019-03-14 DOI: 10.1007/s10910-019-01019-y

Abstract: In this paper, wavelet-based operational matrix methods have been developed to investigate the approximate solutions for amperometric enzyme kinetics problems. The operational matrices of derivatives have been utilized for solving the nonlinear initial value problems. The accuracy of the proposed wavelet-based approximation methods has been confirmed. The main purpose of the proposed method is to get better and more accurate results. Operational matrices of Chebyshev and Legendre wavelets are utilized to obtain a sequence of discrete equations into the systems of algebraic equations and the solutions of algebraic systems lead to the solution of nonlinear initial value problems. Numerical experiments are given to demonstrate the accuracy and efficiency of the proposed method. PubDate: 2019-03-05 DOI: 10.1007/s10910-019-01017-0

Abstract: In this paper and for the first time in this research discipline we form a new multistage multistep complete in phase method with meliorated properties. A theoretical, computational and numerical contemplation is also presented. The sufficiency of the new scheme is tried on using systems of coupled differential equations which represent quantum chemistry problems. PubDate: 2019-03-02 DOI: 10.1007/s10910-019-01011-6

Abstract: We have probed the condition of periodic oscillation in a class of two variable nonlinear dynamical open systems modeled with Lienard–Levinson–Smith (LLS) equation which can be a limit cycle, center or a very slowly decaying center type oscillation. Using a variety of examples of open systems like Glycolytic oscillator, Lotka–Volterra (L–V) model, a generalised van der Pol oscillator and a time delayed nonlinear feedback oscillation as a non-autonomous system, each of which contains a family of periodic orbits, we have solved LLS systems in terms of a multi-scale perturbation theory using Krylov–Bogoliubov (K–B) method and it is utilised to characterise the size and shape of the limit cycle and center as well as the approach to their steady state dynamics. We have shown the condition when the average scaled radius of a center undergoes a power law decay with exponent \(\frac{1}{2}\) . PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0981-7

Abstract: The computational treatment of detailed kinetic reaction mechanisms for combustion is expensive, especially in the case of biodiesel fuels. In this way, great efforts in the search of techniques for the development of reduced kinetic mechanisms have been observed. As Methyl Butanoate (MB, \(C_3H_7COOCH_3\) ) is an essential model frequently used to represent the ester group of reactions in saturated methyl esters of large chain, this paper proposes a reduction strategy and uses it to obtain a reduced kinetic mechanism for the MB. The reduction strategy consists in the use of artificial intelligence to define the main chain and produce a skeletal mechanism, apply the traditional hypotheses of steady-state and partial equilibrium, and justify these assumptions through an asymptotic analysis. The main advantage of the strategy employed here is to reduce the work required to solve the system of chemical equations by two orders of magnitude for MB, since the number of reactions is decreased in the same order. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0984-4

Abstract: In this paper, a new approach is proposed for solving a class of singular boundary value problems of Lane–Emden type. It is well known that the Adomian decomposition method (ADM) fails to provide a convergent series solution to strongly nonlinear boundary value problem in the wider region and the B-spline collocation method yields unsatisfactory approximation in the presence of singularity. To avoid these shortcomings of both methods, we propose a novel numerical method based on a combination of modified Adomian decomposition method and quintic B-spline collocation method to obtain more accurate solution of the problem under consideration. The principal idea of this approach is to decompose the domain of the problem \(D = [0, 1]\) into two subdomains as \(D = D_{1} U D_{2}= [0, \delta ] U [\delta , 1]\) ( \(\delta \) is vicinity of the singularity). In the first domain \(D_1\) , the underlying singular boundary value problem is efficiently tackled by a modified Adomian decomposition method. The intent is to apply the ADM in the smaller domain for finding a satisfactory solution. Finally, in the second domain \(D_2\) , a collocation approach based on quintic B-spline basis function is designed for solving the resulting regular boundary value problem. The error estimation of the quintic B-spline interpolation is supplemented. In addition, six illustrative examples are presented to demonstrate the applicability and accuracy of the new method. It is shown that the resulting solutions appear to be higher accurate when compared to some existing numerical methods. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-00995-x

Abstract: The theoretical equations of Zeeman energy levels, including the zero-field energy levels and the first- and second-order Zeeman coefficients, have been derived for fifteen states of the \(^5T_{2g}\) ground term, considering the axial ligand-field splitting and the spin–orbit coupling. The equations are expressed as the functions of three independent parameters, \(\Delta \) , \(\lambda \) , and \(\kappa \) , where \(\Delta \) is the axial ligand-field splitting parameter, \(\lambda \) is the spin–orbit coupling parameter, and \(\kappa \) is the orbital reduction factor. The equations are useful in simulating magnetic properties (magnetic susceptibility and magnetization) of the complexes with \(^5T_{2g}\) ground terms, e.g. octahedral high-spin iron(II) complexes. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0987-1

Abstract: In this article, the authors approximate solution to the Brusselator model by Galerkin finite element method and present a priori error estimate for the approximation. Further, we study the stability of the model when cross-diffusion is present. We find that the cross-diffusion increases the wave number of the solution. Using Crank–Nicolson method with Predictor–Corrector scheme for discretization of time, we propose an algorithm for numerical simulation of the solution. Lastly, some numerical examples are considered to illustrate the efficiency and accuracy of the method and to plot the Turing patterns of the model. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0982-6

Abstract: A new four stages two-step in phase algorithm with best possible characteristics is formed, for the first time in the bibliography, in this paper. We represent a theoretical and numerical study. The adequacy of the new method is tried out using systems of coupled differential equations. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-00991-1

Abstract: The quantum system having cosinusoidal potential energy is a well known model in which the Schrödinger equation can be cast in the form of the Mathieu equation. The periodic eigenfunctions of the Mathieu equation are then related to the wavefunctions associated with a Hamiltonian having a cosinusoidal potential energy term. These wavefunctions are well known but they are delocalized functions having amplitude in more than one potential energy well. Sometimes a more convenient set of functions are those with amplitude localized in only one of the potential energy wells. This system is analyzed in the context of the group ring algebra associated with the symmetry of the Hamiltonian. Primary results of this work are (i) abstraction to the group ring algebra associated with these systems and (ii) the presentation of general formulae for the localized wavefunctions in terms of linear combinations of the delocalized wavefunctions. Explicit analytic results are presented for cases where the cosinusoidal potential energy has two through nine wells. Discussion of how the group ring approach serves as a path to other types of periodic potentials is also given. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0980-8

Abstract: Granular materials represent a vast category of particle conglomerates with many areas of industrial applications. Here we represent these materials by graphs which capture their topological organization and ordering. Then, using the communicability function—a topological descriptor representing the thermal Green function of a network of harmonic oscillators—we prove the existence of a universal topological melting transition in these graphs. This transition resembles the melting process occurring in solids. We show here that crystalline-like granular materials melts at lower temperatures and display a sharper transition between solid to liquid phases than the random spatial graphs, which represent amorphous granular materials. In addition, we show the evolution mechanism of melting in these granular materials. In the particular case of crystalline materials the process starts by melting a central core of the crystal which then growth until the whole material is in the liquid phase. We provide experimental confirmation from published literature about this process. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0988-0

Abstract: In this work, we review various nonlinear systems with fractional derivatives of the Riesz type in space. Concretely, we consider partial differential equations with fractional Laplacians, which consider potentials of the Fermi–Pasta–Ulam, sine-Gordon, Klein–Gordon and double sine-Gordon. These regimes have the important feature that some energy functional is available in the fractional scenario. For each of these cases, we propose numerical techniques to approximate their solutions, and some discrete functionals are provided in order to approximate the energy dynamics of the systems. The methodologies are energy-preserving (conservative) techniques which are employed then to investigate the presence of nonlinear supratransmission in those systems. It is well known that such process is present in continuous Fermi–Pasta–Ulam, sine-Gordon, Klein–Gordon and double sine-Gordon regimes when the derivatives are of integer order. As one of the most important outcomes of this survey, the computer simulations confirm that this process is also present when the derivatives are of fractional order. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0983-5

Abstract: In this work, we derive an effective non-adiabatic kinetic energy operator for nuclear motion in triatomic molecules on the basis of perturbation theory. For this purpose, we extended the approach of Herman and Asgharian (J Mol Spectr 19:305, 1966) originally developed for diatomic systems. General perturbative-type expressions for effective distance-dependent reduced nuclear masses have been obtained for a triatomic system. It is shown that in the diatomic limit our method reproduces correctly the previous known result of Herman and Asgharian. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-0979-1

Abstract: We obtain the matrix representation of a three-spin Hamiltonian by straightforward application of the Kronecker product. The starting point are the Pauli matrices for spin-1/2 and it is not necessary to resort to ladder operators or any other mathematical property of spin operators. PubDate: 2019-03-01 DOI: 10.1007/s10910-018-00994-y