Error Estimates for Semi-Discrete and Fully Discrete Galerkin Finite Element Approximations of the General Linear Second-Order Hyperbolic Equation

In this article, we derive error estimates for the semi-discrete and fully discrete Galerkin approximations of a general linear second-order hyperbolic partial differential equation with general damping (which includes boundary damping). The results can be applied to a variety of cases (e.g. vibrating systems of linked elastic bodies). The results generalize pioneering work of Dupont and complement a recent article by Basson and Van Rensburg.     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Matrix valued adaptive cross approximation

A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.     

Crystal structure of 5,6‐bis(9H‐carbazol‐9‐yl)benzo[c][1,2,5]thiadiazole: distortion from a hypothetical higher‐symmetry structure

Nucleophilic substitution of F atoms in 5,6‐difluorobenzo[c ][1,2,5]thiadiazole (DFBT) for carbazole could be potentially interesting as a novel way of synthesizing building blocks for new conjugated materials for applications in organic chemistry. The crystal structures of 5,6‐bis(9H‐carbazol‐9‐yl)benzo[c ][1,2,5]thiadiazole (DCBT), C₃₀H₁₈N₄S, and its hydrate, C₃₀H₁₈N₄S·0.125H₂O, were investigated using single‐crystal X‐ray analysis. The hydrate contains two symmetry‐independent DCBT molecules. The dihedral angles between the plane of the central benzothiadiazole fragment and that of the carbazole units vary between 50.8 and 69.9°, indicating conformational flexibility of the DCBT molecule in the crystals, which is consistent with quantum chemical calculations. The analysis of the crystal packing of DCBT revealed that the experimental triclinic structure could be described as a distortion from a hypothetical higher‐symmetry monoclinic structure. The quantum chemical calculations of two possible monoclinic structures, which are related to the experimental structure by a shifting of molecular layers, showed that the proposed structures are higher in energy by 5.4 and 10.1 kcal mol⁻¹. This energy increase is caused by less dense crystal packings of the symmetric structures, which results in a decrease of the number of intermolecular interactions.     

Long-time dynamics for a nonlinear Timoshenko system with delay

This paper is concerned with a nonlinear Timoshenko system with a time delay term in the internal feedback together with initial data and Dirichet boundary conditions. Under some suitable assumptions on the weights of feedback, we obtain the existence of a global attractor with finite fractal dimension for the case of equal speed wave propagation. Furthermore, the existence of exponential attractors is also derived.     

Virtual screening of B-Raf kinase inhibitors: A combination of pharmacophore modelling,molecular docking, 3D-QSAR model and binding free energy calculation studies

B-Raf kinase has been identified as an important target in recent cancer treatment. In order to discover structurally diverse and novel B-Raf inhibitors (BRIs), a virtual screening of BRIs against ZINC database was performed by using a combination of pharmacophore modelling, molecular docking, 3D-QSAR model and binding free energy (ΔG_bind) calculation studies in this work. After the virtual screening, six promising hit compounds were obtained, which were then tested for inhibitory activities of A375 cell lines. In the result, five hit compounds show good biological activities (IC₅₀ < 50 μM). The present method of virtual screening can be applied to find structurally diverse inhibitors, and the obtained five structurally diverse compounds are expected to develop novel BRIs.     

g-C3N4/SnS2 Heterostructure: a Promising Water Splitting Photocatalyst

Graphite-like carbon nitride (g-C₃N₄) based heterostrutures has attracted intensive attention due to their prominent photocatalytic performance. Here, we explore the g-C₃N₄/SnS₂ coupling effect on the electronic structures and optical absorption of the proposed g-C₃N₄/SnS₂ heterostructure through performing extensive hybrid functional calculations. The obtained geometric structure, band structures, band edge positions and optical absorptions clearly reveal that the g-C₃N₄ monolayer weakly couples to SnS₂ sheet, and forms a typical van der Waals heterojunction. The g-C₃N₄/SnS₂ heterostructure can effectively harvest visible light, and its valence band maximum and conduction band minimum locate in energetically favorable positions for both water oxidation and reduction reactions. Remarkably, the charge transfer from the g-C₃N₄ monolayer to SnS₂ sheet leads to the built-in interface polarized electric field, which is desirable for the photogenerated carrier separation. The built-in interface polarized electric field as well as the nice band edge alignment implys that the g-C₃N₄/SnS₂ heterostructure is a promising g-C₃N₄ based water splitting photocatalyst with good performance.     

On the Number of k‐Dominating Independent Sets

We study the existence and the number of k‐dominating independent sets in certain graph families. While the case namely the case of maximal independent sets—which is originated from Erd?s and Moser—is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k‐dominating independent sets in n‐vertex graphs is between and if , moreover the maximum number of 2‐dominating independent sets in n‐vertex graphs is between and . Graph constructions containing a large number of k‐dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes.     

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

We present a second‐order ensemble method based on a blended three‐step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second‐order ensemble method that combines the two‐step BDF timestepping scheme and a special explicit second‐order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 34–61, 2017