Computing topological indices of Sudoku graphs

In QSAR/QSPR study, physico-chemical properties and topological indices such as Randi?, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numeric number. These topological indices correlate certain physico-chemical properties like boiling point, stability, strain energy etc. of chemical compounds. Graph theory has found a considerable use in this area of research. The topological indices of certain interconnection networks were studied recently by Imran et al. (Appl Math Comput 244:936–951, 2014). In this paper, we extend this study to \(n\times n\) Sudoku graphs and derive analytical closed results of general Randi? index \(R_{\alpha }(G)\) for different values of “\(\alpha \)” for Sudoku (SK). We also compute the general Randi?, first Zagreb, ABC, GA, \(ABC_{4}\) and \(GA_{5}\) indices and give closed formulae of these indices for Sudoku graphs.     

A hierarchical multiscale finite element approach for modelling the heat affected zone of welded joints

In the present work a hierarchical multiscale approach, based on finite element method, is applied in order to characterize the heat affected zone (HAZ) of a welded joint with parent material S460M. The metallurgical constituents of particular regions of the HAZ was determined experimentally and, regarding the multiscale approach, a first-order computational homogenization method was implemented. In this context, effective stress-strain-relations in particular spots of the HAZ are determined applying prescribed displacement boundary conditions. The obtained relations were considered in subsequent finite element calculations of butt-welded joints under tensile loading, allowing a comparison of the numerical strain states to experimental results. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic optimization of an urban rail timetable under time‐dependent and uncertain demand

Urban rail planning is extremely complex, mainly because it is a decision problem under different uncertainties. In practice, travel demand is generally uncertain, and therefore, the timetabling decisions must be based on accurate estimation. This research addresses the optimization of train timetable at public transit terminals of an urban rail in a stochastic setting. To cope with stochastic fluctuation of arrival rates, a two‐stage stochastic programming model is developed. The objective is to construct a daily train schedule that minimizes the expected waiting time of passengers. Due to the high computational cost of evaluating the expected value objective, the sample average approximation method is applied. The method provided statistical estimations of the optimality gap as well as lower and upper bounds and the associated confidence intervals. Numerical experiments are performed to evaluate the performance of the proposed model and the solution method.     

On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

In this paper, we study the fractional backward differential formula (FBDF) for the numerical solution of fractional delay differential equations (FDDEs) of the following form: \(\lambda _n {}_0^C D_t^{\alpha _n } y(t - \tau ) + \lambda _{n - 1} {}_0^C D_t^{\alpha _{n - 1} } y(t - \tau ) + \cdots + \lambda _1 {}_0^C D_t^{\alpha _1 } y(t - \tau ) + \lambda _{n + 1} y(t) = f(t), t \in [0,T]\), where \( \lambda _i \in \) \(\mathbb {R}\,(i = 1,\ldots ,n + 1)\,,\,\lambda _{n + 1} \ne 0,\,\, 0 \leqslant \alpha _1< \alpha _2< \cdots< \alpha _n < 1,\,\,T > 0,\) in Caputo sense. We find the Green’s functions for this equation corresponding to periodic/anti-periodic conditions in term of the Mittag-Leffler type. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Finally, some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis     

On the intersection graph of a finite group

For a finite group G, the intersection graph of G which is denoted by Γ(G) is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H ∩ K ≠ 1. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of Aut(Γ(G)).     

Lie algebras with few centralizers

We will characterize all finite dimensional Lie algebras with at most |F|²+|F|+2 centralizers, where F is the underlying field of Lie algebras under consideration.     

Mathematical and numerical modeling of melting in metal-ceramic composite beds

Many of the ceramics used in the manufacture of the metal-ceramic composites are available as powders, which has encouraged investigators to develop tool less manufacturing techniques. Innovative processes like combustion synthesis and laser sintering processes offer great potential for specific material combinations. These processes involve heat and mass transfer in porous matrix with melting, solidification and may be chemical reaction. Since the resulting transport is relatively complex, there are few mathematical models available. In the present study, melting and transport of metal is demonstrated for a one-dimensional bed with uniform heat generation and convectively cooled boundaries. The effects of different Bond, Biot and Prandtl numbers are discussed using constant and temperature varying material properties. At low bond number the transport is controlled by capillary forces and temperature and saturation distribution is symmetric about the center of the bed. For Biot number larger than ten the process is internally controlled. Effect of Prandtl number is opposite of that of the Bond number. Use of constant properties overpredicts the process rate by 13%.     

Efficient Solvent‐Free Synthesis of Benzothiazine‐Fused Pyrrolo[3,4‐c]coumarins: Cycloaddition Reactions between Coumarin‐Based Dihydrobenzothiazoles and Isocyanides

A new and unusual synthesis of benzothiazine‐fused pyrrolo[3,4‐c]coumarins, involving the ring‐opening of coumarin‐based dihydrobenzothiazoles and subsequent [4+1] cycloaddition reaction with isocyanides, was described. Thus, simple heating of various 3‐(2,3‐dihydro‐2‐methylbenzo[d]thiazol‐2‐yl)coumarins with isocyanides produced the title compounds in good yields under solvent‐free conditions.