This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

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2,4‐Diaryl‐2,3‐dihydro‐1H‐1,5‐benzodiazepines readily undergo a ring contraction to generate 2‐aryl‐1‐styrylbenzimidazoles in the presence of some Lewis acids. Copper acetate shows high efficiency compared with other Lewis acids. The ring contraction includes Lewis acid‐catalyzed intramolecular addition, ammonium‐induced ring‐opening of the generated four‐membered azetidine ring, deprotonation, and amine‐promoted nucleophilic styryl 1,2‐shift and elimination. Copper acetate serves as Lewis acid, base, and oxidant. The current reaction provides an efficient method for the convenient synthesis of 2‐aryl‐1‐styrylbenzimidazole derivatives from readily available 2,4‐diaryl‐2,3‐dihydro‐1H‐1,5‐benzodiazepines.

<abstract><p>Traveling salesman problem is a widely studied NP-hard problem in the field of combinatorial optimization. Many and various heuristics and approximation algorithms have been developed to address the problem. However, few studies were conducted on the multi-solution optimization for traveling salesman problem so far. In this article, we propose a circular Jaccard distance based multi-solution optimization (CJD-MSO) algorithm based on ant colony optimization to find multiple solutions for the traveling salesman problem. The CJD-MSO algorithm incorporates "distancing" niching technique with circular Jaccard distance metric which are both proposed in this paper for the first time. Experimental results verify that the proposed algorithm achieves good performance on both quality and diversity of the optimal solutions.</p></abstract>

The finite-difference method is widely used in seismic wave numerical simulation, imaging, and waveform inversion. In the finite-difference method, the finite difference operator is used to replace the differential operator approximately, which can be obtained by truncating the spatial convolution series. The properties of the truncated window function, such as the main and side lobes of the window function’s amplitude response, determine the accuracy of finite-difference, which subsequently affects the seismic imaging and inversion results significantly. Although numerical dispersion is inevitable in this process, it can be suppressed more effectively by using higher precision finite-difference operators. In this paper, we use the krill herd algorithm, in contrast with the standard PSO and CDPSO (a variant of PSO), to optimize the finite-difference operator. Numerical simulation results verify that the krill herd algorithm has good performance in improving the precision of the differential operator.