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The paper by C. Lubich in Numer. Math. 2(52):129–145, 1988 is widely cited for its analysis of convolution quadrature rules for integrals with weakly singular kernels. This analysis depends on a key technical lemma (an upper bound on a discrete convolution sum) whose proof uses some advanced tools. In the present paper it will be shown that this lemma can be quickly proved in an elementary way; moreover, the new proof includes those cases that were excluded from the 1988 paper, and the bounds obtained are shown to be sharp. [ABSTRACT FROM AUTHOR]

Highlights: [•] The tracking consensus analysis method for stochastic MASs is novel. [•] The asymmetric communication delays is focused on in this paper. [•] The delays is time-varying, and one derives its upper bound. [•] This paper also focuses on noise, directed topology, and nonlinear dynamics. [ABSTRACT FROM AUTHOR]

In this paper we carry out a nonlinear self-adjoint classification with differential substitutions of a class of dispersive equations with three arbitrary functions. As a particular case, we obtain the sub families of the C 1 ( m , a + b ) equations which are nonlinearly self-adjoint. Conservation laws for these last families are established using the techniques for finding conserved vectors proposed by Ibragimov. [ABSTRACT FROM AUTHOR]

This letter shows an incorrect application of the chain rule for fractional order derivatives reported in paper (Chen et al., 2014). Due to this misleading application, the proof of Theorem 2 and Theorem 5 in Chen et al., (2014) are incorrect. However, the mentioned Theorem 2 is a straightforward conclusion from results already available in literature (Jarad et al., 2013; Matignon 1996), and consequently there is no need to prove it, as it is stated in this letter. In the same way, although the proof of Theorem 5 in Chen et al. (2014) is not valid, Theorem 5 is indeed true, and a recommendation as to how to prove it is made to the authors. Besides, this letter shows that the proposed Theorem 1 in Chen et al., (2014) is also a straightforward conclusion from well known results available in literature (Jarad et al., 2013; Slotine and Li, 1999), so no demonstration is needed for this result neither. [ABSTRACT FROM AUTHOR]

This paper is devoted in the study of (2 + 1) dimensional KP–BBM equation. A small dispersion of waves is included and the nature of the solutions are examined under this effect using the theory of dynamical system. We prove that in the presence of dispersion effect on the equation, yet there exists bounded traveling wave solutions in different classes in terms of solitary waves, periodic and elliptic functions in certain regions. We obtain the general solution of the equation with or without the dispersion effect in terms of Weirstrass ℘ functions and Jacobi elliptic functions. A new technique based on the application of factorization method and the use of functional transformation yields new form of solutions. Finally, we discuss the stability analysis which shows that the traveling wave speed is a bifurcation parameter, which modulates between different classes of waves. Using phase plane analysis, we show that the solution has a transcritical bifurcation at a critical velocity. [ABSTRACT FROM AUTHOR]

Highlights: [•] Discrete zero-curvature representation from new 4-order matrix spectral problem is introduced. [•] Bargmann symmetry constraint and integrable symplectic map about the 4-field lattice equation in this paper is constructed first. [•] The integrable finite-dimensional systems in the Liouville sense are obtained. [ABSTRACT FROM AUTHOR]