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In the above article, a second-order convergence in the maximum norm of the two-dimensional regularized long-wave equation is proved. However, due to a wrong inequality in Lemma 3.3 used in X. Wang et al.[1], there are some fundamental errors in this paper, in particular the proof of Theorem 3.3 and the convergence Theorem are no longer correct. The present brief paper contains clarifying comments.

The difference between conventional replicator dynamics and pairwise (PW) nonlinear Fermi dynamics can be discerned by studying the evolutionary dynamics of the interactions between the symmetric cyclic structure in the rock–paper–scissors game and inter- and intraspecific competitions. Often, conventional replicator models presume that the payoff difference among species is a linear function (a linear benefit). This study introduces a PW contrast under the properties of the well-known Fermi rule, where species play against one another in pairs. To model a PW nonlinear evolutionary environment (a nonlinear benefit) within this framework, both analytical and numerical approaches are applied. It is determined that the dynamics of the linear and nonlinear benefits can present the same stability conditions at equilibrium. Moreover, it is also demonstrated that, even in an identical equilibrium condition for both dynamics, the numerical result run by a deterministic approach presents a faster stability state for nonlinear benefit dynamics. This study also suggests that introducing mutation as demographic noise can effectively disrupt the phase regions and show the different relationships between linear and nonlinear dynamics. The symmetric bidirectional mutation among all the species reduced to the stable limit cycle by an arbitrary small mutation rate is also explored. Due to the environmental noise, however, linear and nonlinear exhibit the same steady state. Nevertheless, non-linearity illustrates more stable and faster stability situations. Our result suggests that environmental and demographic noise on the evolutionary dynamic framework can serve as a mechanism for supporting PW nonlinear dynamics in multi-species games.

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

The present comment concerns some doubtful mathematical calculation that produced misleading results which are presented in the above paper.

In this paper, we first point out a gap in Yang (2012). Next, strong convergence theorem for the common solution for a finite family of ? -strongly accretive operator equations in Banach spaces is established without any bounded assumption, and we also give an example to show universality of the result of this paper, which improve and extend some recent results.

In the paper the second order half-linear delay differential equation ( r ( t ) | u ' ( t ) | α - 1 u ' ( t ) ) ' + p ( t ) | u ( ? ( t ) ) | α - 1 u ( ? ( t ) ) = 0 , α 1 is studied under the condition ? ∞ r - 1 / α ( t ) d t < ∞ . The oscillation criterion proved by Sun and Meng (2006, Theorem 2.2) is carefully examined and improved by simplifying and sharpening the estimates used in the original proof and removing unnecessary assumptions. Among others, using elementary arguments we show that an arbitrary positive nondecreasing function ? from this criterion can be safely and with no loss of generality replaced by a constant function. An extension to other related equations, such as neutral equations is also provided.

Ensemble clustering has attracted much attention for its robustness, stability, and accuracy in academic and industry communities. In order to yield base clusterings with high quality and diversity simultaneously in ensemble clustering, many efforts have been done by exploiting different clustering models and data information. However, these methods neglect correlation between different base clusterings during the process of base clusterings generation, which is important to obtain a quality and diverse clustering decision. To overcome this deficiency, a sequential ensemble clusterings generation algorithm for mixed data is developed in this paper based on information entropy. The first high quality base clustering is yield by maximizing the entropy-based criterion. Afterward, a sequential paradigm is utilized to incrementally find more base clusterings, in which the diversity between a new base clustering and the former base partitions is measured by the normalized mutual information. Extensive experiments conducted on various data sets have demonstrated the superiority of our proposal as compared to several existing base clusterings generation algorithms.

In this short paper, we introduce and analyze a generalized preconditioned parameterized inexact Uzawa method for solving singular saddle point problems, this method is a generalization of the PPIU method (see, Gao and Kong, 2010), then we give the semi-convergent conditions of this method, which are more practical for the applications.

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u,?v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in Y. Zhang, Y. Hu, 2016 38. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus-Gaddum-type results for Wp(G). Our lower bound on W p ( G ) + W p ( G ? ) is always better than the previous lower bound given by Zhang and Hu.

Recently, some results on the trigonometric approximation of Lip(ω(t), p)-class have been given by Srivastava and Singh (2014). In this note, we point out that one of the assumption conditions in their theorems is invalid in some cases. Thus, all conclusions in the paper based on the condition lose the foundation. Moreover, we obtain the similar results for a class of more extensive functions after deleting the mentioned condition, and weakening another condition in their theorems.

Cognitive processing by intelligent systems involves the deletion of information in favor of higher level abstractions. This process can be addressed through the physics of computation but a formal model that explains this process has not been proposed yet. In this short paper, we propose a model that through physical constraints only generates optimal solution to the collapse of n objects into n sets. A numerical simulation of the model results in a logarithmic function of information loss and condensation that perfectly fits our knowledge of cognitive processes.