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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.

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.

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.

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.

We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature.

In the present paper we studied the Voronovskaya type theorem for modified Gamma operators presented by A. Izgi in the paper ''Voronovskaya type asymptotic approximation by modified Gamma operators'' [Applied Mathematics and Computation 217 (2011) 8061-8067]. We show that this theorem is not true. We present a correct form of this theorem and its proof. Furthermore, we obtain an error estimate for modified Gamma operators.

The authors appreciate Professor Cheng-De Zheng pointing errors in definition of LMI variable [1–8] and would like to correct an error in definition of an LMI variable of the paper [1–8]. In [1–8], Theorems or Corollaries present stability or stabilization criteria for the system with uncertainties or nonlinear perturbations. In [1–8], the LMI variables H in [1–3,5– 8], Hi ði 1⁄4 1;2Þ in [4], and P2 in Theorem 1 of [6] are defined as a positive matrix. However, the aforementioned LMI variables in [1–6] should be positive scalars. Then, all the Theorems and Proofs in [1–8] are still valid without any further changes. Note that the numerical results for systems without uncertainties ([1–3,5,6]) or nonlinear perturbations ([4]) do not change even though the H in [1–3,5–8], Hiði 1⁄4 1;2Þ in [4], and P2 in Theorem 1 of [6] are corrected to a positive scalar. In case that the system has uncertainties or nonlinear perturbations, the numerical results become to change a little. In [1], Example 2 considers the case with uncertainties. The corrected versions for Table 2 and 3 in [1] are given in Tables 1 and 2, respectively. Example 3 in [2] considers the system with uncertainties. The corrected versions for Tables 2–4 in [2] are shown in Tables 3–5, respectively. In [3], Example 2 considers the case with uncertainties. The corrected versions for Tables 2 and 3 in [3] are given in Tables 6 and 7, respectively. In [4], Example 1 and 2 consider the case with nonlinear perturbations. The corrected maximum allowable bound under the same condition in Example 1 is corrected to 2.8687 from 3.4039. The corrected result still gives larger delay bound than one of previous work given in Ref. [6] of paper [4]. And the corrected versions for Table 1 in [4] are given in Table 8. In [5], Example 1, 3, 4, 5, and 6 consider the case with uncertainties. The corrected versions for Tables 1–4, 6 and 7 in [5] are given in Tables 9–14, respectively. In [6], Example 1 and 3 consider the case with uncertainties. The corrected versions for Tables 1, 2 and 4 in [6] are given in Tables 15–17, respectively. In [7], the corrected results are given in Table 18. In [8], the corrected maximum allowable bound under a = 0, hL = 0 and unknown hD changes from 0.7245 to 0.6918, while is the result still larger than the ones of the previous works mentioned in [8]. Also, the corrected results for Tables 1 and 2 [8] are given in Tables 19 and 20, respectively.

Using advanced machine learning techniques as an alternative to conventional double-entry volume equations known as classical allometric models , regression models of the inside-bark volume (dependent variable) for standing Eucalyptus globulus trunks (or main stems) have been built as a function of the following three independent variables: age, height and outside-bark diameter at breast height ( D ). The allometric models of volume, biomass or carbon support the estimation of carbon storage in forests and agroforestry systems. On one hand, this paper presents the construction of allometric models of the inside-bark volume for E. globulus trees. On the other hand, the experimental observed data (age, height, D and inside-bark volume) for 142 trees ( E. globulus ) were measured and a nonlinear model was built using a data-mining methodology based on multivariate adaptive regression splines (MARS) technique and multilayer perceptron networks (MLP) for regression problems. Coefficients of determination and Furnival’s indices indicate the superiority of the MARS technique over the allometric regression models and the MLP network. The agreement of the MARS model with observed data confirmed the good performance of the same one. Finally, conclusions of this innovative research are exposed.

In this paper, we focus on the proposed algorithms to solve a linear programming problem with the convex combination of the max–min and the max–average composition and the max–star composition, respectively. They have been proposed by Ghodousian and Khorram [A. Ghodousian, E. Khorram, Solving a linear programming problem with the convex combination of the max–min and the max–average fuzzy relation equations, Appl. Math. Comput. 180 (2006) 411–418] and Khorram et al. [E. Khorram, A. Ghodousian, A. Abbasi Molai, Solving linear optimization problems with max–star composition equation constraints, Appl. Math. Comput. 179 (2006) 654–661], respectively. Firstly, we show that the “Tabular method algorithm” in the first paper and the “First procedure” in the second paper may not lead to the optimal solutions of the two models in some cases. Secondly, we generalize the proposed algorithm by Abbasi Molai and Khorram [A. Abbasi Molai, E. Khorram, A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian, Appl. Math. Comput. 190 (2007) 1161–1167] to solve the two models. In fact, it modifies the presented algorithms in the two papers. Finally, some numerical examples are given to illustrate the purposes.

Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form (1) d 4 y d x 4 + f ( x ) y = g ( x ) , a ⩽ x ⩽ b , (2) y ( a ) = A 1 , y ( b ) = A 2 , y ′ ( a ) = B 1 , y ′ ( b ) = B 2 , where f ( x ) and g ( x ) are continuous on [ a , b ], and A i and B i ( i = 1, 2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments.

In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations(r(t)@j(x(t))|Z^'(t)|^@a^-^1Z^'(t))^'+q(t)f(x(@s(t)))=0,t>=t"0>0, where Z(t)=x(t)+p(t)x(t-@t) and @a>0,0==t"0>0, where @!"t"""0^~dtr(t)

One of the desirable attributes of real-life examination timetabling solutions is the maximization of paper spread, which is a measure of the amount of study time that each student has between examinations. In this study, we face with a predefined examination schedule that must be modified in order to maximize paper spread in course of examination. We will show how the integer programming can be employed to achieve this aim. The model presents constraints for the quality of feasible examination timetables and all the requirements found in most academic institutions. The approach is tested on real-world exam timetabling problems. The computational experiments and results will be reported.

In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u [ τ ( t ) ] | α - 1 u [ τ ( t ) ] = 0 . The results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper.

In this short note an addition to the paper [A.G. Bratsos, M. Ehrhardt, I.Th. Famelis, A discrete Adomian decomposition method for discrete nonlinear Schrodinger equations, Appl. Math. Comput. 197(1) (2008) 190-205] using the modulus of the terms evaluated from the Adomian decomposition method on p. 194 and their relation to the convergence of the resulting series is presented. Conclusions for the accuracy of the approximated solution are derived.

This paper considers certain neutral differential equation with the distributed delayddt[x(t)+c(t)x([email protected])]+p(t)x(t)+q(t)x([email protected])+r(t)@!"t"-"@d^tx(s)ds=0.Without imposing a nonnegative restriction on coefficients of the equation, we establish a delay-dependent asymptotic stability condition for the equation by introducing a new Lyapunov function. The condition presented here is different from that of Park [J.H. Park, LMI optimization approach to asymptotic stability of certain neutral delay differential equation with time-varying coefficients, Appl. Math. Comput. 160 (2005) 355-361] in the sense that it is dependent on all delays of the equation. Numerical examples illustrate effectiveness and sharpness of our theoretical result.

The aim of this note is to show that the optimization algorithm proposed in [6] may not lead to the optimal solution in some cases. In fact, the optimization problem remains open if we do not apply the branch-and-bound method to solve it or compute all the objective values of minimal solutions of the feasible domain.

A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659–684] are not always true. Some mistakes in the proofs of these theorems are pointed out.

In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function.