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This paper is concerned with some stochastic properties of a novel rock–paper–scissors model. Firstly, the global existence of an unique positive solution of the stochastic model is obtained. Then we demonstrate the positive solution of the model is stochastically bounded. Besides, some sufficient conditions for population to be stochastically permanent and extinct are derived with the use of some appropriate Lyapunov functions. At last, some numerical simulations are carried out to illustrate our theoretical analysis results.

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 aim of this paper is the development of a mathematical model to compute the optimal inventory mix to face stochastic demand at minimum cost in a two-level supply chain. The paper addresses a multi-product dynamic lot-sizing problem under stochastic demand subject to capacity and service level constraints. This model is executed to compare a Make To Order (MTO) strategy to a Vendor Managed Inventory (VMI) partnership between the supplier and their customers. Both strategies provide the demand order to be produced. A schedule of production orders is determined over the planning horizon in order to minimize the inventory holding costs of the supply chain, taking into consideration that the supplier is also responsible of initiating the replenishment orders and deliveries of their customers according to the VMI partnership. The simulation model is illustrated empirically using a real case study: a paper manufacturing company that pursues to improve customer service level and supply chain inventory costs through a proper production planning of their paper machines and a suitable VMI order replenishment schedule.

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.

We highlight some problems in this paper. They lead to question the model itself.

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.

We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s method.

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.

In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form., Comment: 17 pages, no figures

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.

This write-up presents a closure to the misleading comments of Pantokratoras (Int J Appl Comput Math, 2015. doi: 10.1007/s40819-017-0370-O ) on the paper (Manjunatha et al. in Int J Appl Comput Math, 2015. doi: 10.1007/s40819-015-0107-x ) “Effect of Radiation on Flow and Heat Transfer of MHD Dusty Fluid over a Stretching Cylinder Embedded in a Porous Medium in Presence of Heat Source, authored by P. T. Manjunatha, B. J. Gireesha, B. C. Prasannakumara, Int. J. Appl. Comput. Math, Vol. 3(1) 2017 Pp. 293–310”. The authors have addressed each of the comments raised by Pantokratoras (2015) in detail to uphold the correctness of the mathematical formulation together with the pertinent results presented in the published article (Manjunatha et al. 2015).

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.

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral ∫ab ∫-11 f(x, y)/x-y dxdy, where (a,b) ≡ (-1,1) or (a,b) ≡ (a,-1) or (a,b) ≡ (1, b) and f(x,y) is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

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.

This text, which is based on the author's talk in honour of G. Dahlquist at the SciCade05 meeting in Nagoya, describes the two classical papers from 1956 and 1963 of Dahlquist and their enormous impact on the research of decades to come; it also allows the author to present a personal testimony of his never ending admiration for the scientific and personal qualities of this great man.

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.

Each year, the Computational Optimization and Applications (COAP) editorial board selects a paper from the preceding year’s COAP publications for the Best Paper Award. The recipient of the award for papers published in 2007 is Olvi Mangasarian of the University of Wisconsin, Madison and the University of California, San Diego, for his paper “Absolute Value Programming”, published in Volume 36, pages 43–53. This paper [7] as well as subsequent closely related papers [6, 8, 9] deal with the absolute value equation (AVE) Ax + B|x| = b, where A and B are arbitrary m× n real matrices and |x| denotes the vector with absolute values of the n-dimensional real valued vector x. The significance of this class of NP-hard problems arises partly from the fact that when B = I , the identity matrix, AVE is equivalent to the general linear complementarity problem, 0 ≤ x ⊥Mx + q ≥ 0. Even though problems involving absolute values are NP-hard, they share some very interesting properties with those of linear systems. For example, optimization problems with absolute value constraints possess optimality and duality results similar to those of linear programming, even though the problems are inherently nonconvex. Another interesting property that AVE shares with linear inequalities are theorems of the alternative which are established in this paper. The paper also contains a finite successive linearization algorithm for solving absolute value equations that terminates at a necessary optimality condition. This algorithm has solved all random test problems given to it for which mostly m ≥ 2n or n ≥ 2m, up to size (m,n) of (2000,100) and (100,2000). When m = n and B is invertible, which is the case for the linear complementarity problem formulation, a simpler concave minimization

The paper by Rao and Kakehashi [1] uses the Weibull distribution (given by equation (3) in the paper) and a truncated version of it (given by equations (4) and (5)) to model HIV/AIDS data in India. After a careful reading, I have found that all of the results presented in the appendix are incorrect (starting with equation (11) itself). Instead of pointing out all of the specific errors, I have chosen to give the correct formulas (and a brief outline of their derivation) for the $r$th moment of the truncated Weibull distribution.

for their paper “Preconditioning Indefinite Systems in Interior Point Methods for Optimization”, published in Volume 28, pages 149–171. This paper describes a class of indefinite preconditioners for reduced KKT systems arising in quadratic and nonlinear optimization with interior point methods. Spectral analysis of preconditioners is given and the improvements resulting from the use of primal-dual regularization are demonstrated. Computational results are reported for the application of the preconditioner to a variety of medium-size convex quadratic programming problems. Work on the paper began during the Summer of 2001, about a year after the authors had first met when Zilli invited Gondzio to give a series of lectures on interior point methods in Padova in September, 2000. The authors had complementary research backgrounds: Bergamaschi and Zilli worked on the theory and implementation of iterative methods for linear and nonlinear systems of equations [2], while Gondzio worked on the design and implementation of interior point methods for large-scale optimization [3]. This was essential since together, the three authors could tackle in depth a problem which required expertice in several areas. At that time Bergamaschi and Zilli were impressed with the recent development of Lukysan and Vlycek [6], who applied conjugate gradients to indefinite systems, and with the later analysis of Keller, Gould and Wathen [4] of indefinite constraint preconditioners. They convinced Gondzio to look into the issue. In the meantime, Rozlozn´ok and Simoncini [7] came up with an improved understanding of issues connected with indefinite preconditioning (in the context of saddle point problems arising in partial differential equation); these developments led the authors to take a closer look at the problem. Starting in July, 2001, the authors began to incorporate into Gondzio’s interior point code HOPDM (higher order primal dual methods) four iterative techniques: conjugate gradients, BiCGstab, GMRES and QMR. A description of these schemes can be found in Kelley’s book [5] and the references therein. The authors realized that the key difficulty in the solution of augmented systems of Newton equations in interior point methods for quadratic and nonlinear programming (otherwise known as reduced KKT systems) consisted in the loss of sparsity caused by the presence of the Hessian matrix Q in the system � (Q + � 1

In this note, we will modify several gaps in Gurudwan and Sharma [N. Gurudwan, B.K. Sharma, Convergence theorem for the common solution for a finite family of ϕ -strongly accretive operator equations, Appl. Math. Comput. 217 (2011) 6748–6754].

The recent paper mentioned in the title contains a confusing statement on computing conservation laws corresponding to symmetries of nonlinearly self-adjoint differential equations. The present brief article contains clarifying comments.

van Lenthe, Broer, and Rashid made comments on our 2009 paper [Song et al., J. Comput. Chem. 2009, 30, 399] by criticizing that we did not properly reference the work by Broer and Nieuwpoort in 1988 [Broer and Nieuwpoort, Theor. Chim. Acta. 1988, 73, 405], and we favorably compared our valence bond self-consistent field (VBSCF) algorithm with theirs. However, both criticisms are unjustified insignificant. The Broer–Nieuwpoort algorithm, properly cited in our paper, is for the evaluations of matrix elements between determinants of nonorthogonal orbitals. Stating that this algorithm “can be used for an orbital optimization” afterwards [van Lenthe et al., submitted] is not a plausible way to require more credits or even criticize others. While we stand by our statement that our algorithms scales at O(m4) and van Lenthe et al.'s approximate Newton Raphson algorithm scales at O(mN5) (here m and N are the numbers of basis functions and electrons), as we discussed in our original paper, it becomes obvious that any strict comparison among different algorithms is difficult, unproductive, and counteractive. © 2012 Wiley Periodicals, Inc.

Dear colleague, As an Editor-in-Chief, I thank you for your interest in Internet Mathematics, an international, peer reviewed journal focusing on all aspects of complex networks. Internet Mathemati...

A hypergeometric transformation formula is developed that simultaneously simplifies and generalizes arguments and identities in a previous paper of Rao et al. [An entry of Ramanujan on hypergeometric series in his notebooks, J. Comput. Appl. Math. 173(2) (2005) 239–246].

The two papers in this issue deal with differential equations, one with the numerical solution of partial differential equations, and the other one with analytic solutions for ordinary differential equations. In his paper "From Functional Analysis to Iterative Methods", Robert Kirby is concerned with linear systems arising from discretizations of partial differential equations (PDEs). Specifically, the PDEs are elliptic and describe boundary value problems; the discretizations are done via finite elements, and at issue is the convergence rate of iterative methods for solving the linear systems. The author's approach is to go back to the underlying variational problem in a Hilbert space, and to make ample use of the Riesz representation theorem. This point of view results in short and elegant proofs, as well as the construction of efficient preconditioners. The general theory is illustrated with two concrete model problems of PDEs for convection diffusion and planar elasticity. This paper will appeal to anybody who has an interest in the numerical solution of PDEs. In 1963 the mathematician/meteorologist Edward Lorenz formulated a system of three coupled nonlinear ordinary differential equations, whose long-term behavior is described by an attractor with fractal structure. You can see a beautiful rendition of the thus named Lorenz attractor on the cover of this issue. Although it is "easy" to plot solutions of the Lorenz system, it is much harder to determine them mathematically. This is what motivated the paper "Complex Singularities and the Lorenz Attractor" by Divakar Viswanath and Sonmez Sahutoglu. Their idea is to allow the time variable to be complex, rather than real; to focus on singular solutions; and to express these singular solutions in terms of so-called psi series. After all is said and done, the authors end up with a two-parameter family of complex solutions to the Lorenz system. This a highly readable and very enjoyable paper, with concrete steps for future research, and connections to thunderstorms and analytic function theory.

The two papers in this issue are concerned with analysis of differential equations: the first paper discusses the solution of partial differential equations that describe heat transfer, while the second one analyzes dynamical systems whose behavior alternates between continuous and discrete modes. The paper “Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems,” by Sarah Mitchell and Tim Myers, deals with a heat transfer problem, on a semi-infinite region. Consider an experiment on a long metal bar occupying the positive real-axis. The experiment starts by heating the origin, thereby raising the bar above the melting temperature. The bar melts near the origin, and as the heat diffuses, the solid-liquid interface propagates slowly but surely towards infinity. The temperature is modeled by heat equations across the two regions, and by a Stefan condition at their interface. The authors heat balance integral methods to solve for the temperature. These methods reduce a partial differential equation to an ordinary differential equation. The authors investigate different boundary conditions, and different approximating functions for the temperature. Readers interested in heat balance integral methods will find this to be a valuable survey with new results that preserve the simplicity of the method. The second paper is concerned with so-called hybrid dynamical systems. A bouncing ball, for instance, is a hybrid dynamical system. The movement of the ball above ground can be described by Newton's law. However, at the very moment the ball hits the ground and bounces, an instantaneous reversal of velocity occurs along with some dissipation of energy. After the bounce, the ball moves again according to Newton's law until the next bounce, and so on. Mathematically one can show that the time points at which the ball bounces represent a convergent sequence. The convergence of this sequence implies that infinitely many bounces occur in a finite amount of time. This is called “Zeno behavior”: infinitely many switches of mode in a finite amount of time. If Zeno behavior occurs in a control system, a numerical simulation of the system is extremely difficult, if not impossible. In the terminology of dynamical systems, the movement of the ball above ground is a “flow” and the bounce is a “jump.” Hybrid dynamical systems alternate between continuous (flow) and discrete (jump) modes. Rafal Goebel and Andrew Teel in their paper “Preasymptotic Stability and Homogeneous Approximations of Hybrid Dynamical Systems” model hybrid dynamical systems and approximate them by simpler systems obtained from linearization and tangent cones. The authors analyze preasymptotic stability, homogeneity, and convergence. A variety of well-chosen simple examples helps us to understand the general concepts and results.

The aim of this short note is to show on a numerical example that one of the two optimization algorithms proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] may not lead to the optimal solution in some cases. Besides it will be pointed out that the other algorithm (correct, but unfortunately ineffective for larger problems) proposed in [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886] can be extended to a wider class of non-convex optimization problems.