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Abstract: It is shown that the main theorem of Arslan’s paper (Theorem 2, 2011), as stated, is incorrect. Under additional conditions, we present a short proof of the corrected version of the theorem. We also give a proof of a theorem of Rao and Shanbhag (1991) , employed by Arslan, without the use of the Kolmogorov Consistency Theorem. [Copyright &y& Elsevier]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

In this note, we correct an inaccuracy of Eq. (13) in the proof of Theorem 1 in the paper (Higham and Kloeden, 2007). We also give a complete proof of the theorem by applying the Burkholder–Davis–Gundy Inequality. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new shock model called Marshall–Olkin run shock model is defined and studied. According to the model, two components are subject to shocks that may arrive from three different sources, and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3, i = 1 , 2. Reliability and mean residual life functions of such components are studied when the times between shocks follow phase-type distribution. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the n -width problem for the Sobolev space of periodic functions, H per r (0 , 2 π). Building on a theorem of Pinkus we show that it admits optimal even-dimensional spline spaces of all degrees ≥ r − 1. Then, by using a theorem of Karlovitz, we show that it does not admit any optimal spline space of odd dimension > 1. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject. As a corollary we get the main results from the paper of Yang and Liu [H. Yang, X. Liu, The Drazin inverse of the sum of two matrices and its applications, J. Comput. Appl. Math. 235 (2011) 1412–1417]. As an application we give some new representations for the Drazin inverse of a block matrix. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the bounds of the Perron root of a nonnegative irreducible matrix . Two new methods utilizing the relationship between the Perron root of a nonnegative irreducible matrix and its generalized Perron complements are presented. The former method is efficient because it gives the bounds for only by calculating the row sums of the generalized Perron complement or even the row sums of submatrices and . And the latter gives the closest bounds (just in this paper) of . The results obtained by these methods largely improve the classical bounds. Numerical examples are given to illustrate the procedure and compare it with others, which shows that these methods are effective. [Copyright &y& Elsevier]

Abstract: With a brand-new theory, this paper not only provides the differences of attributes in concept, formula expression and function type between fuzzy rough sets and probability statistics, but also introduces their differences in algorithms on target control for better solving the control problem. Some new definitions and theorems concerning fuzzy rough sets and probability statistics are given, but this paper mainly makes a comparison of two control algorithms for the target tracking. The simulation results show that the comprehensive performance of the fuzzy rough sets algorithm is better than that of the probability statistics algorithm, but its control effect is not as good as that of the latter on multisensor target control. Finally, some problems concerning the combination of fuzzy rough sets and the probability statistics phenomenon to be solved and development trends are discussed. By these investigations, we can choose the optimal control algorithms for accomplishing better target control. [Copyright &y& Elsevier]

Abstract: This paper discusses the spectral properties of the nonsymmetric saddle point matrices of the form with A symmetric positive definite, B full rank, and C symmetric positive semidefinite. A new sufficient condition is obtained so that is diagonalizable with all its eigenvalues real and positive. This condition is weaker than that stated in the recent paper [J. Liesen, A note on the eigenvalues of saddle point matrices, Technical Report 10-2006, Institute of Mathematics, TU Berlin, 2006]. [Copyright &y& Elsevier]

Abstract: In this paper we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework. We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity. [Copyright &y& Elsevier]

Abstract: In this paper, high-resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two-dimensional hyperbolic conservation laws. The method is conservative. Moreover, it is stable which is proven numerically in this paper. The computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed. Using this proposed scheme, we compute several two-dimensional model problems and a compressive rate ranging from about 5–10 is observed in all simulations. [Copyright &y& Elsevier]

Abstract: In this paper, we study a non-differentiable minimax fractional programming problem under the assumption of generalized -univex function. In this paper we extend the concept of -invexity [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequalities Pure Appl. Math. 5 (2004) 1–9] and pseudo -invexity [S.K. Mishra, M.A. Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005) 69–75] to -univexity and pseudo -univexity from a view point of generalized convexity. We also introduce the concept of strict pseudo -univex and quasi -univex functions. We derive Karush–Kuhn–Tucker-type sufficient optimality conditions and establish weak, strong and converse duality theorems for the problem and its three different form of dual problems. The results in this paper extend a few known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we improve the preconditioned AOR method of linear systems considered by Evans et al. [The AOR iterative method for new preconditioned linear systems, Comput. Appl. Math. 132 (2001) 461–466]. In Evans’ paper, the coefficient matrix of linear system have to be an L-matrix with , and . When for some , the preconditioned method is invalid. In order to solve the above problem, a new preconditioner is presented. Meanwhile, some recent results are improved. [Copyright &y& Elsevier]

Abstract: In this paper, we study a nondifferentiable minimax fractional programming problem under the assumptions of -invex function. In this paper we utilize the concept of -invexity [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequalities Pure Appl. Math. 5 (2004) 1–9] and pseudo--invexity [S.K. Mishra, M.A. Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005) 69–75]. We also introduce the concept of strict pseudo--invex and quasi--invex functions. We derive Karush–Kuhn–Tucker-type sufficient optimality conditions and establish weak, strong and converse duality theorems for the problem and its three different dual problems. The results in this paper extend several known results in the literature. [Copyright &y& Elsevier]

Abstract: The paper is dealing with the problem of finding the densest packings of equal circles in the unit square. Recently, a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem. With this method it became possible to solve the previously open problems of packing 28, 29, and 30 circles in the numerical sense: tight guaranteed enclosures were given for all the optimal solutions and for the optimum value. The present paper completes the optimality proofs for these cases by determining all the optimal solutions in the geometric sense. Namely, it is proved that the currently best-known packing structures result in optimal packings, and moreover, apart from symmetric configurations and the movement of well-identified free circles, these are the only optimal packings. The required statements are verified with mathematical rigor using interval arithmetic tools. [Copyright &y& Elsevier]

Abstract: This paper shows that Hedstrom''s nonreflecting boundary condition works for some nonsimple-wave solutions in the case where the solution is . Hedstrom''s boundary condition is a pioneering work in the discipline of characteristic nonreflecting boundary conditions and Hedstrom gave a theoretical assurance of the nonreflectivity for simple-wave solutions. In this paper, we extend Hedstrom''s theorem for solutions, that is, we prove that the Hedstrom boundary condition eliminates the reflection if the solution is and there do not exist ingoing waves generated by the nonlinear interactions near the boundary. [Copyright &y& Elsevier]

Abstract: Each member of the family of Gauss hypergeometric functionswhere and z do not depend on n, and (not all equal to zero) satisfies a second order linear difference equation of the formBecause of symmetry relations and functional relations for the Gauss functions, the set of 26 cases (for different values) can be reduced to a set of 5 basic forms of difference equations. In this paper the coefficients , and of these basic forms are given. In addition, domains in the complex z-plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified. The determination of such a pair asks for a detailed study of the asymptotic properties of the Gauss functions for large values of n, and of other Gauss functions outside this group. This will be done in a later paper. [Copyright &y& Elsevier]

Abstract: In this paper the pricing of European-style discrete arithmetic Asian options with fixed and floating strike is studied by deriving analytical lower and upper bounds. In our approach we use a general technique for deriving upper (and lower) bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas et al. (Ins. Math. Econom. 27 (2000) 151–168), and additionally, the ideas of Rogers and Shi (J. Appl. Probab. 32 (1995) 1077–1088) and of Nielsen and Sandmann (J. Financial Quant. Anal. 38(2) (2003) 449–473). We are able to create a unifying framework for European-style discrete arithmetic Asian options through these bounds, that generalizes several approaches in the literature as well as improves the existing results. We obtain analytical and easily computable bounds. The aim of the paper is to formulate an advice of the appropriate choice of the bounds given the parameters, investigate the effect of different conditioning variables and compare their efficiency numerically. Several sets of numerical results are included. We also discuss hedging using these bounds. Moreover, our methods are applicable to a wide range of (pricing) problems involving a sum of dependent random variables. [Copyright &y& Elsevier]

A great variety of causes are implied in the resolution of GPS equations systems for obtaining precise point positioning. Among them, the time interval between observations and the number of observations are relevant to know the influence in positioning unknowns (X,Y,Z). In this paper, we perform a statistical study showing the possible influence of both factors in the obtained solution. From the results of this paper we conclude that the X and Y solution are independent of the above factors; however, Z shows a significant effect in the manner that time interval and the number of observations are chosen. The results provide insight into the most appropriate design for solving the equation systems of GPS observations. [Copyright &y& Elsevier]

In Gordon (1998), Gordon presented alternate proofs of several well known results in elementary real analysis using tagged partitions. In this paper, we provide a set of alternative proofs based on the dyadic partitions. An important difference between tagged and dyadic partitions is that the results based on the dyadic partition can be obtained constructively, i.e. an algorithm is supplied to reach the conclusion of each proof. In addition, the construction involves only concepts and results presented in a first course real analysis, which is the reason why, comparing tagged partition and its theory, the proofs based on dyadic partition are more straightforward and accessible to a wide range of audience. [ABSTRACT FROM AUTHOR]

Padé approximants are rational functions, with a denominator which does not vanish at zero, and whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different barycentric forms, and under a partial fraction form, depending on free parameters. According to the choice of these parameters, Padé-type approximants can be obtained under a barycentric or a partial fraction form. [ABSTRACT FROM AUTHOR]

Abstract: A partition of unity method for the displacement obstacle problem of clamped Kirchhoff plates is considered in this paper. We derive optimal error estimates and present numerical results that illustrate the performance of the method. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to show the usefulness of system signature for computing some important reliability indices of repairable systems. In particular, we obtain signature-based expressions for stationary availability, rate of occurrence of failure, and mean time to the first failure of repairable systems. Using these expressions we compute corresponding reliability indices of all systems with three and four components. Computational results are also presented for consecutive- -within- -out-of- :F and -consecutive- -out-of- :F systems. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a numerical method for the solution of two-dimensional Fredholm integral equations. The method is based on interpolation by Gaussian radial basis function based on Legendre–Gauss–Lobatto nodes and weights. Numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the method. [Copyright &y& Elsevier]

Abstract: This paper is devoted to presenting an explicit expression for the Harish-Chandra c-function for the Lie ball. Our computation technique is based on the expression for the generalized spherical function and its asymptotic behavior. [Copyright &y& Elsevier]

Abstract: In the present paper we construct -Szász–Mirakjan operators that preserve . The rate of global convergence is obtained in the frame of weighted spaces. Furthermore, we obtain a Voronovskaja type theorem for these operators. [Copyright &y& Elsevier]

Abstract: In this paper, the first integral method is used to construct exact solutions of the Hamiltonian amplitude equation and coupled Higgs field equation. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. [Copyright &y& Elsevier]

Abstract: The edge Szeged and edge Wiener indices of graphs are new topological indices presented very recently. It is not difficult to apply a modification of the well-known cut method to compute the edge Szeged and edge Wiener indices of hexagonal systems. The aim of this paper is to propose a method for computing these indices for general graphs under some additional assumptions. [Copyright &y& Elsevier]

Abstract: It is the purpose of this paper to derive two-step hybrid methods for , with oscillatory or periodic solutions, specially tuned to the behaviour of the solution, through the usage of the exponential fitting technique. The construction of two-step exponentially fitted hybrid methods is shown and their properties are discussed. Some numerical experiments confirming the theoretical expectations are provided. [Copyright &y& Elsevier]

Abstract: The use of explicit methods in the numerical treatment of differential equations of fractional order is an area not yet widely investigated. In this paper stability properties of some multistep methods of explicit type are investigated and new methods with larger intervals of stability are proposed. Some numerical experiments are presented in order to validate theoretical results. [Copyright &y& Elsevier]

Abstract: In the present paper we study the computation of the rank of a block bidiagonal Toeplitz (BBT) sequence of matrices. We propose matrix-based, numerical and symbolical, updating and direct methods, computing the rank of BBT matrices and comparing them with classical procedures. The methods deploy the special form of the BBT sequence, significantly reducing the required flops and leading to fast and efficient algorithms. The numerical implementation of the algorithms computes the numerical rank in contrast with the symbolical implementation, which guarantees the computation of the exact rank of the matrix. The combination of numerical and symbolical operations suggests a new approach in software mathematical computations denoted as hybrid computations. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the oscillation of Third-order difference equation with impulses. Some sufficient conditions for the oscillatory behavior of the solutions of Third-order impulsive difference equations are obtained. [Copyright &y& Elsevier]

Abstract: In this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension. In particular, throughout this paper we will focus on the derivation of 6-point interpolating schemes that turn out to be unique in combining vital ingredients like -continuity, simplicity of definition, ease of implementation, user independency, tension control and ability to reproduce salient trigonometric and transcendental curves. [Copyright &y& Elsevier]

Abstract: This paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space. Moreover, the explicit expression for generalized inverse , which is a {2,3}-inverse of having the prescribed rang and null space , is derived. As an application, two numerical examples are given. [Copyright &y& Elsevier]

Abstract: In this paper we present recentered confidence sets for the parameters of a logistic regression model based on preliminary minimum -divergence estimators. Asymptotic coverage probabilities are given as well as a simulation study in order to analyze the coverage probabilities for small and moderate sample sizes. [Copyright &y& Elsevier]

Abstract: This paper, introduces the nearest weighted interval and point approximations of a fuzzy number, and then suggests weighted possibilistic moments about these points of fuzzy numbers. The possibilistic moments play an important role in fuzzy sets and systems, specifically in physics, mathematics and statistics. We provide the definition of the moments of fuzzy numbers as well as the definition of moments in probability theory; some of their applications are mentioned. [Copyright &y& Elsevier]

Abstract: In this paper, we propose some least-squares finite element procedures for linear and nonlinear parabolic equations based on first-order systems. By selecting the least-squares functional properly each proposed procedure can be split into two independent symmetric positive definite sub-procedures, one of which is for the primary unknown variable and the other is for the expanded flux unknown variable . Optimal order error estimates are developed. Finally we give some numerical examples which are in good agreement with the theoretical analysis. [Copyright &y& Elsevier]

Abstract: This paper presents an impulsive optimal control model for solving the optimal designing problem of the trajectory of horizontal wells. We take fully into account the effect of unknown disturbances in drilling. The optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke–Jeeves algorithm. The uniform design technique is incorporated into the revised Hooke–Jeeves algorithm to handle the multimodal objective function. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the fourth-order four-point boundary value problem where and . The upper and lower solution method and a new maximum principle are employed to establish existence results and we release the increasing condition imposed on . [Copyright &y& Elsevier]

Abstract: For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results. [Copyright &y& Elsevier]

Abstract: In this paper, a subspace limited memory BFGS algorithm for solving large-scale bound constrained optimization problems is developed. It is modifications of the subspace limited memory quasi-Newton method proposed by Ni and Yuan [Q. Ni, Y.X. Yuan, A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization, Math. Comput. 66 (1997) 1509–1520]. An important property of our proposed method is that more limited memory BFGS update is used. Under appropriate conditions, the global convergence of the method is established. The implementations of the method on CUTE test problems are presented, which indicate the modifications are beneficial to the performance of the algorithm. [Copyright &y& Elsevier]

Abstract: In this paper we consider constructing some higher-order modifications of Newton’s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units. This construction can be applied to any iteration formula, and per iteration the resulting methods add only one additional function evaluation to increase the order. Some illustrative examples are provided and several numerical results are given to show the performance of the presented methods. [Copyright &y& Elsevier]

Abstract: A new class of functions, termed semistrictly -preinvex functions, is introduced in this paper. The relationships between semistrictly -preinvex functions and -preinvex functions are investigated under mild assumptions. Our results improve and extend the existing ones in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we study the superconvergence of the frictionless Signorini problem. When approximated by bilinear finite elements, by virtue of the information on the contact zone, we can derive a superconvergence rate of under a proper regularity assumption. Finally, a numerical test is given to verify our result. [Copyright &y& Elsevier]

Abstract: This paper proposes a new breakdown-free preconditioning technique, called SAINV-NS, of the AINV method of Benzi and Tuma for nonsymmetric positive definite matrices. The resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for QMR, BiCGSTAB and GMRES(m) methods. The preconditoner is reliable (pivot breakdown can not occur) and effective at reducing the number of iterations. Some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the AINV-A algorithm. [Copyright &y& Elsevier]

Abstract: Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterised by the existence of two optimisation problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimisation problem. In this paper we focus on the class of bilevel problems in which the upper level objective function is linear multiplicative, the lower level one is linear and the common constraint region is a bounded polyhedron. After replacing the lower level problem by its Karush–Kuhn–Tucker conditions, the existence of an extreme point which solves the problem is proved by using a penalty function approach. Besides, an algorithm based on the successive introduction of valid cutting planes is developed obtaining a global optimal solution. Finally, we generalise the problem by including upper level constraints which involve both level variables. [Copyright &y& Elsevier]

Abstract: Modern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance , attempt to advance the integration selecting the size of each step so that some measure of the local error is . Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff, J. Schröder (Eds.), Numerical Treatment of Differential Equations, Proceedings of Oberwolfach, 1976, Lecture Notes in Mathematics, vol. 631, Springer, Berlin, 1978, pp. 188–200; Tolerance proportionality in ODE codes, in: R. März (Ed.), Proceedings of the Second Conference on Numerical Treatment of Ordinary Differential Equations, Humbold University, Berlin, 1980, pp. 109–123] and the extensions of Higham [Global error versus tolerance for explicit Runge–Kutta methods, IMA J. Numer. Anal. 11 (1991) 457–480; The tolerance proportionality of adaptive ODE solvers, J. Comput. Appl. Math. 45 (1993) 227–236; The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge–Kutta codes the global errors behave asymptotically as some rational power of . This step-size policy, for a given IVP, determines at each grid point a new step-size so that is a continuous function of . In this paper a study of the tolerance proportionality property under a discontinuous step-size policy that does not allow to change the size of the step if the step-size ratio between two consecutive steps is close to unity is carried out. This theory is applied to obtain global error estimations in a few problems that have been solved with the code Gauss2 [S. Gonzalez-Pinto, R. Rojas-Bello, Gauss2, a Fortran 90 code for second order initial value problems, http://www.netlib.org/ode/ ], based on an adaptive two stage Runge–Kutta–Gauss method with this discontinuous step-size policy. [Copyright &y& Elsevier]

Abstract: Static condensation of internal degrees of freedom, partial orthogonalization of basis functions, and ILU preconditioning are techniques used to facilitate the solution of discrete problems obtained in the hp-FEM. This paper shows that for symmetric linear (not necessarily positive-definite) problems, under mild technical assumptions, these three techniques are completely equivalent. In fact, the same matrices can be obtained by the same arithmetic operations. The study can be extended to nonsymmetric problems naturally. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a higher-order Mond–Weir dual for a set-valued optimization problem by virtue of higher-order contingent derivatives and discuss their weak duality, strong duality and converse duality properties. [Copyright &y& Elsevier]

Abstract: In this paper, we study the single-population evolutionary game and construct an algorithm to find evolutionarily stable strategies. Finally, by an example, we illuminate the computing process of algorithm. [Copyright &y& Elsevier]