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In this paper, we obtain some gap functions for generalized mixed quasi variational inequality problems in terms of regularized gap function and D-gap function. Further, by using these gap functions we obtain global error bounds for the solution of generalized mixed quasi variational inequality problems in Hilbert spaces. The results obtained in this paper generalize and improve some corresponding known results in literatures.

We propose a matrix-pencil method to identify monomial-exponential sums.In the univariate case, it is based on the QR factorization of Hankel matrices.In the bivariate case, it assures the same accuracy of the univariate case. In this paper we propose a matrix-pencil method for the numerical identification of the parameters of monomial-exponential sums in one and two variables. While in the univariate case the proposed method is a variant of that developed by the authors in a preceding paper, the bivariate case is treated for the first time here. In the bivariate case, the method we propose, easily extendible to more variables, reduces the problem to a pair of univariate problems and subsequently to the solution of a linear system. As a result, the relative errors in the univariate and in the bivariate case are almost of the same order.

In the paper 1], Jaklic and ?agar studied curvature variation minimizing cubic Hermite interpolants. To match planar two-point G 1 Hermite data, they obtained the optimal cubic curve by minimizing an approximate form of the curvature variation energy. In this paper, we present a simple method for this problem by minimizing the jerk energy, which is also an approximate form of the curvature variation energy. The unique solution can be easily obtained since the jerk energy is represented as a quadratic polynomial of two unknowns and is strictly convex. Finally, we prove that our method is equivalent to their method.

This paper investigates the delay-dependent synchronization schemes for the non-diffusively coupled time-varying complex dynamical networks. The outer coupling configuration matrix in our network model may be non-diffusive, time-varying, uncertain, asymmetric and irreducible. Different time-varying coupling delays for different nodes are also put into consideration in this paper. Besides, the nodes may have different state dimensions. Furthermore, only the common bound of the outer coupling coefficients (CBOCC) is used to design the synchronization controllers. If the CBOCC is known, our delay-dependent synchronization scheme can guarantee the network achieving exponential synchronization. And when the CBOCC is uncertain, the adaptive synchronization scheme, where only one adaptive law is needed, is proposed to guarantee the network realizing asymptotic synchronization. Simulation examples are provided to verify the effectiveness and feasibility of our theoretical results.

Based on an elementary inequality and two Gronwall-Bellman type integral inequality systems, this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain fractional systems with Hadamard derivative established in a recent paper by the authors. The major generalizations come from linearization method and de-singular approach after overcoming some difficulties in the Hadamard type singular kernel.

In this paper, the initial-boundary-value problems for linear and non-linear multi-term fractional diffusion equations with the Riemann-Liouville time-fractional derivatives are considered. To guarantee the uniqueness of solutions, we employ the weak and the strong maximum principles for the equations under consideration that are formulated and proved in this paper for the first time. An essential element of our proof of the maximum principles is an estimation for the value of the Riemann-Liouville fractional derivative of a function at its maximum point that is established in this paper for a wider space of functions compared to those used in our previous publications. In the linear case, the solutions to the problems under consideration are constructed in form of the Fourier series with respect to the eigenfunctions of the corresponding eigenvalue problems.

An adaptive fuzzy controller is presented for a class of stochastic nonlinear systems.An quadratic function is constructed to give the condition of the stability of systems.The scheme guarantees the stability of closed-loop system in the mean square sense.The drawback of the quartic moment approach is overcome and the method is simplified. This paper addresses the problem of adaptive tracking control for a class of stochastic strict-feedback nonlinear time-varying delays systems using fuzzy logic systems (FLS). In this paper, quadratic functions are used as Lyapunov functions to analyze the stability of systems, other than the fourth moment approach proposed by H. Deng and M. Krstic, and the hyperbolic tangent functions are introduced to deal with the Hessian terms. This approach overcomes the drawback of the traditional quadratic moment approach and reduce the complexity of design procedure and controller. Based on the backstepping technique, the appropriate Lyapunov-Krasovskii functionals and the FLS, the adaptive fuzzy controller is well designed. The proposed adaptive fuzzy controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error can converge to a small residual set around the origin in the mean square sense.

In this paper, we consider the existence of infinitely many solutions to following nonlinear Kirchhoff-Schrodinger-Poisson system a + b ? R 3 | ? u | 2 + V ( x ) u 2 - Δ u + V ( x ) u + λ l ( x ) ? u = f ( x , u ) , x ? R 3 , - Δ ? = λ l ( x ) u 2 , x ? R 3 , where constants a 0 , b ? 0 and λ ? 0 . When f has sublinear growth in u, we obtain infinitely many solutions under certain assumption that V do not have a positive lower bound. The technique we use in this paper is the symmetric mountain pass theorem established by Kajikiya (2005).

This paper deals with Cauchy problem for a class of impulsive partial hyperbolic differential equations involving the Caputo derivative. Our first purpose is to show that the formula of solutions in cited papers are incorrect. Next, we reconsider a class of impulsive fractional partial hyperbolic differential equations and introduce a correct formula of solutions for Cauchy problem in R n . Further, some sufficient conditions for existence of the solutions are established by applying fixed point method. At last, we consider the Cauchy problem in a Banach space via the technique of measures of noncompactness and Monch's fixed point theorem. Some examples are given to illustrate our results.

This paper deals with a concept and description of a RC network as an electro-analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions. It is based on well-known analogy between heat and electrical conduction. In the paper are compared analytical solution together with numerical solution and experimentally measured data. For the first time a fractional order model of diffusion process and its modeling via lumped RC network has been used. Simple examples of simulations, measurements and their comparison are shown.

In this paper, we establish some sufficient conditions for the complete controllability of linear and semilinear impulsive matrix Lyapunov systems. For the semilinear systems, we assume that nonlinearities are Lipschitz type or monotone type. Few illustrative examples are given to compare and substantiate the results of the paper.

In this paper, a new split-step ? (SS?) method for stochastic age-dependent population equations with Poisson jumps is constructed. The main aim of this paper is to investigate the convergence of the SS? method for stochastic age-dependent population equations with Poisson jumps. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory, the results show that the SS? method has better accuracy compared to the Euler method.

Real life optimization problems involve a number of constraints arising out of user requirements, physical laws, statutory requirements, resource limitations etc. Such constraints are routinely evaluated using computationally expensive analysis i.e., solvers relying on finite element methods, computational fluid dynamics, computational electro magnetic, etc. Existing optimization approaches adopt a full evaluation policy, i.e., all the constraints corresponding to a solution are evaluated throughout the course of search. Furthermore, a common sequence of constraint evaluation is used for all the solutions. In this paper, we introduce a novel scheme for constraint handling, wherein every solution is assigned a random sequence of constraints and the evaluation process is aborted whenever a constraint is violated. The solutions are sorted based on two measures i.e., the number of satisfied constraints and the violation measure. The number of satisfied constraints takes precedence over the amount of violation and the most efficient sequence of constraint evaluation is evolved during the course of search. The performance of the proposed scheme is rigorously compared with other state of the art constraint handling methods using single objective inequality constrained test problems of CEC-2006 and CEC-2010. The constraint handling approach is generic and can be easily used for the solution of constrained multiobjective optimization problems or even problems with equality constraints. A bi-objective welded beam design problem, a tri-objective car side impact problem and an equality constrained optimization problem of CEC-2006 is solved as an illustration. The results clearly highlight potential savings offered by the proposed strategy and more importantly the paper provides a detailed insight on why such a strategy performs better than others.

This paper is devoted to analyzing the instantaneous availability of a typical software system with intrusion tolerance. By formulating the system with a couple of ordinary differential and partial differential equations, this paper describes the system as a time-delay partial differential equation. Based on the time-delay model, both steady-state availability and instantaneous availability are investigated. The optimal policy for preventive patch management to maximize the steady-state availability of the software system is obtained, and its related availability criterions are also presented. Employing the finite difference scheme and Trotter-Kato theorem, we converted the time-delay partial equation into a time-delay ordinary equation. As a result, the instantaneous availability of the system is derived. Some numerical results are given to show the effectiveness of the method presented in the paper.

Multiquadric (MQ) quasi-interpolation is a popular method for the numerical solution of differential equations. However, MQ quasi-interpolation is not well suited for the equations with periodic solutions. This is mainly due to the fact that its kernel (the MQ function) is not a periodic function. A reasonable way of overcoming the difficulty is to use a quasi-interpolant whose kernel itself is also periodic in these cases. The paper constructs such a quasi-interpolant. Error estimates of the quasi-interpolant are also provided. The quasi-interpolant possesses many fair properties of the MQ quasi-interpolant (i.e., simplicity, efficiency, stability, etc). Moreover, it is more suitable (than the MQ quasi-interpolant) for periodic problems since the quasi-interpolant as well as its derivatives are periodic. Examples of solving both linear and nonlinear partial differential equations (whose solutions are periodic) by the quasi-interpolant and the MQ quasi-interpolant are compared at the end of the paper. Numerical results show that the quasi-interpolant outperforms the MQ quasi-interpolant for periodic problems.

In this paper, we first recall some results concerning the construction and the properties of quadratic B-splines over a refinement Δ of a quadrangulation ◊ of a planar domain introduced recently by Lamnii et al. Then we introduce the B-spline representation of Hermite interpolant, in the special space S 2 1 , 0 ( Δ ) , of any polynomial or any piecewise polynomial over refined quadrangulation Δ of ◊ . After that, we use this B-representation for constructing several superconvergent discrete quasi-interpolants. The new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper.

In this paper, the problem of finite-time stability analysis for a class of non-autonomous neural networks with heterogeneous proportional delays is considered. By introducing a novel constructive approach, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. As a result, we also obtain conditions for the power-rate global stability of the system. Numerical examples are given to demonstrate the effectiveness and less restrictiveness of the results obtained in this paper.

This paper presents a proposed approach for modeling the life data for system components that have failure modes by different Weibull models. This approach is applied for censored, grouped and ungrouped samples. To support the main idea, numerical applications with exact failure times and censored data are implemented. The parameters are obtained by different computational statistical methods such as graphic method based on Weibull probability plot (WPP), maximum likelihood estimates (MLE), Bayes estimators, non-linear Benard’s median rank regression. This paper also presents a parametric estimation method depends on expectation–maximization (EM) algorithm for estimation the parameters of finite Weibull mixture distributions. GOF is used to determine the best distribution for modeling life data. The performance of the proposed approach to model lifetime data is assessed. It’s an efficient approach for moderate and large samples especially with a heavily censored data and few exact failure times.

In this paper we investigate the existence of homoclinic solutions for the following fourth order nonautonomous differential equations u(4)+wu″+a(x)u=f(x,u),(FDE) wherew is a constant, a∈C(R,R) and f∈C(R×R,R). The novelty of this paper is that, when (FDE) is nonperiodic, i.e., a and f are nonperiodic in x and assuming that a does not fulfil the coercive conditions and f satisfies some more general (AR) condition, we establish one new criterion to guarantee that (FDE) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.

In this paper, we introduce and study a new viscosity approximation method by modify the hybrid steepest descent method for finding a common solution of split variational inclusion problem and fixed point problem of a countable family of nonexpansive mappings. Under suitable conditions, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of the split variational inclusion problem and fixed point problem for a countable family of nonexpansive mappings which is the unique solution of the variational inequality problem. The results present in this paper are the supplement, extension and generalization of the previously known results in this area. Numerical results demonstrate the performance and convergence of our result that the algorithm converges to a solution to a concrete split variational inclusion problem and fixed point problem.

This paper deals with a degenerate nonlinear diffusion equation with gradient absorption. We at first determine finite time blow-up of solutions both in the L ∞ -norm and an integral measure, and then estimate a lower bound of the blow-up time by using the differential inequality technique. It is mentioned that the blowing up of solutions to nonlinear PDEs is usually defined in the L ∞ -norms, while the lower bounds of blow-up time are all determined via some measures in the form of energy integrals. So, in general, to estimate the lower bounds of blow-up time, it has to be assumed that the solutions do blow up in finite time with the involved integral measure before establishing their lower bounds of blow-up time. Such assumptions are unnecessary in this paper.

In this paper, we establish quantum analogue of classical integral identity. Using this identity, we derive some quantum estimates for Hermite-Hadamard inequalities for q-differentiable convex functions and q-differentiable quasi convex functions. Results obtained present refinement and improvement of the known results. The ideas and techniques of this paper may stimulate further research.

In this paper, the author presents a graphical tool that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool gives the information as previous tools such as Feigenbaum diagrams and Lyapunov exponents for every initial point. The convergence plane can be used, inter alia, to find the elements of a family that have good convergence properties, to see how the basins of attraction changes along the elements of the family, to study two-point methods such as Secant method or even to study two-parameter families of iterative methods. To show the applicability of the tool an example of the dynamics of the Damped Newton's method applied to a cubic polynomial is presented in this paper.

Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. In this paper, we propose applying the genetic algorithm to determine a good shape parameter of radial basis functions for the solution of partial differential equations. We use meshless collocation method based on the radial basis function (Kansa's method) to solve partial differential equations. Due to the severely ill-conditioned matrix arising from using RBF, we also consider the truncated singular value decomposition method (TSVD) for solving system of linear equations which is obtained from Kansa's method. Numerical results show that the proposed algorithm based on the genetic optimization is effective and provides a reasonable shape parameter along with acceptable accuracy of the solution.

The paper concerns the decompositions of polynomials onto iterated integrals. This is a continuation of our previous paper (Lorenc, submitted for publication), in which the existence of such decomposition for the Faulhaber polynomials is proven. In the current paper we prove the basic theorem (Theorem 2) presenting the necessary and sufficient conditions for the existence of such decomposition. We discuss these conditions in the wider context of theory of the real and complex polynomials. A number of exemplary decompositions onto iterated integrals of the known classical kinds of polynomials are also presented.

The purpose of this paper is to introduce the mixed Agarwal-O'Regan-Sahu type iterative scheme (Agarwal et al., 2007) for finding a common fixed point of the multi-valued nonexpansive mappings in the setting of hyperbolic spaces. Under suitable conditions, some Δ -convergence theorems of the iterative sequence generated by the proposed scheme to approximate a common fixed point of multi-valued nonexpansive mappings are proved. The results presented in the paper extend and improve some recent results announced in the current literature.

This paper is a contribution to the study of the automorphism groups of 2 - ( v , k , 1 ) designs. Our aim is to classify pairs ( D , G ) in which D is a 2 - ( v , k , 1 ) design and G is a block-transitive group of automorphisms of D . It is clear that if one wishes to study the structure of a finite group acting on a 2 - ( v , k , 1 ) design then describing the socle is an important first step. Let G act as a block-transitive and point-primitive automorphism group of a 2 - ( v , k , 1 ) design D . Set k 2 = ( k , v - 1 ) . In this paper we prove that when q = p a for some prime power and q is "large", specifically, q � 2 2 ( k 2 k - k 2 + 1 ) a , then the socle of G is not F 4 ( q ) .

This paper presents adaptive wavelet collocation methods for the numerical solutions to partial integro-differential equations (PIDEs) arising from option pricing in a market driven by jump-diffusion process. The first contribution of this paper lies in the formulation of the wavelet collocation schemes: the integral and differential operators are formulated in the collocation setting exactly and efficiently in both adaptive and non-adaptive wavelet settings. The wavelet compression technique is employed to replace the full matrix corresponding to the nonlocal integral term by a sparse matrix. An adaptive algorithm is developed, which automatically obtains the solution on a near-optimal grid. The second contribution of this paper is the theoretical analysis of the wavelet collocation schemes: due to the possible degeneracy of the parabolic operators, classical solutions of the jump-diffusion models may not exist. In this paper we first prove the convergence and stability of the proposed numerical schemes under the framework of viscosity solution theory, and then the numerical experiments demonstrate the accuracy and computational efficiency of the methods we developed.

This paper concerns with the problem of exponential stabilization for a class of non-autonomous neural networks with mixed discrete and distributed time-varying delays. Two cases of discrete time-varying delay, namely (i) slowly time-varying; and (ii) fast time-varying, are considered. By constructing an appropriate Lyapunov-Krasovskii functional in case (i) and utilizing the Razumikhin technique in case (ii), we establish some new delay-dependent conditions for designing a memoryless state feedback controller which stabilizes the system with an exponential convergence of the resulting closed-loop system. The proposed conditions are derived through solutions of some types of Riccati differential equations. Applications to control a class of autonomous neural networks with mixed time-varying delays are also discussed in this paper. Some numerical examples are provided to illustrate the effectiveness of the obtained results.

The paper is concerned with the properties of approximate sampled-data models and their zero dynamics, as the sampling period tends to zero, composed of a fractional order hold (FROH), a continuous-time multivariable plant and a sampler in cascade. The emphasis of this paper is the stability of discrete zero dynamics with the generalized gain β of the FROH, where we also present a condition to assure the stability of the sampling zero dynamics, which they have no counterpart in the underlying continuous-time system, of the resulting model. Similar to the linear case, the parameter β is the only factor in affecting the stability of discrete zero dynamics, and the appropriate β is determined to obtain the FROH that provides zero dynamics as stable as possible, or with improved stability properties even when unstable, for a given continuous-time multivariable plant. The study is also shown that the stability of the sampling zero dynamics is improved compared with a zero-order hold (ZOH).

In this paper, the generation of fractal curves and surfaces along with their properties, using ternary 4-point interpolatory subdivision scheme with one parameter, are analyzed. The relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples. The specific range of the tension parameter has also been depicted, which provides a clear way to generate fractal curves. Since the fractal scheme introduces, in the paper, have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve.

Log aesthetic curve (LAC) has been explored extensively by many researchers since 2005. At first, Gaussian-Kronrod has been proposed to evaluate LAC as the formulation of LAC involves double integration. Recently, Incomplete Gamma Function (IGF) has been proposed to represent LAC analytically which decreases the computation time up to 13 times. This paper embarks on the representation of LAC using adaptive Runge-Kutta methods to decrease the LAC computation time. The famous adaptive methods such as Runge-Kutta Fehlberg, Dormand-Prince, Sarafyan and Kutta-Merson are employed to evaluate LAC so that a desired accuracy can be achieved. This paper ends with a detailed investigation on performance metric of IGF and adaptive RK methods to compute LAC. These methods will be compared in terms of computation time and truncation error. Numerical results indicate that the computation time of LAC can be greatly improved and at the same time preserving the LAC's family.

There have been numerous proposals in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra. The present paper reviews the quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform and it studies a number of its properties. In the first part of this paper, we establish a generalized Riemann-Lebesgue lemma for the (right-sided) QLCT. This lemma prescribes the asymptotic behaviour of the QLCT extending and refining the classical Riemann-Lebesgue lemma for the Fourier transform of 2D quaternion signals. We then introduce the QLCT of a probability measure, and we study some of its basic properties such as linearity, reconstruction formula, continuity, boundedness, and positivity. Finally, we extend the classical Bochner-Minlos theorem to the QLCT setting showing the applicability of our approach.

How to define products and powers of distributions is a difficult and not completely understood problem, and has been investigated from several points of views since Schwartz established the theory of distributions around 1950. Many fields, such as differential equations or quantum mechanics, require such operations. In this paper, we use Caputo fractional derivatives and the following generalized Taylor's formula for 0 < α < 1 ? ( t ) = ? i = 0 m C D ? 0 , t i α ? ( 0 ) ? ( i α + 1 ) t i α + C D ? 0 , t ( m + 1 ) α ? ( ? ) ? ( ( m + 1 ) α + 1 ) t ( m + 1 ) α to give meaning to the distributions ? k ( x ) and ( ? ' ) k ( x ) for all k ? R . These can be regarded as powers of Dirac delta functions and have applications to quantum theory. At the end of this paper, the distributions log ? ( t ) and ? ( t 2 ) are given by the ?-sequence and the neutrix limit.

In this paper, for impulsive differential equations with fractional-order q ? ( 0 , 1 ) , we show that the formula of solutions in cited papers are incorrect. Secondly, we find out a formula of the general solution for impulsive Cauchy problem with Caputo fractional derivative q ? ( 0 , 1 ) . Further, for a kind of impulsive fractional differential equations system with special initial value, we come to an existence result for it by applying fixed point methods.

This paper deals with the controllability of fractional integro-differential inclusions.The results are obtained by using Bohnenblust-Karlin's fixed point theorem.Further, we extend the result to study the controllability concept with nonlocal conditions.Finally, an example is also given to illustrate our main results. In this paper, we consider a class of fractional integro-differential inclusions in Banach spaces. This paper deals with the controllability for fractional integro-differential control systems. First, we establishes a set of sufficient conditions for the controllability of fractional semilinear integro-differential inclusions in Banach spaces via resolvent operators. We use Bohnenblust-Karlin's fixed point theorem to prove our main results. Further, we extend the result to study the controllability concept with nonlocal conditions. An example is also given to illustrate our main results.

Formulas and identities involving many well-known special functions (such as the Gamma and Beta functions, the Gauss hypergeometric function, and so on) play important roles in themselves and in their diverse applications. Various families of generating functions have been established by a number of authors in many different ways. In this paper, we aim at establishing some (presumably new) generating functions for the generalized Gauss type hypergeometric type function F p ( α , β ; ? , µ ) ( a , b ; c ; z ) which is introduced here. We also present some special cases of the main results of this paper.

This paper represents the continuation of the analysis from papers Milosevic (2011) [10] and Milosevic (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function.

In this paper, we consider a new system of extended general quasi variational inequalities involving six nonlinear operators. Using projection operator technique, we show that the system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. Using this alternative equivalent formulation, we propose and analyze some parallel schemes for solving a system of extended general quasi variational inequalities. The convergence of these new schemes is discussed under some mild conditions. Several special cases are discussed. Results obtained in this paper continue to hold for these problems. The ideas and techniques of this paper may stimulate further research in this dynamic field.

In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in details. The advantage of the IMLS method is that the meshless method which is constructed based on the IMLS method can apply the essential boundary conditions directly and easily. A simpler expression of the approximation function of the IMLS method is obtained. Then the error estimate of the approximation function and its first and second order derivatives of the IMLS method are presented in one-dimensional case in this paper. The theoretical results show that if the order of the polynomial basis functions is big enough and the original function is sufficiently smooth, then the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, some selected numerical examples are given to prove the theories in this paper.

In the commented paper, the authors claim to have proved the existence of heteroclinic and homoclinic orbits of Silnikov type in two-Lorenz like systems, the so-called Lu and Zhou systems. According to them, they have analytically demonstrated that both systems exhibit Smale horseshoe chaos. Unfortunately, we show that the results they obtain are incorrect. In the proof, they use the undetermined coefficient method, introduced by Zhou et al. in [Chen’s attractor exists, Int. J. Bifurcation Chaos 14 (2004) 3167–3178], a paper that presents very serious shortcomings. However, it has been cited dozens of times and its erroneous method has been copied in lots of papers, including the commented paper where a misuse of a time-reversibility property leads the authors to use an odd (even) expression for the first component of the heteroclinic (homoclinic) connection. It is evident that this odd (even) expression cannot represent the first component of a Silnikov heteroclinic (homoclinic) connection, an orbit which is necessarily non-symmetric. Consequently, all their results, stated in Theorems 3–5, are invalid.

Numerical solutions of partial differential equations are traditional topics that have been studied by many researchers. During the last decade, support vector machine (SVM) has been widely used for approximation problems. The contribution of this paper is two folds. One is to combine the reproducing kernel-SVM method with the Tikhonov regularization method, called the SVM-Tik methods, in which the kernels K λ and K λ ? (see below) are newly developed. In the paper they are respectively phrased as the SVM-Tik- K λ and SVM-Tik- K λ ? methods. The second contribution is to use the two models, SVM-Tik- K λ and SVM-Tik- K λ ? , to solve the Dirichlet problem. The methods are meshless. They produce sparse representations in the linear combination form of specific functions (the K λ and K λ ? kernels). The generalization bound result in learning theory is used to give an estimation of the approximation errors. With the illustrative examples the sparseness and robustness properties, as well as the effectiveness of the methods are presented. The proposed methods are compared with currently the most commonly used finite difference method (FDM) showing promising results.

Determine desirable low voltage solution for multi-LVS at maximum loading point.Compute the exact optimal multiplier for optimal multiplier load flow method.Calculating the maximum loading point (MLP).Using the polar coordinate system based on the second order load flow method. This paper utilises concavity properties in the optimal multiplier load flow method (OMLFM) to find the most suitable low voltage solution (LVS) for the systems having multiple LVS at the maximum loading point. In the previous method, the calculation of the optimal multiplier is based on only one remaining low voltage solution at the vicinity of voltage collapse point. However, this does not provide the best convergence for multi-low voltage solutions at the maximum loading point. Therefore, in this paper, concavity properties of the cost function in OMLFM are presented as the indicator to find the most suitable optimal multiplier in order to determine the most suitable low voltage solutions at the maximum loading point. The proposed method uses polar coordinate system instead of the rectangular coordinate system, which simplifies the task further and by keeping PV type buses. The polar coordinate in this method is based on the second order load flow equation in order to reduce the calculation time. The proposed method has been validated by the results obtained from the tests on the IEEE 57, 118 and 300-bus systems for well-conditioned systems and at the maximum loading point.

A novel class of quasi-polynomials orthogonal with respect to the fractional integration operator has been developed in this paper. The related Gaussian quadrature formulas for numerical evaluation of fractional order integrals have also been proposed. By allowing the commensurate order of quasi-polynomials to vary independently of the integration order, a family of fractional quadrature formulas has been developed for each fractional integration order, including novel quadrature formulas for numerical approximation of classical, integer order integrals. A distinct feature of the proposed quadratures is high computational efficiency and flexibility, as will be demonstrated in the paper. As auxiliary results, the paper also presents methods for Lagrangian and Hermitean quasi-polynomial interpolation and Hermitean fractional quadratures. The development is illustrated by numerical examples.

In this paper, the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian systems is considered. With the help of numerical analysis, by using bifurcation theory of planar dynamical systems and the method of detection function, we show that a Z 6 -equivariant planar perturbed Hamiltonian vector field of degree 7 has at least 37 limit cycles. The paper also shows the configuration of compound eyes of that Z 6 -equivariant system.

This paper considers a generalized variation of the Weber problem (GVWP) on the plane in which a straight line divides the plane into two regions and different gauges are employed to measure the distances in different regions. GVWP is a generalized problem of some well-studied variations of Weber problem (VWP) in the following aspects: (1) both the unconstrained and constrained problems are taken into consideration; (2) the more general distance measuring functions, gauges, are employed to measure distances instead of the usually used l p -norms. Therefore, the GVWP is more practical and applicable in practice. GVWP is nonconvex as VWP, and as a generalized problem it is more complex than the latter. In this paper the GVWP is divided into three subproblems which are optimally solved: two subproblems are reformulated into monotone linear variational inequalities (LVIs) and then solved by a projection–contraction method, while the third subproblem is divided into some convex problems and the golden section search is used to solve them. By dividing the GVWP into three subproblems and solving these subproblems optimally, an algorithm which can obtain the global optimal solution of GVWP is proposed. Preliminary numerical results are reported to verify the evident effectiveness of the proposed algorithm.

Similar to fuzzy set on a possibility space, uncertain set is a set-valued function on an uncertainty space, and attempts to model unsharp concepts. Entropy provides a quantitative measurement of the uncertainty associated with an uncertain set. This paper presents a formula for calculating the entropy of an uncertain set via its inverse membership function. Based on the formula, the entropy operator is shown to satisfy positive linearity property. In addition, this paper proposes a concept of relative entropy to describe the divergence between the membership functions of two uncertain sets.

For many engineering problems, it is always too difficult to get the exact solutions, one usually tries to find a numerical solution to the problems. Many discrete methods have been proposed and the finite element method may be a good one. When the number of the discrete elements is very great, the numerical method will also encounter some difficulties (Aliabadi, 2002) [1], for example, it always takes a lot of time for calculation. A kernel solution of the beam system is proposed in the paper, using the proposed method one can obtain the solution to the bending problem of beam systems only by getting the solution to the boundary equations of beam systems. An example of the numerical solution is also presented in the paper.

In this paper, we present a new semilocal convergence analysis for an inverse free Broyden’s method in a Hilbert space setting. In the analysis, we apply our new idea of recurrent functions concepts of divided differences of order one and Lipschitz/center–Lipschitz conditions on the operator involved. Our analysis extends the applicability of Broyden’s method in cases not covered before. Finally, we give an example to illustrate the main result in this paper.

In this paper, using the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, some sufficient conditions for the existence and global exponential stability of almost automorphic solutions for a class of neutral type high-order Hopfield neural networks (HHNNs) with delays in leakage terms on time scales are obtained. Finally, a numerical example is provided to illustrate the feasibility of our results and our results also show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new even when the time scale T = R or Z and show that the delays in the leakage term do harm to the existence and global exponential stability of almost automorphic solutions.