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This paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model, analytically and numerically. Firstly, the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic. Secondly, the asymptotic behavior of non-oscillatory analytic solution is verified. Thirdly, the θ-methods are applied to the mentioned equation, some conditions under which the numerical solution oscillates are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to the steady state of the model. Finally, some numerical simulations for verifying the theoretical findings are provided.

This paper studies the problem of underdetermined blind source separation with the nonstrictly sparse condition. Different from current approaches in literature, we propose a new and more effective algorithm to estimate the mixing matrices resulted from noise output data sets. After we introduce a clustering prototype of orthogonal complement space and give an extension of the normal vector clustering prototype, a new method combing the fuzzy clustering and eigenvalue decomposition technique to estimate the mixing matrix is presented in order to deal with the nonstrictly sparse situation. A convergent algorithm for estimating the mixing matrices is established, and numerical simulations are given to demonstrate the effectiveness of the proposed approach.

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ-methods are used to solve the above-mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1-16, 2014

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. We study the finite-step realizability of the joint/generalized spectral radius of a pair of n × n square matrices. Let Σ = {A, B} where A,B are n × n matrices and B is a rank-one matrix. Then we have ρ(Σ)= max:t,s ρ(AtBs)1/(s+t). That is to say, Σ have the finiteness property where the maximum is attained at (t, s) with the optimal sequence AtBs.

In this paper, we consider the oscillations of numerical solutions for the nonlinear delay differential equations in a hematopoiesis model. Using two θ-methods, namely the linear θ-method and the one-leg θ-method, several conditions, under which the numerical solutions oscillate, are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to an equilibrium point of the original system. Some numerical experiments are provided to support the theoretical results. MSC:65L05, 65L20.

This paper focuses on the stability and oscillations of Euler-Maclaurin method for linear differential equations with piecewise constant arguments ${u}^{\prime }\mathbf{\left(}t\mathbf{\right)} \mathbf{=} au\mathbf{\left(}t\mathbf{\right)} \mathbf{+} bu\mathbf{\left(}\mathbf{\left[}t\mathbf{\right]}\mathbf{\right)}$ . The necessary and sufficient conditions under which the numerical stability region contains the analytical stability region are given. Furthermore, the conditions of oscillation for the Euler-Maclaurin method are obtained. We prove that the Euler-Maclaurin method preserves the oscillations of the analytic solution. Moreover, the relationships between stability and oscillations are discussed for analytic solution and numerical solution, respectively. Finally, some numerical experiments for verifying the theoretical analysis are also provided.

Numerical oscillation of Runge-Kutta methods for differential equations with piecewise constant arguments is considered in this paper. The conditions of oscillation for the Runge-Kutta methods are obtained. It is proven that the numerical oscillation on the integer nodes are equivalent to the numerical oscillation on the any nodes and oscillation of the analytic solution is preserved by the Runge-Kutta methods. Moreover, the relationship between stability and oscillation is discussed for analytic solution and numerical solution, respectively. At last, several numerical simulations are carried out to support the theoretical analysis of the research.

Based on Bernoulli-Gaussian model and the sparseness of source signals, a new algorithm for estimating the mixing matrix is proposed in this paper. It estimates the mixing matrix by searching the cluster points which are found through the density of points in the region. In order to enhance the precision of the algorithm, the cost function is constructed to search the cluster points. The last simulations show the good performance of the proposed algorithm.

This paper proposes a new kind of constraint handling method for optimization. In the proposed method, a transformation of constraint functions to be another objective function will be discussed in details. With this technique we can transform the constraint problem to be unconstraint multiobjective optimization and use the multi-objective optimization methods to find feasible solutions. The performance of the algorithm is tested on 13 benchmark functions in the literature. The results show that we can find very good solutions by this approach.

In this paper, the difference of satisfactions for fuzzy yield and fuzzy risk from investors is viewed as a fuzzy number with parabolic membership function, and a fuzzy decision securities investment combination model bases on parabolic satisfaction degree is constructed with fuzzy set and fuzzy decision theories. The experimental results on real-word data show that our new approach is very efficient and effective for the investment decision.

For the problem of Underdetermined Blind Source Separation (UBSS) under nonstrictly sparse condition, we propose a new algorithm to estimate mixing matrix. Firstly, a clustering prototype of orthogonal complement space is introduced; Secondly, a fuzzy EVD clustering method which combines fuzzy clustering and Eigenvalue Decomposition (EVD) is presented. Based on these two methods, the algorithm proposed in this paper is robust against noise without losing convergence speed.

In this paper, we propose a decomposition-based evolutionary algorithm with boundary search and archive for constrained multi-objective optimization problems (CMOPs), named CM2M. It decomposes a CMOP into a number of optimization subproblems and optimizes them simultaneously. Moreover, a novel constraint handling scheme based on the boundary search and archive is proposed. Each subproblem has one archive, including a subpopulation and a temporary register. Those individuals with better objective values and lower constraint violations are recorded in the subpopulation, while the temporary register consists of those individuals ever found before. To improve the efficiency of the algorithm, the boundary search method is designed. This method makes the feasible individuals with a higher probability to perform genetic operator with the infeasible individuals. Especially, when the constraints are active at the Pareto solutions, it can play its leading role. Compared with two algorithms, i.e. CMOEA/D-DE-CDP and Gary’s algorithm, on 18 CMOPs, the results show the effectiveness of the proposed constraint handling scheme.