The concept of soft sets introduced by Molodtsov is a general mathematical tool for dealing with uncertainty. Just as the conventional set-theoretic operations of intersection, union, complement, and difference, some corresponding operations on soft sets have been proposed. Unfortunately, such operations cannot keep all classical set-theoretic laws true for soft sets. In this paper, we redefine the intersection, complement, and difference of soft sets and investigate the algebraic properties of these operations along with a known union operation. We find that the new operation system on soft sets inherits all basic properties of operations on classical sets, which justifies our definitions.

Derivatives of eigenvectors with respect to structural parameters play an important role in structural design, identification, and optimization. Particularly, calculation of eigenvector sensitivity is considered when the eigenvalues are repeated. A relaxation factor embedded in the combined approximations (CA) method makes it effective to the structural response at various modified designs. The proposed method is feasible after overcoming the defection of irreversibility of the characteristic matrix. Numerical examples show that it is easy to implement the computational procedure, and the method presented in this paper is efficient for the general linear vibration damped systems with repeated frequencies.

Let $K$ be a nonempty, closed, and convex subset of a real uniformly convex Banach space $E$. Let $\{T_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $\{S_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ be two infinite families of asymptotically nonexpansive mappings from $K$ to itself with $F:=\{x\in K:T_{\lambda }x=x=S_{\lambda }x,\lambda \in \mathrm{\Lambda }\}\ne \varnothing$. For an arbitrary initial point $x_0\in K$, $\{x_n\}$ is defined as follows: $x_n={\alpha }_n{x}_{n-1}+{\beta }_n(T_{n-1}^{*}{)}^{m_{n-1}}{x}_{n-1}+{\gamma }_n(T_n^{*}{)}^{m_n}{y}_n$, $y_n={\alpha }_n^{\prime }{x}_n+{\beta }_n^{\prime }(S_{n-1}^{*}{)}^{m_{n-1}}{x}_{n-1}+{\gamma }_n^{\prime }(S_n^{*}{)}^{m_n}{x}_n$, $n=\mathrm{1,2},3,\dots$, where $T_n^{*}=T_{{\lambda }_{i_n}}$ and $S_n^{*}=S_{{\lambda }_{i_n}}$ with $i_n$ and $m_n$ satisfying the positive integer equation: $n=i+(m-1)m/2$, $m\ge i$; $\{T_{{\lambda }_i}{\}}_{i=1}^{\infty }$ and $\{S_{{\lambda }_i}{\}}_{i=1}^{\infty }$ are two countable subsets of $\{T_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $\{S_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }},$ respectively; $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\alpha }_n^{\prime }\}$, $\{{\beta }_n^{\prime }\}$, and $\{{\gamma }_n^{\prime }\}$ are sequences in $[\delta ,1-\delta ]$ for some $\delta \in (\mathrm{0,1})$, satisfying ${\alpha }_n+{\beta }_n+{\gamma }_n=1$ and ${\alpha }_n^{\prime }+{\beta }_n^{\prime }+{\gamma }_n^{\prime }=1$. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings $\{T_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $\{S_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

Multiple zeta values are the numbers defined by the convergent series $\zeta (s_1,s_2,\dots ,s_k)={\sum }_{n_1>n_2>\cdots >n_k>0}(1/n_1^{s_1}\hspace{0.17em}{n}_2^{s_2}\cdots n_k^{s_k})$, where $s_1$, $s_2$, $\dots$, $s_k$ are positive integers with $s_1>1$. For $k\le n$, let $E(2n,k)$ be the sum of all multiple zeta values with even arguments whose weight is $2n$ and whose depth is $k$. The well-known result $E(2n,2)=3\zeta (2n)/4$ was extended to $E(2n,3)$ and $E(2n,4)$ by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers $E(2n,k)$ and then gave a direct formula for $E(2n,k)$ for arbitrary $k\le n$. In this paper we apply a technique introduced by Granville to present an algorithm to calculate $E(2n,k)$ and prove that the direct formula can also be deduced from Eisenstein's double product.

We present a new improved parsimonious multivariate Markov chain model. Moreover, we find a new convergence condition with a new variability to improve the prediction accuracy and minimize the scale of the convergence condition. Numerical experiments illustrate that the new improved parsimonious multivariate Markov chain model with the new convergence condition of the new variability performs better than the improved parsimonious multivariate Markov chain model in prediction.

We give some refinements of generalized Aczél's inequality and Bellman's inequality proposed by Tian. As applications, some refinements of integral type of generalized Aczél's inequality and Bellman's inequality are given.

The thermomechanical behavior of a material is expressed mathematically by means of one or more constitutive equations representing the response of the body to the history of its deformation and temperature. These settings induce a set of connections which can express local properties. We replace two of them by a second order connection and prove that the holonomity of this connection classifies our materials.

We investigate the existence and multiplicity of solutions to a boundary value problem for impulsive differential equations. By using critical point theory, some criteria are obtained to guarantee that the impulsive problem has at least one solution, at least two solutions, and infinitely many solutions. Some examples are given to illustrate the effectiveness of our results.

Invasion of an exotic species initiated by its local introduction is considered subject to intraguild predation (IGP). Mathematically, the system dynamics is described by three nonlinear diffusion-reaction equations in two spatial dimensions. The key factors that determine successful invasion are investigated by means of extensive numerical simulations. The results reveal high asymmetry. An exotic species can invade successfully if it acted as the top predator and engaged in IGP, and the IGP interactions of the postinvasion web will be kept. While the exotic species were introduced as the intraguild prey (IGprey), they invade and spread through patchy invasion which corresponds to the invasion at the edge of extinction. Increase of the IGprey's dispersal rate and decrease of the IGpredator's may make the IGprey invade. But the interactions of the postinvasion web will change from IGP to competition, which is absolutely different from the first case. Finally, the common existence of IGP was explored once again from the perspective of biological invasion.

An analysis has been carried out to study the flow and heat transfer in a liquid film over a permeable stretching sheet. Using similarity transformations, the time-dependent boundary layer equations are reduced to a set of nonlinear ordinary differential equations. The resulting parameter problem and velocity as well as temperature fields are solved using the homotopy analysis method (HAM). Analytic series solutions are given, and numerical results for velocity and the temperature profiles are presented through graphs of different values for pertinent parameter. The effects of unsteadiness parameter and permeability parameter on the velocity and temperature profiles are explored for different values of blowing or suction parameter.

A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature.

A comparative study is presented about the Adomian’s decomposition method (ADM), variational iteration method (VIM), and fractional variational iteration method (FVIM) in dealing with fractional partial differential equations (FPDEs). The study outlines the significant features of the ADM and FVIM methods. It is found that FVIM is identical to ADM in certain scenarios. Numerical results from three examples demonstrate that FVIM has similar efficiency, convenience, and accuracy like ADM. Moreover, the approximate series are also part of the exact solution while not requiring the evaluation of the Adomian’s polynomials.

A new certain differential operator $T_{\alpha ,\beta ,\delta ,\lambda }^{m}f(z)$ and a subclass $S_{m,w}^{*}(\alpha ,\beta ,\gamma ,\delta ,\lambda )$ are introduced for functions of the form $f(z)=(z-w)-{\sum }_{n=2}^{\infty }{a}_n{(z-w)}^n$ which are univalent in the unit disc . In this paper, we obtain coefficient inequalities, distortion theorem, closure theorems, and class preserving integral operators of functions belonging to the class $S_{m,w}^{*}(\alpha ,\beta ,\gamma ,\delta ,\lambda )$.

The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash equilibrium problem (NEP), in which each player's strategy set may depend on the rival player's strategies. In this paper, we present two descent type methods. The algorithms are based on a reformulation of the generalized Nash equilibrium using Nikaido-Isoda function as unconstrained optimization. We prove that our algorithms are globally convergent and the convergence analysis is not based on conditions guaranteeing that every stationary point of the optimization problem is a solution of the GNEP.

In order to effectively solve the dynamic vehicle routing problem with time windows, the mathematical model is established and an improved variable neighborhood search algorithm is proposed. In the algorithm, allocation customers and planning routes for the initial solution are completed by the clustering method. Hybrid operators of insert and exchange are used to achieve the shaking process, the later optimization process is presented to improve the solution space, and the best-improvement strategy is adopted, which make the algorithm can achieve a better balance in the solution quality and running time. The idea of simulated annealing is introduced to take control of the acceptance of new solutions, and the influences of arrival time, distribution of geographical location, and time window range on route selection are analyzed. In the experiment, the proposed algorithm is applied to solve the different sizes' problems of DVRP. Comparing to other algorithms on the results shows that the algorithm is effective and feasible.

This paper presents a novel, general method that is aimed at identifying both the harmonic excitations and nonlinear damping of forced nonlinear oscillation systems. An inverse formalism is developed for the identification method, which is shown to be well posed. That is, the inverse formalism has a unique solution and it depends continuously on the data. The unique solution is derived in the closed form. In addition to the mathematical analysis, a numerical example involving a highly nonlinear system is examined for demonstrating the workability of the proposed method for simultaneously identifying both the two aforementioned excitations and nonlinear damping.

We present a new diagonal quasi-Newton update with an improved diagonal Jacobian approximation for solving large-scale systems of nonlinear equations. In this approach, the Jacobian approximation is derived based on the quasi-Cauchy condition. The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as to carry some additional information from the functions. The effectiveness of our proposed scheme is appraised through numerical comparison with some well-known Newton-like methods.

The Biconjugate $A$-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate $A$-orthonormalization procedure may possibly tend to suffer from two sources of numerical instability, known as two kinds of breakdowns, similarly to those of the Biconjugate Gradient (BCG) method. This paper naturally exploits the composite step strategy employed in the development of the composite step BCG (CSBCG) method into the BiCOR method to cure one of the breakdowns called as pivot breakdown. Analogously to the CSBCG method, the resulting interesting variant, with only a minor modification to the usual implementation of the BiCOR method, is able to avoid near pivot breakdowns and compute all the well-defined BiCOR iterates stably on the assumption that the underlying biconjugate $A$-orthonormalization procedure does not break down. Another benefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergence history of the norm of the residuals, which is justified by numerical experiments. In addition, the exhibited method inherits the promising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG method so that it outperforms the CSBCG method to some extent.

In the paper titled “Lattices generated by two orbits of subspaces under finite classical group” by Wang and Guo. The subspaces in the lattices are characterized and the geometricity is classified. In this paper, the result above is generalized to singular symplectic space. This paper characterizes the subspaces in these lattices, classifies their geometricity, and computes their characteristic polynomials.

This paper deals with a discrete-time bulk-service $Geo/Geo/1$ queueing system with infinite buffer space and multiple working vacations. Considering an early arrival system, as soon as the server empties the system in a regular busy period, he leaves the system and takes a working vacation for a random duration at time $n$. The service times both in a working vacation and in a busy period and the vacation times are assumed to be geometrically distributed. By using embedded Markov chain approach and difference operator method, queue length of the whole system at random slots and the waiting time for an arriving customer are obtained. The queue length distributions of the outside observer’s observation epoch are investigated. Numerical experiment is performed to validate the analytical results.

A novel robust hybrid metaheuristic optimization approach, which can be considered as an improvement of the recently developed bat algorithm, is proposed to solve global numerical optimization problems. The improvement includes the addition of pitch adjustment operation in HS serving as a mutation operator during the process of the bat updating with the aim of speeding up convergence, thus making the approach more feasible for a wider range of real-world applications. The detailed implementation procedure for this improved metaheuristic method is also described. Fourteen standard benchmark functions are applied to verify the effects of these improvements, and it is demonstrated that, in most situations, the performance of this hybrid metaheuristic method (HS/BA) is superior to, or at least highly competitive with, the standard BA and other population-based optimization methods, such as ACO, BA, BBO, DE, ES, GA, HS, PSO, and SGA. The effect of the HS/BA parameters is also analyzed.

We consider the generalized minimax programming problem (P) in which functions are locally Lipschitz ($G$, $\beta$)-invex. Not only $G$-sufficient but also $G$-necessary optimality conditions are established for problem (P). With $G$-necessary optimality conditions and ($G$, $\beta$)-invexity on hand, we construct dual problem (DI) for the primal one (P) and prove duality results between problems (P) and (DI). These results extend several known results to a wider class of programs.

This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

Stability of the first-order neutral delay equation $x^{\prime }(t)+a{x}^{\prime }(t-\tau )=bx(t)+cx(t-\tau )$ with complex coefficients is studied, by analyzing the existence of stability switches.

The objective of this study is to use a new style of waved flow channel instead of the plane surface channel in the proton exchange membrane fuel cell (PEMFC). The velocity, concentration, and electrical performance with the waved flow channel in PEMFC are investigated by numerical simulations. The results show that the waved channel arises when the transport benefits through the porous layer and improves the performance of the PEMFC. This is because the waved flow channel enhances the forced convection and causes the more reactant gas flow into the gas diffusion layer (GDL). The performance which was compared to a conventional straight gas flow channel increases significantly with the small gap size when it is smaller than 0.5 in the waved flow channel. The performance is decreased at the high and low velocities as the force convection mechanism is weakened and the reactant gas supply is insufficient. The pressure drop is increased as the gap size becomes smaller, and the wave number decreases. (gap size) δ > 0.3 has a reasonable pressure drop. Consequently, compared to a conventional PEMFC, the waved flow channel improves approximately 30% of power density.

A new multiscale finite element method is presented for solving the elliptic equations with rapidly oscillating coefficients. The proposed method is based on asymptotic analysis and careful numerical treatments for the boundary corrector terms by virtue of the recovery technique. Under the assumption that the oscillating coefficient is periodic, some superconvergence results are derived, which seem to be never discovered in the previous literature. Finally, some numerical experiments are carried out to demonstrate the efficiency and accuracy of this method, and it is seen that they agree very well with the analytical result.

We present a simple numerical method to find the optimal exercise boundary in an American put option. We formulate an intermediate function with the fixed free boundary that has Lipschitz character near optimal exercise boundary. Employing it, we can easily determine the optimal exercise boundary by solving a quadratic equation in time-recursive way. We also present several numerical results which illustrate a comparison to other methods.

We consider the existence of positive solutions of nonlinear p-Laplacian dynamic equations with derivative on time scales. Applying the Avery-Peterson fixed point theorem, we obtain at least three positive solutions to the problem. An example is also presented to illustrate the applications of the obtained results.

This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in ($1+1$)-dimensions. The integrability aspect as well as the bifurcation analysis is studied in this paper. The numerical simulations are also given where the finite difference approach was utilized. There are a few constraint conditions that naturally evolve during the course of derivation of the soliton solutions. These constraint conditions must remain valid in order for the soliton solution to exist. For the bifurcation analysis, the phase portraits are also given.

The stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. The existence of the Hopf bifurcation with delay $\tau$ at the endemic equilibria is established by analyzing the distribution of the characteristic values. The explicit formulae which determine the direction of the bifurcations, stability, and the other properties of the bifurcating periodic solutions are given by using the normal form theory and the center manifold theorem. Numerical simulation verifies the theoretical results.

This paper investigates the stabilization of networked control systems (NCSs) with random delays and random sampling periods. Sampling periods can randomly switch between three cases according to the high, low, and medium types of network load. The sensor-to-controller (S-C) random delays and random sampling periods are modeled as Markov chains. The transition probabilities of Markov chains do not need to be completely known. A state feedback controller is designed via the iterative linear matrix inequality (LMI) approach. It is shown that the designed controller is two-mode dependent and depends on not only the current S-C delay but also the most recent available sampling period at the controller node. The resulting closed-loop systems are special discrete-time jump linear systems with two modes. The sufficient conditions for the stochastic stability are established. An example of the cart and inverted pendulum is given to illustrate the effectiveness of the theoretical result.

Quality management is very important for many manufacturing industries. Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. Hence, the lifetime performance index $C_L$ is utilized to measure the performance of product, where $L$ is the lower specification limit. This study constructs a Bayesian estimator of $C_L$ under a Rayleigh distribution with the upper record values. The Bayesian estimations are based on squared-error loss function, linear exponential loss function, and general entropy loss function, respectively. Further, the Bayesian estimators of $C_L$ are utilized to construct the testing procedure for $C_L$ based on a credible interval in the condition of known $L$. The proposed testing procedure not only can handle nonnormal lifetime data, but also can handle the upper record values. Moreover, the managers can employ the testing procedure to determine whether the lifetime performance of the Rayleigh products adheres to the required level. The hypothesis testing procedure is a quality performance assessment system in enterprise resource planning (ERP).

A three-dimensional hybrid cellular automata (CA) model is developed to study the dynamic process of multicellular tumour spheroid (MTS) growth by introducing hypoxia as an important microenvironment factor which influences cell migration and cell phenotype expression. The model enables us to examine the effects of different hypoxic environments on the growth history of the MTS and to study the dynamic interactions between MTS growth and chemical environments. The results include the spatial distribution of different phenotypes of tumour cells and associated oxygen concentration distributions under hypoxic conditions. The discussion of the model system responses to the varied hypoxic conditions reveals that the improvement of the resistance of tumour cells to a hypoxic environment may be important in the tumour normalization therapy.

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted by $h$ and $k$, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to the $L_1$ norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values of $k$ and $h$. Two optimisation techniques are then implemented to find the optimal values of $k$ when $h=0.02$ for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

We study exponential attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains. First, we give the existence of the exponential attractor in $L^2(\Omega )\times L^2(\mathrm{\Gamma })$ by proving that the corresponding semigroup satisfies the enhanced flattering property. Second, we apply asymptotic a priori estimate and obtain the exponential attractor in $L^p(\Omega )\times L^q(\mathrm{\Gamma })$. Finally, we show the exponential attractor in $(H^1(\Omega )\cap L^p(\Omega ))\times L^q(\mathrm{\Gamma })$.

A new method for computing the approximation of bivariate matrix function is introduced. It uses the construction of bivariate Newton-Thiele type matrix rational interpolants on a rectangular grid. The rational interpolant is of the form motivated by Tan and Fang (2000), which is combined by Newton interpolant and branched continued fractions, with scalar denominator. The matrix quotients are based on the generalized inverse for a matrix which is introduced by C. Gu the author of this paper, and it is effective in continued fraction interpolation. The algorithm and some other important conclusions such as divisibility and characterization are given. In the end, two examples are also given to show the effectiveness of the algorithm. The numerical results of the second example show that the algorithm of this paper is better than the method of Thieletype matrix-valued rational interpolant in Gu (1997).

In power systems, determining the values of time dial setting (TDS) and the plug setting (PS) for directional overcurrent relays (DOCRs) is an extremely constrained optimization problem that has been previously described and solved as a nonlinear programming problem. Optimization coordination problems of near-end faults and far-end faults occurring simultaneously in circuits with various topologies, including fixed and variable network topologies, are considered in this study. The aim of this study was to apply the Nelder-Mead (NM) simplex search method and particle swarm optimization (PSO) to solve this optimization problem. The proposed NM-PSO method has the advantage of NM algorithm, with a quicker movement toward optimal solution, as well as the advantage of PSO algorithm in the ability to obtain globally optimal solution. Neither a conventional PSO nor the proposed NM-PSO method is capable of dealing with constrained optimization problems. Therefore, we use the gradient-based repair method embedded in a conventional PSO and the proposed NM-PSO. This study used an IEEE 8-bus test system as a case study to compare the convergence performance of the proposed NM-PSO method and a conventional PSO approach. The results demonstrate that a robust and optimal solution can be obtained efficiently by implementing the proposal.

This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction number ${ℛ}_0$ and the other value ${ℛ}_c$ which is larger than ${ℛ}_0$. If ${ℛ}_0$ and ${ℛ}_c$ are both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.

A periodic discrete time three species competition system is investigated. With the aid of differential equations with piecewise constant arguments, a discrete analogue of continuous nonautonomous three species competition system is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive periodic solutions of the model are obtained.

An analysis is carried out to study the steady two-dimensional stagnation-point flow of an incompressible viscous fluid towards a stretching vertical sheet. It is assumed that the sheet is stretched nonlinearly, with prescribed surface heat flux. This problem is governed by three parameters: buoyancy, velocity exponent, and velocity ratio. Both assisting and opposing buoyant flows are considered. The governing partial differential equations are transformed into a system of ordinary differential equations and solved numerically by finite difference Keller-box method. The flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Dual solutions are found in the opposing buoyant flows, while the solution is unique for the assisting buoyant flows.

We propose a new iterative method to find the bisymmetric minimum norm solution of a pair of consistent matrix equations $A_{1}X{B}_1=C_1$, $A_{2}X{B}_2=C_2$. The algorithm can obtain the bisymmetric solution with minimum Frobenius norm in finite iteration steps in the absence of round-off errors. Our algorithm is faster and more stable than Algorithm 2.1 by Cai et al. (2010).

Cellular manufacturing (CM) is a production approach directed towards reducing costs, as well as increasing system's flexibility in today's small-to-medium lot production environment. Many structural and operational issues should be considered for a successful CM design and implementation such as cell formation (CF), production planning, and facility layout. Most researchers have addressed these issues sequentially or independently, instead of jointly optimizing a combination of these issues. In order to attain better results to ensure that the system will be capable of remaining efficient in unknown future situations, these issues should be addressed simultaneously. In this paper, a mathematical model is developed using an integrated approach for production planning and cell formation problems in a CM. A set of numerical examples are provided from existing the literature in order to test and illustrate the proposed model. In order to evaluate and verify the performance of the proposed model, it is compared with a well-known cell formation methods (rank order clustering and direct clustering analysis), using group capability index (GCI) measure. The results and comparisons indicate that the proposed model has a significantly higher and satisfactory performance and it is reliable for the design and the analysis of CM systems.

We derive certain sufficient conditions for starlikeness and convexity of order $\alpha$ of analytic functions in the unit disk. Applications are indicated for the subordination results to electromagnetic cloaking.

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable.

We introduce a new form of Laplace decomposition algorithm (LDA). By this form a new iterative method was achieved in which there is no need to calculate Adomian polynomials, which require so much computational time for higher-order approximations. We have implemented this method for the solutions of different types of nonlinear pantograph equations to support the proposed analysis.

We present a fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option based on the extension principle. In order to keep consistent with the real market, we assume that the interest rate, the spot exchange rate, and the volatility are fuzzy numbers in the FG-K formula. The conditions of a basic proposition about the fuzzy-valued functions of fuzzy subsets are modified. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. To simplify the trade, the weighted possibilistic mean (WPM) value with a weighting function is adopted to defuzzify the fuzzy price to a crisp price. The numerical example shows our method makes the α-level set of fuzzy price smaller, which decreases the fuzziness. The example also indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. Inspired by this example, we provide a method, which can identify the optimal parameter.

The fuzzy Sylvester matrix equation $A\tilde{X}+\tilde{X}B=\tilde{C}$ in which $A,B$ are $m\times m$ and $n\times n$ crisp matrices, respectively, and $\tilde{C}$ is an $m\times n$ LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.

We propose a new class of mathematical structures called $(m,n)$-semirings (which generalize the usual semirings) and describe their basic properties. We define partial ordering and generalize the concepts of congruence, homomorphism, and so forth, for $(m,n)$-semirings. Following earlier work by Rao (2008), we consider systems made up of several components whose failures may cause them to fail and represent the set of such systems algebraically as an $(m,n)$-semiring. Based on the characteristics of these components, we present a formalism to compare the fault-tolerance behavior of two systems using our framework of a partially ordered $(m,n)$-semiring.

We introduced and studied $\mathrm{GF}$-regular modules as a generalization of $\pi$-regular rings to modules as well as regular modules (in the sense of Fieldhouse). An $R$-module $M$ is called $\mathrm{GF}$-regular if for each $x\in M$ and $r\in R$, there exist $t\in R$ and a positive integer $n$ such that $r^{n}t{r}^{n}x=r^{n}x$. The notion of $G$-pure submodules was introduced to generalize pure submodules and proved that an $R$-module $M$ is $\mathrm{GF}$-regular if and only if every submodule of $M$ is $G$-pure iff $M_{𝔐}$ is a $\mathrm{GF}$-regular $R_{𝔐}$-module for each maximal ideal $𝔐$ of $R$. Many characterizations and properties of $\mathrm{GF}$-regular modules were given. An $R$-module $M$ is $\mathrm{GF}$-regular iff $R/\text{a}\text{n}\text{n}\left(x\right)$ is a $\pi$-regular ring for each $0\ne x\in M$ iff $R/\text{a}\text{n}\text{n}\left(M\right)$ is a $\pi$-regular ring for finitely generated module $M$. If $M$ is a $\mathrm{GF}$-regular module, then $J\left(M\right)=0$.

An accurate forecasting method for power generation of the wind energy conversion system (WECS) is urgently needed under the relevant issues associated with the high penetration of wind power in the electricity system. This paper proposes a hybrid method that combines orthogonal least squares (OLS) algorithm and genetic algorithm (GA) to construct the radial basis function (RBF) neural network for short-term wind power forecasting. The RBF neural network is composed of three-layer structures, which contain the input, hidden, and output layers. The OLS algorithm is used to determine the optimal number of nodes in a hidden layer of RBF neural network. With an appropriate RBF neural network structure, the GA is then used to tune the parameters in the network, including the centers and widths of RBF and the connection weights in second stage. To demonstrate the effectiveness of the proposed method, the method is tested on the practical information of wind power generation of a WECS installed in Taichung coast of Taiwan. Comparisons of forecasting performance are made to the persistence method and back propagation neural network. The good agreements between the realistic values and forecasting values are obtained; the test results show the proposed forecasting method is accurate and reliable.

This paper is concerned with a double nonlinear dispersive equation: the Sharma-Tasso-Olver equation. We propose an improved $G^{\prime }/G$-expansion method which is employed to investigate the solitary and periodic traveling waves of this equation. As a result, some new traveling wave solutions involving hyperbolic functions, the trigonometric functions, are obtained. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions, and the periodic wave solutions are derived from the trigonometric function solutions. The improved $G^{\prime }/G$-expansion method is straightforward, concise and effective and can be applied to other nonlinear evolution equations in mathematical physics.

The objective of this paper is to extend some results of pioneers for the nonlinear equation $m_t=\mathrm{(1}/\mathrm{2)}{(1/m^k)}_{xxx}-\mathrm{(1}/\mathrm{2)}{(1/m^k)}_x$ introduced by Qiao. The equivalent relationship of the traveling wave solutions between the integrable equation and the generalized KdV equation is revealed. Moreover, when $k=-\mathrm{(p}/\mathrm{q)}(p\ne q$ and $p,q\in {\mathbb{Z}}^{+})$, we obtain some explicit traveling wave solutions by the bifurcation method of dynamical systems.

This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of order n. Based on the proposed algorithm, the eigenvalues that have the largest and smallest modulus, real parts, or absolute values of imaginary parts can be extracted, respectively, as well as the corresponding eigenvectors. Although the ordinary differential equation on which our proposed algorithm is built is only n-dimensional, it can succeed to extract n-dimensional complex eigenvectors that are indeed 2n-dimensional real vectors. Moreover, we show that extracting eigen-pairs of general real matrices can be reduced to those of real normal matrices by employing the norm-reducing skill. Numerical experiments verified the computational capability of the proposed algorithm.

This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.

We study the asymptotic behavior of the parabolic Monge-Ampère equation $\partial \phi (x,t)/\partial t=\text{log}(\text{det}(g(x)+\text{Hess}\phi (x,t))/\text{det}g(x))-\lambda \phi (x,t)$ in $𝕄\times (0,\infty )$, $\phi (x,0)={\phi }_0(x)$ in $𝕄$, where $𝕄$ is a compact complete Riemannian manifold, λ is a positive real parameter, and ${\phi }_0(x):𝕄\to ℝ$ is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.

A class of fuzzy neural networks (FNNs) with time-varying delays and impulses is investigated. With removing some restrictions on the amplification functions, a new differential inequality is established, which improves previouse criteria. Applying this differential inequality, a series of new and useful criteria are obtained to ensure the existence of global robust attracting and invariant sets for FNNs with time-varying delays and impulses. Our main results allow much broader application for fuzzy and impulsive neural networks with or without delays. An example is given to illustrate the effectiveness of our results.

The synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems via sliding mode controller is investigated. By designing an active sliding mode controller and choosing proper control parameters, the drive and response systems are synchronized. Synchronization between the fractional-order Chen chaotic system and the integer-order Chen chaotic system and between integer-order hyperchaotic Chen system and fractional-order hyperchaotic Rössler system is used to illustrate the effectiveness of the proposed synchronization approach. Numerical simulations coincide with the theoretical analysis.

We investigate the existence of solutions and positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form $x^{(4)}(t)=f(t,x(t),x^{\prime }(t),x^{\prime \prime }(t),x^{\prime \prime \prime }(t))$, $t\in [0,1]$, $x(0)=x^{\prime }(1)=0$, $x^{\prime \prime }(0)={\int }_0^{1}h(s,x(s),x^{\prime }(s),x^{\prime \prime }(s))\text{d}s$, $x^{\prime \prime \prime }(1)=0$, where $f\in C([0,1]\times {ℝ}^4)$, $h\in C([0,1]\times {ℝ}^3)$. By using a fixed point theorem due to D. O'Regan, the existence of solutions and positive solutions for the previous boundary value problems is obtained. Meanwhile, as applications, some examples are given to illustrate our results.

Fractional-order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, in order to solve the fractional advection-diffusion equation, the fractional characteristic finite difference method is presented, which is based on the method of characteristics (MOC) and fractional finite difference (FD) procedures. The stability, consistency, convergence, and error estimate of the method are obtained. An example is also given to illustrate the applicability of theoretical results.

We present the greatest value $p$ such that the inequality $P(a,b)>L_p(a,b)$ holds for all $a,b>0$ with $a\ne b$, where $P(a,b)$ and $L_p(a,b)$ denote the Seiffert and $p$th generalized logarithmic means of $a$ and $b$, respectively.

The concept of fuzzy modular space is first proposed in this paper. Afterwards, a Hausdorff topology induced by a $\beta$-homogeneous fuzzy modular is defined and some related topological properties are also examined. And then, several theorems on $\mu$-completeness of the fuzzy modular space are given. Finally, the well-known Baire’s theorem and uniform limit theorem are extended to fuzzy modular spaces.

Soft set theory is a newly emerging tool to deal with uncertain problems. Based on soft sets, soft rough approximation operators are introduced, and soft rough sets are defined by using soft rough approximation operators. Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model. This paper is devoted to investigating soft rough approximation operations and relationships among soft sets, soft rough sets, and topologies. We consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are investigated, and their related properties are given. We show that Pawlak's rough set model can be viewed as a special case of soft rough sets, obtain the structure of soft rough sets, give the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space.

This paper is concerned with stochastic differential equations (SDEs) with multi-Markovian switching. The existence and uniqueness of solution are investigated, and the pth moment of the solution is estimated. The classical theory of SDEs with single Markovian switching is extended.

A coepidemic arises when the spread of one infectious disease stimulates the spread of another infectious disease. Recently, this has happened with human immunodeficiency virus (HIV) and tuberculosis (TB). The density of individuals infected with latent tuberculosis is structured by age since latency. The host population is divided into five subclasses of susceptibles, latent TB, active TB (without HIV), HIV infectives (without TB), and coinfection class (infected by both TB and HIV). The model exhibits three boundary equilibria, namely, disease free equilibrium, TB dominated equilibrium, and HIV dominated equilibrium. We discuss the local or global stabilities of boundary equilibria. We prove the persistence of our model. Our simple model of two synergistic infectious disease epidemics illustrates the importance of including the effects of each disease on the transmission and progression of the other disease. We simulate the dynamic behaviors of our model and give medicine explanations.

From the classical point of view, it is important to determine if in a Markov decision process (MDP), besides their existence, the uniqueness of the optimal policies is guaranteed. It is well known that uniqueness does not always hold in optimization problems (for instance, in linear programming). On the other hand, in such problems it is possible for a slight perturbation of the functional cost to restore the uniqueness. In this paper, it is proved that the value functions of an MDP and its cost perturbed version stay close, under adequate conditions, which in some sense is a priority. We are interested in the stability of Markov decision processes with respect to the perturbations of the cost-as-you-go function.

We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameter $\sigma$ associated with the intensity of harvesting is explored. As $\sigma$ grows, the number of periodic solutions drops from two to zero. We provide bounds for the bifurcation parameter whose value in practice can be efficiently approximated numerically.

A general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

We study a system of second-order dynamic equations on time scales $(p_1{u}_1^{\nabla }{)}^{\mathrm{\Delta }}(t)-q_1(t)u_1(t)+\lambda f_1(t,u_1(t),u_2(t))=0,t\in (t_1,t_n),(p_2{u}_2^{\nabla }{)}^{\mathrm{\Delta }}(t)-q_2(t)u_2(t)+\lambda f_2(t,u_1(t)$, $u_2(t))=0$, satisfying four kinds of different multipoint boundary value conditions, $f_i$ is continuous and semipositone. We derive an interval of $\lambda$ such that any $\lambda$ lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.

By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker spacetime and for hypersurfaces in hyperbolic space are given.

The limit $q$-Bernstein operator $B_q$ emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the $q$-boson theory to describe the energy distribution in a $q$-analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the $q$-operators. Over the past years, the limit $q$-Bernstein operator has been studied widely from different perspectives. It has been shown that $B_q$ is a positive shape-preserving linear operator on $C[\mathrm{0,1}]$ with $\parallel B_q\parallel =1$. Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit $q$-Bernstein operator related to the approximation theory. A complete bibliography is supplied.

We prove that a dynamical system is chaotic in the sense of Martelli and Wiggins, when it is a transitive distributively chaotic in a sequence. Then, we give a sufficient condition for the dynamical system to be chaotic in the strong sense of Li-Yorke. We also prove that a dynamical system is distributively chaotic in a sequence, when it is chaotic in the strong sense of Li-Yorke.

We propose some similarity measures between two triangular fuzzy numbers (TFNs) based on the vector similarity measures in vector space, which can be used to aggregate the decision information with TFNs. A methodology for multiple criteria group decision-making (MCGDM) problems with triangular fuzzy information is proposed; the criteria values take the form of linguistic values, which can be converts to TFNs. According to the weighted similarity measures between each alternative and ideal alternative, it is easy to rank alternatives and select the most desirable alternative. Finally, we apply the proposed methods to an illustrative example of MCGDM; the numerical results show that our method is effective and practical. For comparison, we also apply our similarity measures method to solve the fuzzy decision-making problem in Wei (2011); our method has simpler computation and gets the same results more rapidly than the FLOWHM method.

This paper was concerned to simulate both wet and dry bed dam break problems. A high-resolution finite volume method (FVM) was employed to solve the one-dimensional (1D) and two-dimensional (2D) shallow water equations (SWEs) using an unstructured Voronoi mesh grid. In this attempt, the robust local Lax-Friedrichs (LLxF) scheme was used for the calculating of the numerical flux at cells interfaces. The model named V-Break was run under the asymmetry partial and circular dam break conditions and then verified by comparing the model outputs with the documented results. Due to a precise agreement between those output and documented results, the V-Break could be considered as a reliable method for dealing with shallow water (SW) and shock problems, especially those having discontinuities. In addition, statistical observations indicated a good conformity between the V-Break and analytical results clearly.

The boundary layer problem for power-law fluid can be recast to a third-order $p$-Laplacian boundary value problem (BVP). In this paper, we transform the third-order $p$-Laplacian into a new system which exhibits a Lie-symmetry SL$(3,ℝ)$. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of $r\in [0,1]$. The present SL$(3,ℝ)$ Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order $p$-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4) method to obtain a quite accurate numerical solution of the $p$-Laplacian.

This paper discusses why the selection of a finite planning horizon is preferable to an infinite one for a replenishment policy of production inventory models. In a production inventory model, the production rate is dependent on both the demand rate and the inventory level. When there is an exponentially decreasing demand, the application of an infinite planning horizon model is not suitable. The emphasis of this paper is threefold. First, while pointing out questionable results from a previous study, we propose a corrected infinite planning horizon inventory model for the first replenishment cycle. Second, while investigating the optimal solution for the minimization problem, we found that the infinite planning horizon should not be applied when dealing with an exponentially decreasing demand. Third, we developed a new production inventory model under a finite planning horizon for practitioners. Numerical examples are provided to support our findings.

The universal form of univariate Quasi-Bézier basis functions with multiple shape parameters and a series of corresponding Quasi-Bézier curves were constructed step-by-step in this paper, using the method of undetermined coefficients. The series of Quasi-Bézier curves had geometric and affine invariability, convex hull property, symmetry, interpolation at the endpoints and tangent edges at the endpoints, and shape adjustability while maintaining the control points. Various existing Quasi-Bézier curves became special cases in the series. The obvious geometric significance of shape parameters made the adjustment of the geometrical shape easier for the designer. The numerical examples indicated that the algorithm was valid and can easily be applied.

We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions.

This paper addresses the consensus of second-order multiagent systems with general topology and time delay based on the nearest neighbor rule. By using the Laplace transform technique, it is proved that the second-order multi-agent system in the presence of time-delay can reach consensus if the network topology contains a globally reachable node and time delay is bounded. The bound of time-delay only depends on eigenvalues of the Laplacian matrix of the system. The main contribution of this paper is that the accurate state of the consensus center and the upper bound of the communication delay to make the agents reach consensus are given. Some numerical simulations are given to illustrate the theoretical results.

This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear $q$-difference equations with three-point $q$-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

This paper is devoted to multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions. We classify the linear Hamiltonian systems by the index theory and obtain the existence and multiplicity of solutions for the Hamiltonian systems, based on an application of the classical symmetric mountain pass lemma.

This paper investigates the synchronization and antisynchronization problems of a hyperchaotic complex Chen system with unknown parameters based on the properties of a passive system. The essential conditions are derived under which the synchronization or antisynchronization error dynamical system could be equivalent to a passive system and be globally asymptotically stabilized at a zero equilibrium point via smooth state feedback. Corresponding parameter estimation update laws are obtained to estimate the unknown parameters as well. Numerical simulations verify the effectiveness of the theoretical analysis.

A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term $\nabla ·(a(\mathbf{x},t)\nabla u)$ is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical $\mathbf{H}(\text{d}\text{i}\text{v};\mathrm{\Omega })$ space and the hyperbolic part $d(\mathbf{x})(\partial u/\partial t)+\mathbf{c}(\mathbf{x},t)·\nabla u$ is handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in $L^{\mathrm{2}}$- and $H^{\mathrm{1}}$-norms for the scalar unknown $u$ and a priori error estimates in $(L^{\mathrm{2}}{)}^{\mathrm{2}}$-norm for its gradient $\lambda$ and its flux $\sigma$ (the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.

Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar (IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally, pumping lemma for intuitionistic fuzzy context-free languages is investigated.

It is well known that the Steiner minimal tree problem is one of the classical nonlinear combinatorial optimization problems. A visualization experiment approach succeeds in generating Steiner points automatically and showing the system shortest path, named Steiner minimum tree, physically and intuitively. However, it is difficult to form stabilized system shortest path when the number of given points is increased and irregularly distributed. Two algorithms, geometry algorithm and geometry-experiment algorithm (GEA), are constructed to solve system shortest path using the property of Delaunay diagram and basic philosophy of Geo-Steiner algorithm and matching up with the visualization experiment approach (VEA) when the given points increase. The approximate optimizing results are received by GEA and VEA for two examples. The validity of GEA was proved by solving practical problems in engineering, experiment, and comparative analysis. And the global shortest path can be obtained by GEA successfully with several actual calculations.

A Hamilton-Poisson realization and some stability problems for a dynamical system arisen from Chua system are presented. The stability and dynamics of a linearized smooth version of the Chua system are analyzed using the Hamilton-Poisson formalism. This geometrical approach allows to deduce the nonlinear stabilization near different equilibria.

A modified parallel variable distribution (PVD) algorithm for solving large-scale constrained optimization problems is developed, which modifies quadratic subproblem $Q{P}_l$ at each iteration instead of the $Q{P}_l^{\mathrm{0}}$ of the SQP-type PVD algorithm proposed by C. A. Sagastizábal and M. V. Solodov in 2002. The algorithm can circumvent the difficulties associated with the possible inconsistency of $Q{P}_l^{\mathrm{0}}$ subproblem of the original SQP method. Moreover, we introduce a nonmonotone technique instead of the penalty function to carry out the line search procedure with more flexibly. Under appropriate conditions, the global convergence of the method is established. In the final part, parallel numerical experiments are implemented on CUDA based on GPU (Graphics Processing unit).

This paper discusses a predator-prey system with Holling-($n+1$) functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition. The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution $(\tilde{u},\tilde{v})$ is asymptotically stable when the parameter k satisfies some conditions.

A discrete-time predator-prey model is proposed with Leslie-type numerical response, and the asymmetrical influence of Allee effect on the proposed system is investigated. By mathematical analysis, locally stable conditions for the equilibrium points of the considered systems with or without Allee effect are obtained firstly. Furthermore, numerical simulation is used to verify the results and detect some new outcomes. The results show that Allee effect on predator leads the system to its stable state in much longer time. Conversely, the prey population with Allee effect makes it much faster. In particular, a large value of Allee effect on prey results in periodic dynamics of the system.

We present a block preconditioner and consider block preconditioned SSOR iterative methods for solving linear system $Ax=b$. When $A$ is an $H$-matrix, the convergence and some comparison results of the spectral radius for our methods are given. Numerical examples are also given to illustrate that our methods are valid.

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique is demonstrated through several linear and nonlinear test examples.

The permutation behavior of Dickson polynomials of the first kind has been extensively studied, while such behavior for Dickson polynomials of the second kind is less known. Necessary and sufficient conditions for a polynomial of the second kind to be a permutation over some finite fields have been established by Cohen, Matthew, and Henderson. We introduce a new way to define these polynomials and determine the number of their fixed points.

The requirements for interpolation of scattered data are high accuracy and high efficiency. In this paper, a piecewise bivariate Hermite interpolant satisfying these requirements is proposed. We firstly construct a triangulation mesh using the given scattered point set. Based on this mesh, the computational point ($x,y$) is divided into two types: interior point and exterior point. The value of Hermite interpolation polynomial on a triangle will be used as the approximate value if point ($x,y$) is an interior point, while the value of a Hermite interpolation function with the form of weighted combination will be used if it is an exterior point. Hermite interpolation needs the first-order derivatives of the interpolated function which is not directly given in scatted data, so this paper also gives the approximate derivative at every scatted point using local radial basis function interpolation. And numerical tests indicate that the proposed piecewise bivariate Hermite interpolations are economic and have good approximation capacity.

A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP). We firstly convert the problem (LMP) into an equivalent programming one by introducing $p$ auxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients $\delta =\gamma =\mathrm{1}$ and retarded argument $\mathrm{\Delta }\equiv \mathrm{0}$ in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

The sliding mode control and optimization are investigated for a class of nonlinear neutral systems with the unmatched nonlinear term. In the framework of Lyapunov stability theory, the existence conditions for the designed sliding surface and the stability bound ${\alpha }^{\ast }$ are derived via twice transformations. The further results are to develop an efficient sliding mode control law with tuned parameters to attract the state trajectories onto the sliding surface in finite time and remain there for all the subsequent time. Finally, some comparisons are made to show the advantages of our proposed method.

We are concerned with the nonnegative constraints optimization problems. It is well known that the conjugate gradient methods are efficient methods for solving large-scale unconstrained optimization problems due to their simplicity and low storage. Combining the modified Polak-Ribière-Polyak method proposed by Zhang, Zhou, and Li with the Zoutendijk feasible direction method, we proposed a conjugate gradient type method for solving the nonnegative constraints optimization problems. If the current iteration is a feasible point, the direction generated by the proposed method is always a feasible descent direction at the current iteration. Under appropriate conditions, we show that the proposed method is globally convergent. We also present some numerical results to show the efficiency of the proposed method.

We consider positive operators on the real line $ℝ$ with property of interpolation, and we show the weighted $L_p$-convergence of the operators. We will construct an analogical operator of one which is studied by Knopfmacher (1986). Furthermore, we treat the Shepard-type interpolatory operator (cf. Xie et al. (1998)).

This paper investigates finite-time synchronization of the singular hybrid coupled networks. The singular systems studied in this paper are assumed to be regular and impulse-free. Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled networks under a state feedback controller by using finite-time stability theory. A numerical example is finally exploited to show the effectiveness of the obtained results.

Singular two-point boundary value problems (BVPs) are investigated using a new technique, namely, optimal homotopy asymptotic method (OHAM). OHAM provides a convenient way of controlling the convergence region and it does not need to identify an auxiliary parameter. The effectiveness of the method is investigated by comparing the results obtained with the exact solution, which proves the reliability of the method.

Feature selection is an essential process in data mining applications since it reduces a model’s complexity. However, feature selection with various types of costs is still a new research topic. In this paper, we study the cost-sensitive feature selection problem of numeric data with measurement errors. The major contributions of this paper are fourfold. First, a new data model is built to address test costs and misclassification costs as well as error boundaries. It is distinguished from the existing models mainly on the error boundaries. Second, a covering-based rough set model with normal distribution measurement errors is constructed. With this model, coverings are constructed from data rather than assigned by users. Third, a new cost-sensitive feature selection problem is defined on this model. It is more realistic than the existing feature selection problems. Fourth, both backtracking and heuristic algorithms are proposed to deal with the new problem. Experimental results show the efficiency of the pruning techniques for the backtracking algorithm and the effectiveness of the heuristic algorithm. This study is a step toward realistic applications of the cost-sensitive learning.

The concept of the exponentially stable limit cycle (ESLC) is introduced, and the ESLC phenomenon for a class of nonlinear systems is explored. Based on time-domain approach with differential inequality, the existence and uniqueness of the ESLC for such nonlinear systems can be guaranteed. Besides, the period of oscillation, the amplitude of oscillation, and guaranteed convergence rate can be accurately estimated. Finally, two numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result.

We investigate multiple criteria group decision-making problems in which there are priority relationships between the decision elements (criteria and experts), and decision information provided by decision makers takes the form of multigranular uncertain linguistic information. Firstly, some operational laws and possibility degree of multi-granular uncertain linguistic variables are introduced. Then, some new linguistic aggregation operators based on the prioritized aggregation operator, such as the multigranular uncertain linguistic prioritized weighted average (MULPWA) operator and the multigranular uncertain linguistic prioritized ordered weighted average (MULPOWA) operator, are developed and their desirable properties are studied. The prominent characteristics of these proposed operators are that they can aggregate directly the uncertain linguistic variables whose values form the linguistic term sets with different granularities and convey the prioritization phenomenon among the aggregated arguments. Furthermore, based on the MULPWA and MULPOWA operators, an approach to deal with multiple criteria group decision-making problems under multi-granular uncertain linguistic environments is developed. Finally, a practical example is provided to illustrate the multiple criteria group decision-making process.

We study the dynamics of aone-predator, two-preysystem in which the predator has an indirect effect on the preys. We show that, in presence of the indirect effect term, the system admits coexistence of the three populations while, if we disregard it, at least one of the populations goes to extinction.

By M-matrix theory, inequality techniques, and Lyapunov functional method, certain sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of periodic solution for a new type of high-order BAM neural networks with continuously distributed delays and impulses. These novel conditions extend and improve some previously known results in the literature. Finally, an illustrative example and its numerical simulation are given to show the feasibility and correctness of the derived criteria.

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variable $u$ and the auxiliary variable $\sigma$ with respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

A distribution generation (DG) multiobjective optimization method based on an improved Pareto evolutionary algorithm is investigated in this paper. The improved Pareto evolutionary algorithm, which introduces a penalty factor in the objective function constraints, uses an adaptive crossover and a mutation operator in the evolutionary process and combines a simulated annealing iterative process. The proposed algorithm is utilized to the optimize DG injection models to maximize DG utilization while minimizing system loss and environmental pollution. A revised IEEE 33-bus system with multiple DG units was used to test the multiobjective optimization algorithm in a distribution power system. The proposed algorithm was implemented and compared with the strength Pareto evolutionary algorithm 2 (SPEA2), a particle swarm optimization (PSO) algorithm, and nondominated sorting genetic algorithm II (NGSA-II). The comparison of the results demonstrates the validity and practicality of utilizing DG units in terms of economic dispatch and optimal operation in a distribution power system.

The paper deals with the order of convergence of the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system, the $n$ roots of a complex number with modulus one. The supremum norm of the error of interpolation is obtained for analytic functions as well as the corresponding asymptotic constants.

By using the time-switched method and the comparison theorem, we derived a criterion of delay-dependent stability for the switched large-scale time-delay systems. To guarantee the exponential stability for the switched large-scale time-delay systems with stability margin $\lambda$, the total activation time ratio of the switching law is determined. An example is used to illustrate the effectiveness of our result.

We consider a hybrid Dai-Yuan conjugate gradient method. We confirm that its numerical performance can be improved provided that this method uses a practical steplength rule developed by Dong, and the associated convergence is analyzed as well.

We propose a sequence of highly accurate higher order convergent iterative schemes by embedding the quasilinearization algorithm within a spectral collocation method. The iterative schemes are simple to use and significantly reduce the time and number of iterations required to find solutions of highly nonlinear boundary value problems to any arbitrary level of accuracy. The accuracy and convergence properties of the proposed algorithms are tested numerically by solving three Falkner-Skan type boundary layer flow problems and comparing the results to the most accurate results currently available in the literature. We show, for instance, that precision of up to 29 significant figures can be attained with no more than 5 iterations of each algorithm.

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the quasireversibility regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

A repairable computer system model which consists of hardware and software in series is established in this paper. This study is devoted to discussing the unique existence of the solution and the stability of the studied system. In view of $c_0$ semigroup theory, we prove the existence of a unique nonnegative solution of the system. Then by analyzing the spectra distribution of the system operator, we deduce that the transient solution of the system strongly converges to the nonnegative steady-state solution which is the eigenvector corresponding to eigenvalue 0 of the system operator. Finally, some reliability indices of the system are provided at the end of the paper with a new method.

We propose a fourth-order total bounded variation regularization model which could reduce undesirable effects effectively. Based on this model, we introduce an improved split Bregman iteration algorithm to obtain the optimum solution. The convergence property of our algorithm is provided. Numerical experiments show the more excellent visual quality of the proposed model compared with the second-order total bounded variation model which is proposed by Liu and Huang (2010).

Using Matrix-Forest theorem and Matrix-Tree theorem, we present some invariants for weighted digraphs under state in-splittings or out-splittings.

It remained prevalent in the past years to obtain the precommitment strategies for Markowitz's mean-variance portfolio optimization problems, but not much is known about their time-consistent strategies. This paper takes a step to investigate the time-consistent Nash equilibrium strategies for a multiperiod mean-variance portfolio selection problem. Under the assumption that the risk aversion is, respectively, a constant and a function of current wealth level, we obtain the explicit expressions for the time-consistent Nash equilibrium strategy and the equilibrium value function. Many interesting properties of the time-consistent results are identified through numerical sensitivity analysis and by comparing them with the classical pre-commitment solutions.

Based on the F-expansion method, and the extended version of F-expansion method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation. With the aid of Maple, more exact solutions expressed by Jacobi elliptic function are obtained. When the modulus m of Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained.

This paper is concerned with the dynamical stability analysis for almost periodic solution of memristive neural networks with time-varying delays. Under the framework of Filippov solutions, by applying the inequality analysis techniques, the existence and asymptotically almost periodic behavior of solutions are discussed. Based on the differential inclusions theory and Lyapunov functional approach, the stability issues of almost periodic solution are investigated, and a sufficient condition for the existence, uniqueness, and global exponential stability of the almost periodic solution is established. Moreover, as a special case, the condition which ensures the global exponential stability of a unique periodic solution is also presented for the considered memristive neural networks. Two examples are given to illustrate the validity of the theoretical results.

Nowadays, many distribution networks deal with the distribution and storage of perishable products. However, distribution network design models are largely based on assumptions that do not consider time limitations for the storage of products within the network. This study develops a model for the design of a distribution network that considers the short lifetime of perishable products. The model simultaneously determines the network configuration and inventory control decisions of the network. Moreover, as the lifetime is strictly dependent on the storage conditions, the model develops a trade-off between enhancing storage conditions (higher inventory cost) to obtain a longer lifetime and selecting those storage conditions that lead to shorter lifetimes (less inventory cost). To solve the model, an efficient Lagrangian relaxation heuristic algorithm is developed. The model and algorithm are validated by sensitivity analysis on some key parameters. Results show that the algorithm finds optimal or near optimal solutions even for large-size cases.

A SIR epidemic model is proposed to understand the impact of limited medical resource on infectious disease transmission. The basic reproduction number is identified. Existence and stability of equilibria are obtained under different conditions. Bifurcations, including backward bifurcation and Hopf bifurcation, are analyzed. Our results suggest that the model considering the impact of limited medical resource may exhibit vital dynamics, such as bistability and periodicity when the basic reproduction number ${ℝ}_0$ is less than unity, which implies that the basic reproductive number itself is not enough to describe whether the disease will prevail or not and a subthreshold number is needed. It is also shown that a sufficient number of sickbeds and other medical resources are very important for disease control and eradication. Considering the costs, we provide a method to estimate a suitable treatment capacity for a disease in a region.

We introduce the weighted mixed almost unbiased ridge estimator (WMAURE) based on the weighted mixed estimator (WME) (Trenkler and Toutenburg 1990) and the almost unbiased ridge estimator (AURE) (Akdeniz and Erol 2003) in linear regression model. We discuss superiorities of the new estimator under the quadratic bias (QB) and the mean square error matrix (MSEM) criteria. Additionally, we give a method about how to obtain the optimal values of parameters $k$ and $\mathrm{w}$. Finally, theoretical results are illustrated by a real data example and a Monte Carlo study.

In this paper, we propose an iterative spectral method for solving differential equations with initial values on large intervals. In the proposed method, we first extend the Legendre wavelet suitable for large intervals, and then the Legendre-Guass collocation points of the Legendre wavelet are derived. Using this strategy, the iterative spectral method converts the differential equation to a set of algebraic equations. Solving these algebraic equations yields an approximate solution for the differential equation. The proposed method is illustrated by some numerical examples, and the result is compared with the exponentially fitted Runge-Kutta method. Our proposed method is simple and highly accurate.

This paper consists of main two sections. In the first section, we prove a common fixed point theorem in modified intuitionistic fuzzy metric space by combining the ideas of pointwise $R$-weak commutativity and reciprocal continuity of mappings satisfying contractive conditions. In the second section, we prove common fixed point theorems in modified intuitionistic fuzzy metric space from the class of compatible continuous mappings to noncompatible and discontinuous mappings. Lastly, as an application, we prove fixed point theorems using weakly reciprocally continuous noncompatible self-mappings on modified intuitionistic fuzzy metric space satisfying some implicit relations.

Erratum to “Common Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces”dx.doi.org/10.1155/2014/486102

Given n objects and an $n\times n$ symmetric dissimilarity matrix D with zero main diagonal and nonnegative off-diagonal entries, the least-squares unidimensional scaling problem asks to find an arrangement of objects along a straight line such that the pairwise distances between them reflect dissimilarities represented by the matrix D. In this paper, we propose an improved branch-and-bound algorithm for solving this problem. The main ingredients of the algorithm include an innovative upper bounding technique relying on the linear assignment model and a dominance test which allows considerably reducing the redundancy in the enumeration process. An initial lower bound for the algorithm is provided by an iterated tabu search heuristic. To enhance the performance of this heuristic we develop an efficient method for exploring the pairwise interchange neighborhood of a solution in the search space. The basic principle and formulas of the method are also used in the implementation of the dominance test. We report computational results for both randomly generated and real-life based problem instances. In particular, we were able to solve to guaranteed optimality the problem defined by a $36\times 36$ Morse code dissimilarity matrix.

Nonsmooth mechanical systems, which are mechanical systems involving dry friction and rigid unilateral contact, are usually described as differential inclusions (DIs), that is, differential equations involving discontinuities. Those DIs may be approximated by ordinary differential equations (ODEs) by simply smoothing the discontinuities. Such approximations, however, can produce unrealistic behaviors because the discontinuous natures of the original DIs are lost. This paper presents a new algebraic procedure to approximate DIs describing nonsmooth mechanical systems by ODEs with preserving the discontinuities. The procedure is based on the fact that the DIs can be approximated by differential algebraic inclusions (DAIs), and thus they can be equivalently rewritten as ODEs. The procedure is illustrated by some examples of nonsmooth mechanical systems with simulation results obtained by the fourth-order Runge-Kutta method.

Nonautonomous systems with periodic solutions are encountered frequently in applications. In this paper, we will consider simple systems whose solutions are periodic with a known period. Their transformation under linearized collocation methods is investigated, using a technique called stroboscopic sampling, a discrete version of the well-known Poincaré map. It is shown that there is an inextricable relationship between AN stability (or BN stability) of the numerical methods and the correct qualitative behaviour of solutions.

This paper presents a novel and general approach, which is based on the composite control method, to synthesize the controller and observer-based state feedback to stabilize the singularly perturbed time-delay systems. First, the equivalent models of the original systems and the subsystems reduced via singular perturbation techniques are derived. Through these equivalent models, $O(\epsilon )$ approximation of the stabilization and observer design for the original systems can be achieved through separate analyses for the slow and fast subsystems via a transformation of block diagonalization.

This paper investigates the synchronization of complex dynamical networks with coupling delays and external disturbances by applying local feedback injections to a small fraction of nodes in the whole network. Based on $H_{\infty }$ control theory, some delay-independent and -dependent synchronization criteria with a prescribed $H_{\infty }$ disturbances attenuation index are derived for such controlled networks in terms of linear matrix inequalities (LMIs), which guarantee that by placing a small number of feedback controllers on some nodes, the whole network can be pinned to reach network synchronization. A simulation example is included to validate the theoretical results.

The spatiotemporal dynamics of a phytoplankton-zooplankton model with an alternative prey and refuge effect is investigated mathematically and numerically. The stability of the equilibrium point and the traveling wave solution of the phytoplankton-zooplankton model are described based on theoretical mathematical work, which provides the basis of the numerical simulation. The numerical analysis shows that refuges have a strong effect on the spatiotemporal dynamics of the model according to the pattern formation. These results may help us to understand prey-predator interactions in water ecosystems. They are also relevant to research into phytoplankton-zooplankton ecosystems.

The stabilized Gauge-Uzawa method (SGUM), which is a 2nd-order projection type algorithm used to solve Navier-Stokes equations, has been newly constructed in the work of Pyo, 2013. In this paper, we apply the SGUM to the evolution Boussinesq equations, which model the thermal driven motion of incompressible fluids. We prove that SGUM is unconditionally stable, and we perform error estimations on the fully discrete finite element space via variational approach for the velocity, pressure, and temperature, the three physical unknowns. We conclude with numerical tests to check accuracy and physically relevant numerical simulations, the Bénard convection problem and the thermal driven cavity flow.

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.

We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained. This approach can be suitable for solving fractional partial differential equations with more general forms than the method proposed by S. Zhang and H.-Q. Zhang (2011).

A radial basis function (RBF) neural network adaptive sliding mode control system is developed for the current compensation control of three-phase active power filter (APF). The advantages of the adaptive control, neural network control, and sliding mode control are combined together to achieve the control task; that is, the harmonic current of nonlinear load can be eliminated and the quality of power system can be well improved. Sliding surface coordinate function and sliding mode controller are used as input and output of the RBF neural network, respectively. The neural network control parameters are online adjusted through gradient method and Lyapunov theory. Simulation results demonstrate that the adaptive RBF sliding mode control can compensate harmonic current effectively and has strong robustness to disturbance signals.

The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely, the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

The motion of the viscous, incompressible fluid through a porous medium with heat and mass transfer over a shrinking sheet is investigated. The cross-diffusion effect between temperature and concentration is considered. This phenomenon is modulated mathematically by a set of partial differential equations which govern the continuity, momentum, heat, and mass. These equations are transformed to a set of ordinary differential equations by using similarity solutions. The analytical solutions of these equations are obtained. The velocity, temperature, and concentration of the fluid as well as the heat and mass transfer with shear stress at the sheet are obtained as a function of the physical parameters of the problem. The effects of Prandtl number, mass transfer parameter, the wall shrinking parameter, the permeability parameter, and Dufour and Soret numbers on temperature and concentration are studied. Also, the effects of mass transfer parameter, permeability parameter, and shrinking strength on the velocity and shear stress are discussed. These effects are illustrated graphically through a set of figures.

The positive definite solutions of the nonlinear matrix equation $X^s+A^{\ast }f(X)A=Q$ are discussed. A necessary and sufficient condition for the existence of positive definite solutions for this equation is derived. Then, the uniqueness of the Hermitian positive definite solution is studied based on an iterative method proposed in this paper. Lastly the perturbation analysis for this equation is discussed.

We introduce two powerful methods to solve the Davey-Stewartson equations: one is the homotopy perturbation method (HPM) and the other is the homotopy analysis method (HAM). HAM is a strong and easy to use analytic tool for nonlinear problems. Comparison of the HPM results with the HAM results, and compute the absolute errors between the exact solutions of the DS equations with the HPM solutions and HAM solutions are obtained.

Emergency incidents, including natural disasters, terrorist attacks, public health outbreaks, and industrial and mining accidents, and so forth, result in severe human casualties and property losses. Emergency facilities, which provide relief materials and services, play an important role in rescue management. The decision of where to locate the emergency rescue facilities is very important, as it determines the efficiency and effectiveness of the emergency management process. This paper develops a modified $p$-median problem model that accounts for rescue time limitations. A variable neighbourhood search- (VNS-) based algorithm is developed for the model considered. The modified VNS algorithm exhibits good performance on $p$-median benchmark problems. A case from Western China is studied, and a reasonable location decision is then made for emergency rescue facilities using the modified VNS algorithm. The paper also compares the results with and without considering the rescue time limitation.

In this paper, an HIV infection model including an eclipse stage of infected cells is considered. Some quicker cells in this stage become productively infected cells, a portion of these cells are reverted to the uninfected class, and others will be latent down in the body. We consider CTL-response delay in this model and analyze the effect of time delay on stability of equilibrium. It is shown that the uninfected equilibrium and CTL-absent infection equilibrium are globally asymptotically stable for both ODE and DDE model. And we get the global stability of the CTL-present equilibrium for ODE model. For DDE model, we have proved that the CTL-present equilibrium is locally asymptotically stable in a range of delays and also have studied the existence of Hopf bifurcations at the CTL-present equilibrium. Numerical simulations are carried out to support our main results.

The notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada index of CSQG whose diameter does not exceed two. Additionally, we show that our results improve most of the results obtained by Güngör and Bozkurt (2009) and Zaferani (2008).

Annual Crop Planning (ACP) is an NP-hard-type optimization problem in agricultural planning. It involves finding optimal solutions concerning the seasonal allocations of a limited amount of agricultural land amongst the various competing crops that are required to be grown on it. This study investigates the effectiveness of employing three new local search (LS) metaheuristic techniques in determining solutions to an ACP problem at a new Irrigation Scheme. These three new LS metaheuristic techniques are the Best Performance Algorithm (BPA), Iterative Best Performance Algorithm (IBPA), and the Largest Absolute Difference Algorithm (LADA). The solutions determined by these LS metaheuristic techniques are compared against the solutions of two other well-known LS metaheuristic techniques in the literature. These techniques are Tabu Search (TS) and Simulated Annealing (SA). The comparison with TS and SA was to determine the relative merits of the solutions found by BPA, IBPA, and LADA. The results show that TS performed as the overall best. However, LADA determined the best solution that was the most economically feasible.

This paper is concerned with accurate and efficient numerical methods for solving viscous and nonviscous wave problems. The paper first introduces a new second-order PR-ADI like scheme. For an efficient simulation, the scheme is also extended to a high-order compact PRADI like method. Both of them have the advantages of unconditional stability, less impact of the perturbing terms on the accuracy, and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and costs less in computational time. Numerical results are presented to show the accuracy and efficiency of the new algorithms.

A numerical method for nonlinear fractional-order differential equations with constant or time-varying delay is devised. The order here is an arbitrary positive real number, and the differential operator is with the Caputo definition. The general Adams-Bashforth-Moulton method combined with the linear interpolation method is employed to approximate the delayed fractional-order differential equations. Meanwhile, the detailed error analysis for this algorithm is given. In order to compare with the exact analytical solution, a numerical example is provided to illustrate the effectiveness of the proposed method.

Cell formation (CF) is a crucial aspect in the design of cellular manufacturing (CM) systems. This paper develops a comprehensive mathematical programming model for the cell formation problem, where product demands, cell size limits, sequence of operations, multiple units of identical machines, machine capacity, or machine cost are all considered. In this model, the intercell moves are restricted to be unidirectional from one cell to the downstream cells, without backtracking. The proposed model is investigated through several numerical examples. To evaluate the solution quality of the proposed model, it is compared with some well-known cell formation methods from the literature, by using group capability index (GCI) as a performance measure. The results and comparisons indicate that the proposed model produces solution with a higher performance.

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of int-soft positive implicative ideals is introduced, and related properties are investigated. Relations between an int-soft ideal and an int-soft positive implicative ideal are established. Characterizations of an int-soft positive implicative ideal are obtained. Extension property for an int-soft positive implicative ideal is constructed. The $\wedge$-product and $\vee$-product of int-soft positive implicative ideals are considered, and the soft intersection (resp., union) of int-soft positive implicative ideals is discussed.