The output tracking problem for a class of uncertain strict-feedback nonlinear systems with unknown Duhem hysteresis input is investigated. In order to handle the undesirable effects caused by unknown hysteresis, the properties in respect to Duhem model are used to decompose it as a nonlinear smooth term and a nonlinear bounded “disturbance-like” term, which makes it possible to deal with the unknown hysteresis without constructing inverse in the controller design. By combining robust control and dynamic surface control technique, an adaptive controller is proposed in this paper to avoid “the explosion complexity” in the standard backstepping design procedure. The negative effects caused by the unknown hysteresis can be mitigated effectively, and the semiglobal uniform ultimate boundedness of all the signals in the closed-loop system is obtained. The effectiveness of the proposed scheme is validated through a simulation example.

Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if $X$ is approximately compact and $X$ is 2-strictly convex, then metric generalized inverses of bounded linear operators in $X$ are upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, a sufficient condition for set-valued mapping $T^{\partial }$ to be continuous mapping is given.

The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée-Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which transform the sequence belonging to the space of absolutely convergent double series into the space of generalized almost convergence.

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

The Bäcklund transformation of fractional Riccati equation with nonlinear superposition principle of solutions is employed to establish the infinite sequence solutions of nonlinear fractional partial differential equations in the sense of modified Riemann-Liouville derivative. To illustrate the reliability of the method, some examples are provided.

We introduce the notion of h-stability for fractional differential systems. Then we investigate the boundedness and h-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give examples to illustrate our results.

We deal with the stability of the exponential Cauchy functional equation $F(x+y)=F(x)F(y)$ in the class of functions $F:G\to L$ mapping a group ($G$, +) into a Riesz algebra $L$. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.

Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction number $R_0$. Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that if $R_0\le 1$, the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. If $R_0>1$, a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.

The existence of uncountably many positive solutions and Mann iterative approximations for a nonlinear three-dimensional difference system are proved by using the Banach fixed point theorem. Four illustrative examples are also provided.

We will determine the nonspectrality of self-affine measure ${\mu }_{B,D}$ corresponding to $B=\mathrm{diag}[p_1,p_2,p_3]\text{(}{p}_1\in (2\mathbb{Z}+1)\setminus \{\pm 1\}$, $p_2\in 2\mathbb{Z}\setminus \{0\})$, and $D=\{0,e_1,e_2,e_3\}$ in the space ${ℝ}^3$ is supported on $T(B,D)$, where $e_1,e_2$, and $e_3$ are the standard basis of unit column vectors in ${ℝ}^3$, and there exist at most $4$ mutually orthogonal exponential functions in $L^2({\mu }_{B,D})$, where the number $4$ is the best. This generalizes the known results on the spectrality of self-affine measures.

This paper considers the pricing of turbo warrants under a hybrid stochastic and local volatility model. The model consists of the constant elasticity of variance model incorporated by a fast fluctuating Ornstein-Uhlenbeck process for stochastic volatility. The sensitive structure of the turbo warrant price is revealed by asymptotic analysis and numerical computation based on the observation that the elasticity of variance controls leverage effects and plays an important role in characterizing various phases of volatile markets.

We firstly prove that $\beta$-times integrated $\alpha$-resolvent operator function ($(\alpha ,\beta )$-ROF) satisfies a functional equation which extends that of $\beta$-times integrated semigroup and $\alpha$-resolvent operator function. Secondly, for the inhomogeneous $\alpha$-Cauchy problem ${\mathrm{}}^c{D}_t^{\alpha }u(t)=Au(t)+f(t)$, $\mathrm{}\mathrm{}t\in (\mathrm{0},T)$, $\mathrm{}u(\mathrm{0})=x_{\mathrm{0}}$, $\mathrm{}\mathrm{}u\mathrm{\text{'}}(\mathrm{0})=x_{\mathrm{1}},$ if $A$ is the generator of an $(\alpha ,\beta )$-ROF, we give the relation between the function $v(t)=S_{\alpha ,\beta }(t)x_{\mathrm{0}}+(g_{\mathrm{1}}\mathrm{*}{S}_{\alpha ,\beta })(t)x_{\mathrm{1}}+(g_{\alpha -\mathrm{1}}\mathrm{*}{S}_{\alpha ,\beta }\mathrm{*}f)(t)$ and mild solution and classical solution of it. Finally, for the problem ${\mathrm{}}^c{D}_t^{\alpha }v(t)=Av(t)+g_{\beta +\mathrm{1}}(t)x$, $t>\mathrm{0}$, $\mathrm{}{v}^{(k)}(\mathrm{0})=\mathrm{0}$, $k=\mathrm{0,1},\dots ,N-\mathrm{1},$ where $A$ is a linear closed operator. We show that $A$ generates an exponentially bounded $(\alpha ,\beta )$-ROF on a Banach space $X$ if and only if the problem has a unique exponentially bounded classical solution $v_x$ and $A{v}_x\in L_{\text{l}\text{o}\text{c}}^{\mathrm{1}}({ℝ}^{+},X).$ Our results extend and generalize some related results in the literature.

This paper investigates the finite-time distributed $L_2\text{–}{L}_{\infty }$ consensus control problem of multiagent systems with parameter uncertainties. The relative states of neighboring agents are used to construct the control law and some agents know their own states. By substituting the control input into multiagent systems, an augmented closed-loop system is obtained. Then, we analyze its finite-time boundedness (FTB) and finite-time $L_2\text{–}{L}_{\infty }$ performance. A sufficient condition for the existence of the designed controller is given with the form of linear matrix inequalities (LMIs). Finally, simulation results are described.

We investigate the value distribution of difference product $f(z{)}^n{\sum }_{i=\mathrm{1}}^k\mathrm{‍}{a}_{i}f(z+c_i)$, for $n\ge \mathrm{2}$ and $n=\mathrm{1}$, respectively, where $f(z)$ is a transcendental entire function of finite order and $a_i,\mathrm{}{c}_i$ are constants satisfying ${\sum }_{i=\mathrm{1}}^k\mathrm{‍}{a}_{i}f(z+c_i)\not\equiv \mathrm{0}$.

We study existence of solutions for the fractional Laplacian equation ${\left(-\Delta \right)}^{s}u+V\left(x\right)u={\left|u\right|}^{2^{*}\left(s\right)-2}u+f\left(x, u\right)$ in ${ℝ}^N$, $u\in H^s(R^N)$, with critical exponent $2^{*}\left(s\right)=2N/(N-2s)$, $N>2s$, $s\in \left(0, 1\right)$, where $V\left(x\right)\ge 0$ has a potential well and $f:{ℝ}^N\times ℝ\to ℝ$ is a lower order perturbation of the critical power ${\left|u\right|}^{2^{*}\left(s\right)-2}u$. By employing the variational method, we prove the existence of nontrivial solutions for the equation.

The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of the $C^{\infty }$ solutions in non-Maxwellian and strong singularity cases.

This paper is concerned with the existence and nonexistence of positive solutions for a nonlinear higher-order three-point boundary value problem. The existence results are obtained by applying a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O’Regan.

This paper studies the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations of order $\mathrm{\alpha ,\beta }\in (\mathrm{4,5}]$ with antiperiodic boundary conditions. Our results are based on the nonlinear alternative of Leray-Schauder type and the contraction mapping principle. Two illustrative examples are also presented.

Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.

A new approach is presented for obtaining the solutions to Yakubovich-$j$-conjugate quaternion matrix equation $X-A\widehat{X}B=CY$ based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix $A$. The closed form solution is established and the equivalent form of solution is given for this Yakubovich-$j$-conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation $X-A\overline{X}B=CY$ is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich-$j$-conjugate quaternion matrix equation $X-A\widehat{X}B=CY$. Numerical example shows the effectiveness of the proposed results.

By using some lattice-valued Kowalsky’s dual diagonal conditions, some weaker regularities for Jäger’s generalized stratified $L$-convergence spaces and those for Boustique et al’s stratified $L$-convergence spaces are defined and studied. Here, the lattice $L$ is a complete Heyting algebra. Some characterizations and properties of weaker regularities are presented. For Jäger’s generalized stratified $L$-convergence spaces, a notion of closures of stratified $L$-filters is introduced and then a new $p$-regularity is defined. At last, the relationships between $p$-regularities and weaker regularities are established.

This paper considers the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. First, the concept of finite time stability is extended to stochastic hybrid systems. Then, by using common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of stochastic hybrid systems are presented. Furthermore, a new notion called stochastic minimum dwell time is proposed and then, combining it with the method of multiple Lyapunov functions, a sufficient condition for finite time stability of stochastic hybrid systems is given. Finally, a numerical example is provided to illustrate the theoretical results.

We put side by side the methodology of two comparatively new analytical techniques to get to the bottom of the system of nonlinear fractional modified Kawahara equation. The technique is described and exemplified with a numerical example. The dependability of both methods and the lessening in computations give these methods a wider applicability. In addition, the computations implicated are very simple and undemanding.

The molecular compressibility, which is a macroscopic quantity to reveal the microcompressibility by additivity of molecular constitutions, is considered as a fixed value for specific organic liquids. In this study, we introduced two calculated expressions of molecular adiabatic compressibility to demonstrate its pressure and temperature dependency. The first one was developed from Wada’s constant expression based on experimental data of density and sound velocity. Secondly, by introducing the 2D fitting expressions and their partial derivative of pressure and temperature, molecular compressibility dependency was analyzed further, and a 3D fitting expression was obtained from the calculated data of the first one. The third was derived with introducing the pressure and temperature correction factors based on analogy to Lennard-Jones potential function and energy equipartition theorem. In wide range of temperatures and pressures , which represent the typical values used in dynamic injection process for diesel engines, the calculated results consistency of three formulas demonstrated their effectiveness with the maximum 0.5384% OARD; meanwhile, the dependency on pressure and temperature of molecular compressibility was certified.

We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay. The Bezier curves method can solve boundary value problems for singularly perturbed differential-difference equations. The approximation process is done in two steps. First we divide the time interval, into $k$ subintervals; second we approximate the trajectory and control functions in each subinterval by Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degree $n$ and determined Bezier curves on any subinterval by $n+1$ control points. The proposed method is simple and computationally advantageous. Several numerical examples are solved using the presented method; we compared the computed result with exact solution and plotted the graphs of the solution of the problems.

We study the existence and uniqueness of positive solutions for a class of quasilinear elliptic equations. This model has been proposed in the self-channeling of a high-power ultrashort laser in matter.

The existence of three weak solutions for the following nonlocal fractional equation $(-\mathrm{\Delta }{)}^{s}u-\lambda u=\mu f(x,u)$ in $\mathrm{}\mathrm{}\mathrm{\Omega },\mathrm{}u=\mathrm{0}\mathrm{}$ in ${ℝ}^n\setminus \mathrm{\Omega },$ is investigated, where $s\in (\mathrm{0,1})$ is fixed, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplace operator, $\lambda$ and $\mu$ are real parameters, $\mathrm{\Omega }$ is an open bounded subset of ${ℝ}^n$, $n>\mathrm{2}s$, and the function $f$ satisfies some regularity and natural growth conditions. The approach is based on a three-critical-point theorem for differential functionals.

We establish the necessary condition of optimality for optimal control problem governed by some pseudoparabolic differential equations involving monotone graphs. Some approximating control process and examples are given.

Double Laplace transform method was applied to evaluate the exact value of double infinite series. Further we generalize the current existing methods and provide some examples to illustrate and verify that the present method is a more general technique.

By the weighted ergodic space, we propose a new class of functions called weighted Stepanov-like pseudoperiodic function and explore its properties. Furthermore, the existence and uniqueness of the weighted pseudoperiodic solution to fractional integro-differential equations and nonautonomous differential equations are investigated. Some interesting examples are presented to illustrate the main findings.

The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate $Q$ and monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given.

Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.

We consider the three-dimensional Boussinesq equations and obtain a regularity criterion involving the pressure gradient in the Morrey-Companato space $M_{p,q}$. This extends and improves the result of Gala (Gala 2013) for the Navier-Stokes equations.

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

In this paper we established some vector-valued inequalities of Gronwall type in ordered Banach spaces. Our results can be applied to investigate systems of real-valued Gronwall-type inequalities. We also show that the classical Gronwall-Bellman-Bihari integral inequality can be generalized from composition operators to a variety of operators, which include integral operators, maximal operators, geometric mean operators, and geometric maximal operators.

In this work we establish some existence results for abstract second order Cauchy problems modeled by a retarded differential inclusion involving nonlocal and impulsive conditions. Our results are obtained by using fixed point theory for the measure of noncompactness.

A model of the interactions among a host population, an insect-vector population, which transmits virus from hosts to hosts, and a vector predator population is proposed based on virus-host, host-vector, and prey (vector)-enemy theories. The model is investigated to explore the indirect effect of natural enemies on host-virus dynamics by reducing the vector densities, which shows the basic reproduction numbers $R_{01}$ (without predators) and $R_{02}$ (with predators) that provide threshold conditions on determining the uniform persistence and extinction of the disease in a host population. When the model is absent from predator, the disease is persistent if $R_{01}>1$; in such a case, by introducing predators of a vector, then the insect-transmitted disease will be controlled if . From the point of biological control, these results show that an additional predator population of the vector may suppress the spread of vector-borne diseases. In addition, there exist limit cycles with persistence of the disease or without disease in presence of predators. Finally, numerical simulations are conducted to support analytical results.

In a recent paper (Zhang (2013)), the author claims that he has proposed two rules to modify Ibragimov’s theorem on conservation laws to “ensure the theorem can be applied to nonlinear evolution equations with any mixed derivatives.” In this letter, we analysis the paper. Indeed, the so-called “modification rules” are needless and the theorem of Ibragimov can be applied to construct conservation laws directly for nonlinear equations with any mixed derivatives as long as the formal Lagrangian is rewritten in symmetric form. Moreover, the conservation laws obtained by the so-called “modification rules” in the paper under discussion are equivalent to the one obtained by Ibragimov’s theorem.

We study the existence of solutions and optimal controls for some fractional impulsive equations of order . By means of Gronwall’s inequality and Leray-Schauder’s fixed point theorem, the sufficient condition for the existence of solutions and optimal controls is presented. Finally, an example is given to illustrate our main results.

In the first part of the paper, following the works of Pehlivan et al. (2004), we study the set of all A-statistical cluster points of sequences in m-dimensional spaces and make certain investigations on the set of all A-statistical cluster points of sequences in m-dimensional spaces. In the second part of the paper, we apply this notion to study an asymptotic behaviour of optimal paths and optimal controls in the problem of optimal control in discrete time and prove a general version of turnpike theorem in line of the work of Mamedov and Pehlivan (2000). However, all results of this section are presented in terms of a more general notion of $ℐ$-cluster points.

The drive system can synchronize with the response system by the scaling factor in the traditional projective synchronization. This paper proposes a novel adaptive hybrid dislocated synchronization with uncertain parameters scheme for chaos synchronization using the Lyapunov stability theory. The drive system is synchronized by the sum of hybrid dislocated state variables for the response system. By designing effective hybrid dislocated adaptive controller and hybrid dislocated adaptive law of the parameters estimation, we investigate the synchronization of two identical memristor chaotic oscillator systems and two different memristor chaotic oscillator systems with uncertain parameters. Finally, the numerical simulation examples are provided to show the effectiveness of our method.

Let $f\left(z\right)$ be a meromorphic function in $ℂ$, and let $\alpha \left(z\right)=R\left(z\right)h\left(z\right)\not\equiv 0$, where $h\left(z\right)$ is a nonconstant elliptic function and $R\left(z\right)$ is a rational function. Suppose that all zeros of $f\left(z\right)$ are multiple except finitely many and $T\left(r,\alpha \right)=o\left\{T\left(r,f\right)\right\}$ as $r\to \infty$. Then $f^{\text{'}}\left(z\right)=\alpha \left(z\right)$ has infinitely many solutions.

The tripled fixed point is a generalization of the well-known concept of “coupled fixed point.” In this paper, tripled coincidence and common fixed point results for two hybrid pairs consisting of multivalued and single valued mappings on a metric space are proved. We give examples to illustrate our results. In the process, several comparable coincidence and fixed point results in the existing literature are improved, unified, and generalized.

A perfect achievement has been made for wavelet density estimation by Dohono et al. in 1996, when the samples without any noise are independent and identically distributed (i.i.d.). But in many practical applications, the random samples always have noises, and estimation of the density derivatives is very important for detecting possible bumps in the associated density. Motivated by Dohono's work, we propose new linear and nonlinear wavelet estimators ${\widehat{f}}_{\text{lin}}^{(m)},{\widehat{f}}_{\text{non}}^{(m)}$ for density derivatives $f^{(m)}$ when the random samples have size-bias. It turns out that the linear estimation $E(\parallel {\widehat{f}}_{\text{lin}}^{(m)}-f^{(m)}{\parallel }_p)$ for $f^{(m)}\in B_{r,q}^s(A,L)$ attains the optimal covergence rate when $r\ge p$, and the nonlinear one $E(\parallel {\widehat{f}}_{\text{lin}}^{(m)}-f^{(m)}{\parallel }_p)$ does the same if .

This paper investigates the problem of state-feedback stabilization for a class of stochastic high-order nonlinear systems with time-varying delays. Under the weaker conditions on the power order and the nonlinear growth, by using the method of adding a power integrator, a state-feedback controller is successfully designed, and the global asymptotic stability in the probability of the resulting closed-loop system is proven with the help of an appropriate Lyapunov-Krasovskii functional. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.

The purpose of this paper is to introduce a new Kirk type iterative algorithm called Kirk multistep iteration and to study its convergence. We also prove some theorems related to the stability results for the Kirk multistep and Kirk-SP iterative processes by employing certain contractive-like operators. Our results generalize and unify some other results in the literature.

This paper investigates the combination-combination synchronization of four nonlinear complex chaotic systems. Based on the Lyapunov stability theory, corresponding controllers to achieve combination-combination synchronization among four different nonlinear complex chaotic systems are derived. The special cases, such as combination synchronization and projective synchronization, are studied as well. Numerical simulations are given to illustrate the theoretical analysis.

Maneuvering target tracking is a challenge. Target’s sudden speed or direction changing would make the common filtering tracker divergence. To improve the accuracy of maneuvering target tracking, we propose a tracking algorithm based on spline fitting. Curve fitting, based on historical point trace, reflects the mobility information. The innovation of this paper is assuming that there is no dynamic motion model, and prediction is only based on the curve fitting over the measured data. Monte Carlo simulation results show that, when sea targets are maneuvering, the proposed algorithm has better accuracy than the conventional Kalman filter algorithm and the interactive multiple model filtering algorithm, maintaining simple structure and small amount of storage.

We investigate further the invariance properties of the bounded linear operator product ${AC}^{\left(1\right)} B^{\left(1\right)} D$ and its range with respect to the choice of the generalized inverses $X$ and $Y$ of bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses.

We introduce the modified iterations of Mann's type for nonexpansive mappings and asymptotically nonexpansive mappings to have the strong convergence in a uniformly convex Banach space. We study approximation of common fixed point of asymptotically nonexpansive mappings in Banach space by using a new iterative scheme. Applications to the accretive operators are also included.

A Mahalanobis hyperellipsoidal learning machine class incremental learning algorithm is proposed. To each class sample, the hyperellipsoidal that encloses as many as possible and pushes the outlier samples away is trained in the feature space. In the process of incremental learning, only one subclassifier is trained with the new class samples. The old models of the classifier are not influenced and can be reused. In the process of classification, considering the information of sample’s distribution in the feature space, the Mahalanobis distances from the sample mapping to the center of each hyperellipsoidal are used to decide the classified sample class. The experimental results show that the proposed method has higher classification precision and classification speed.

We characterize the boundedness and compactness of the Hankel operator with conjugate analytic symbols on the weighted $L^P$-Bergman spaces with exponential type weights.

We study the existence of periodic solutions to a discrete $p$-Laplacian system. By using the Clark duality method and computing the critical groups, we find a simple condition that is sufficient to ensure the existence of nonconstant periodic solutions to the system.

The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.

This paper investigates the global stabilization problem for a class of nonholonomic systems in chained form with input delay. A particular transformation is introduced to convert the original time-delay system into a delay-free form. Then, by using input-state-scaling technique and the method of sliding mode control, a constructive design procedure for state feedback control is given, which can guarantee that all the system states globally asymptotically converge to the origin. An illustrative example is also provided to demonstrate the effectiveness of the proposed scheme.

The problem of finite-time boundedness for a class of switched linear systems with time-varying delay and external disturbance is investigated. First of all, the multiply Lyapunov function of the system is constructed. Then, based on the Jensen inequality approach and the average dwell time method, the sufficient conditions which guarantee the system is finite-time bounded are given. Finally, an example is employed to verify the validity of the proposed method.

The problem of fuzzy-based direct adaptive tracking control is considered for a class of pure-feedback stochastic nonlinear systems. During the controller design, fuzzy logic systems are used to approximate the packaged unknown nonlinearities, and then a novel direct adaptive controller is constructed via backstepping technique. It is shown that the proposed controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages lie in that the proposed controller structure is simpler and only one adaptive parameter needs to be updated online. Simulation results are used to illustrate the effectiveness of the proposed approach.

The exact boundary controllability of a class of enhanced oil recovery systems is discussed in this paper. With a simple transformation, the enhanced oil recovery model is first affirmed to be neither genuinely nonlinear nor linearly degenerate. It is then shown that the enhanced oil recovery system with nonlinear boundary conditions is exactly boundary controllable by applying a constructed method. Moreover, an interval of the control time is presented to not only give the optimal control time but also show the time for avoiding the blowup of the controllable solution. Finally, an example is given to illustrate the effectiveness of the proposed criterion.

This paper is concerned with the eigenvalues of perturbed higher-order discrete vector boundary value problems. A suitable admissible function space is first introduced, a new variational formula of eigenvalues is then established under certain nonsingularity conditions, and error estimates of eigenvalues of problems with small perturbation are finally given by using the variational formula. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the nonsingularity conditions. In addition, two special perturbed cases are discussed.

We show a necessary and sufficient condition for the existence of metric projection on a class of half-space $K_{x_0^{*},c}=\{x\in X:x^{*}(x)\le c\}$ in Banach space. Two representations of metric projections $P_{K_{x_0^{*},c}}$ and $P_{K_{x_0,c}}$ are given, respectively, where $K_{x_0,c}$ stands for dual half-space of $K_{x_0^{*},c}$ in dual space $X^{*}$. By these representations, a series of continuity results of the metric projections $P_{K_{x_0^{*},c}}$ and $P_{K_{x_0,c}}$ are given. We also provide the characterization that a metric projection is a linear bounded operator.

We define new subclasses of meromorphic $p$-valent functions by using certain differential operator. Combining the differential operator and certain integral operator, we introduce a general $p$-valent meromorphic function. Then we prove the sufficient conditions for the function in order to be in the new subclasses.

We study optimization problems involving eigenvalues of symmetric matrices. We present a nonsmooth optimization technique for a class of nonsmooth functions which are semi-infinite maxima of eigenvalue functions. Our strategy uses generalized gradients and $𝒰𝒱$ space decomposition techniques suited for the norm and other nonsmooth performance criteria. For the class of max-functions, which possesses the so-called primal-dual gradient structure, we compute smooth trajectories along which certain second-order expansions can be obtained. We also give the first- and second-order derivatives of primal-dual function in the space of decision variables $R^m$ under some assumptions.

Let $({ℍ}^n,g)$ be the hyperbolic space of dimension $n$. By our previous work (Theorem 2.3 of (Yang (2012))), for any , there exists a constant $\tau >\mathrm{0}$ depending only on $n$ and $\alpha$ such that where ${\alpha }_n=n{\omega }_{n-\mathrm{1}}^{\mathrm{1}/(n-\mathrm{1})}$, ${\omega }_{n-\mathrm{1}}$ is the measure of the unit sphere in ${ℝ}^n$, and ${∥u∥}_{1,\tau }={∥{\nabla }_{g}u∥}_{L^n({ℍ}^n)}+\tau {∥u∥}_{L^n({ℍ}^n)}$. In this note we shall improve the above mentioned inequality. Particularly, we show that, for any and any $\tau >\mathrm{0}$, the above mentioned inequality holds with the definition of ${∥u∥}_{1,\tau }$ replaced by $({\int }_{{ℍ}^n}\mathrm{‍}(|{\nabla }_{g}u{|}^n+\tau |u{|}^n)d{v}_g{)}^{\mathrm{1}/n}$. We solve this problem by gluing local uniform estimates.

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems $\ddot{u}+a\left(t\right)W_u\left(u\right)=0$, (HS) where , $u=\left(u_1,u_2, \dots ,u_N\right)\in {ℝ}^N\left(N\ge 3\right)$, $a:ℝ\to ℝ$ is a continuous bounded function, and the potential $W:{ℝ}^N\setminus \{\xi \}\to ℝ$ has a singularity at $0\ne \xi \in {ℝ}^N$, and $W_u\left(u\right)$ is the gradient of $W$ at $u$. The novelty of this paper is that, for the case that $N$ $\ge 3$ and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of $W$. Different from the cases that (HS) is autonomous $\left(a\left(t\right)\equiv 1\right)$ or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and $N$ $\ge 3$. Besides the usual conditions on $W$, we need the assumption that for all $t\in ℝ$ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

In this communication, we characterize a measure of information of types $\alpha$, $\beta$, and $\gamma$ by taking certain axioms parallel to those considered earlier by Havrda and Charvat along with the recursive relation $H_n$($p_1,\dots ,p_n$; $\alpha$, $\beta$, $\gamma$)$-H_{n-1}$ $(p_1+p_2$, $p_3,\dots ,$ $p_n$; $\alpha$, $\beta$, $\gamma$)$=$ ($A_{(\mathrm{\alpha },\mathrm{\gamma })}/$($A_{(\mathrm{\alpha },\mathrm{\gamma })}-A_{(\mathrm{\beta },\mathrm{\gamma })}$))${\left(p_1+p_2\right)}^{\mathrm{\alpha }/\mathrm{\gamma }}{H}_2(p_1/(p_1+p_2)$, $p_2/(p_1+p_2)$; $\alpha$,$\gamma )+$($A_{(\mathrm{\beta },\mathrm{\gamma })}/{\mathrm{(A}}_{(\mathrm{\beta },\mathrm{\gamma })}-A_{(\mathrm{\alpha },\mathrm{\gamma })}$))${(p_1+p_2)}^{(\mathrm{\beta }/\mathrm{\gamma })}{H}_2(p_1/(p_1+p_2)$, $p_2/(p_1+p_2)$; $\gamma$, $\beta$), $\alpha \ne \gamma \ne \beta$, $\alpha$, $\beta$, $\gamma >0$. Some properties of this measure are also studied. This measure includes Shannon’s information measure as a special case.

Feature combination is a powerful approach to improve object classification performance. While various combination algorithms have been proposed, average combination is almost always selected as the baseline algorithm to be compared with. In previous work we have found that it is better to use only a sample of the most powerful features in average combination than using all. In this paper, we continue this work and further show that the behaviors of features in average combination can be integrated into the k-Nearest-Neighbor (kNN) framework. Based on the kNN framework, we then propose to use a selection based average combination algorithm to obtain the best classification performance from average combination. Our experiments on four diverse datasets indicate that this selection based average combination performs evidently better than the ordinary average combination, and thus serves as a better baseline. Comparing with this new and better baseline makes the claimed superiority of newly proposed combination algorithms more convincing. Furthermore, the kNN framework is helpful in understanding the underlying mechanism of feature combination and motivating novel feature combination algorithms.

We consider a Schrödinger-Poisson system in ${ℝ}^3$ with a strongly indefinite potential and a general nonlinearity. Its variational functional does not satisfy the global linking geometry. We obtain a nontrivial solution and, in case of odd nonlinearity, infinitely many solutions using the local linking and improved fountain theorems, respectively.

This paper is concerned with the problem of general output feedback stabilization for fractional order linear time-invariant (FO-LTI) systems with the fractional commensurate order . The objective is to design suitable output feedback controllers that guarantee the stability of the resulting closed-loop systems. Based on the slack variable method and our previous stability criteria, some new results in the form of linear matrix inequality (LMI) are developed to the static and dynamic output feedback controllers synthesis for the FO-LTI system with . Furthermore, the results are extended to stabilize the FO-LTI systems with . Finally, robust output feedback control is discussed. Numerical examples are given to illustrate the effectiveness of the proposed design methods.

The object of this paper is to establish some nonlinear retarded inequalities on time scales which can be used as handy tools in the theory of integral equations with time delays.

New notions of ϵ-equilibrium flow and ${\xi }_{k_0}$-ϵ-equilibrium flow of multicriteria network equilibrium problem are introduced; an equivalent relation between vector ϵ-equilibrium pattern flow and ${\xi }_{k_0}$-ϵ-equilibrium flow is established. Then, the well-posedness of multicriteria network equilibrium problem is discussed.

The existence results of positive $\omega$-periodic solutions are obtained for the third-order ordinary differential equation with delays $u^{\mathrm{\prime \prime \prime }}(t)+a(t)u(t)=f(t,u(t-{\tau }_{\mathrm{0}}),u^{\prime }(t-{\tau }_{\mathrm{1}}),u^{\mathrm{\prime \prime }}(t-{\tau }_{\mathrm{2}})),t\in \mathrm{ℝ,}$ where $a\in C(ℝ,(\mathrm{0},\infty ))$ is $\omega$-periodic function and $f:ℝ\times [0,\infty )\times {ℝ}^2\to [0,\infty )$ is a continuous function which is $\omega$-periodic in $t,\text{and\hspace{0.17em}}{\tau }_{\mathrm{0}},{\tau }_{\mathrm{1}},{\tau }_{\mathrm{2}}$ are positive constants. The discussion is based on the fixed-point index theory in cones.

This paper studies the problem of finite-time $H_{\infty }$ control for time-delayed Itô stochastic systems with Markovian switching. By using the appropriate Lyapunov-Krasovskii functional and free-weighting matrix techniques, some sufficient conditions of finite-time stability for time-delayed stochastic systems with Markovian switching are proposed. Based on constructing new Lyapunov-Krasovskii functional, the mode-dependent state feedback controller for the finite-time $H_{\infty }$ control is obtained. Simulation results illustrate the effectiveness of the proposed method.

The main purpose of this paper is using the properties of Gauss sums and the estimate for character sums to study a mean value problem related to the primitive roots $\mathrm{mod}\hspace{0.17em}p$ and the different forms of Golomb’s conjectures and propose an interesting asymptotic formula for it.

General asynchronously coupled consensus algorithm associated with adjacent compensations, is proposed to solve the dynamical consensus problem of second-order multiagent network with communication delay under leader-following coordination control framework. Based on frequency-domain analysis, firstly, delay-independent consensus convergence is proved for the second-order multiagent systems with a spanning tree topology that has the leader root and then delay-dependent consensus condition is obtained for the multiagent systems with communication delay under a general leader-following interconnection topology. Simulation illustrates the correctness of the results.

This paper studies the finite-time stability of neutral fractional time-delay systems. With the generalized Gronwall inequality, sufficient conditions of the finite-time stability are obtained for the particular class of neutral fractional time-delay systems.

We will give the growth properties of harmonic functions of order greater than one in a half space, which generalize the result obtained by B. Levin in a half plane.

Recently, Zhang et al. proposed a sufficient descent Polak-Ribière-Polyak (SDPRP) conjugate gradient method for large-scale unconstrained optimization problems and proved its global convergence in the sense that ${\text{lim\hspace{0.17em}inf}}_{k\to \infty }\parallel \nabla f(x_k)\parallel =0$ when an Armijo-type line search is used. In this paper, motivated by the line searches proposed by Shi et al. and Zhang et al., we propose two new Armijo-type line searches and show that the SDPRP method has strong convergence in the sense that ${\text{lim}}_{k\to \infty }\parallel \nabla f(x_k)\parallel =0$ under the two new line searches. Numerical results are reported to show the efficiency of the SDPRP with the new Armijo-type line searches in practical computation.

A kind of the well-known matrix Riccati equations which arise in certain stochastic optimal problems is investigated. With the aid of the operator spectrum and the generalized Lyapunov equation approach, we give a sufficient condition for existence and uniqueness of the strong solution related to the critical mean square stabilization of stochastic linear controlled systems, which proves Conjecture 10 in (Zhang et al. (2008)) to a large extent. In addition, we get some properties of the strong solution. At last, we give a kind of stochastic system which has only a strong solution by an example.

A kind of the Euler-Maruyama schemes in discrete forms for stochastic differential equations with variable and distributed delays is proposed. The linear interpolation method is applied to deal with the values of the solutions at the delayed instants. The assumptions of this paper on the coefficients and related parameters are somehow weaker than those imposed by the related past literature. The error estimations for the Euler-Maruyama schemes are given, which are proved to be the same as those for the fundamental Euler-Maruyama schemes.

This paper addresses a new approach for modeling of versatile controllers in industrial automation and process control systems such as pneumatic controllers. Some fractional order dynamical models are developed for pressure and pneumatic systems with bellows-nozzle-flapper configuration. In the light of fractional calculus, a fractional order derivative-derivative (FrDD) controller and integral-derivative (FrID) are remodeled. Numerical simulations illustrate the application of the obtained theoretical results in simple examples.

We extend the notion of (α ψ, β $\phi$)-contractive mapping, a very recent concept by Berzig and Karapinar. This allows us to consider contractive conditions that generalize a wide range of nonexpansive mappings in the setting of metric spaces provided with binary relations that are not necessarily neither partial orders nor preorders. Thus, using this kind of contractive mappings, we show some related fixed point theorems that improve some well known recent results and can be applied in a variety of contexts.

By virtue of the modified Riesz kernel introduced by Qiao (2012), we give the integral representations for solutions of the Neumann problems in a half space.

This paper considers a stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise. We first prove the existence and uniqueness of strong probabilistic solution of an initial-boundary value problem with homogeneous Dirichlet boundary conditions. Then we give an asymptotic behavior of the solution.

We study the quadratic cost optimal control problems for the viscous Dullin-Gottwald-Holm equation. The main novelty of this paper is to derive the necessary optimality conditions of optimal controls, corresponding to physically meaningful distributive observations. For this we prove the Gâteaux differentiability of nonlinear solution mapping on control variables. Moreover by making use of the second order Gâteaux differentiability of solution mapping on control variables, we prove the local uniqueness of optimal control. This is another novelty of the paper.

We introduce a class of Fourier multiplier operators $M_b$ on n-complex unit sphere, where the symbol $b\in H^s(S_{\omega })$. We obtained the Sobolev boundedness of $M_b$. Our result implies that the operators $M_b$ take a role of fractional differential operators on $\partial 𝔹$.

We point out in this paper that the claims made by Kim et al. in the commented paper are incorrect and no new exact solution was obtained.

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.

We study the following nonhomogeneous Kirchhoff equation: $-(a+b{\int }_{R^{\mathrm{3}}}\mathrm{‍}|\nabla u{|}^{\mathrm{2}}dx)\mathrm{\Delta }u+u=k(x)f(u)+h(x),x\in R^{\mathrm{3}},u\in H^{\mathrm{1}}(R^{\mathrm{3}}),u>\mathrm{0},x\in R^{\mathrm{3}}$, where $f$ is asymptotically linear with respect to $t$ at infinity. Under appropriate assumptions on $k,f$, and $h$, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.

We investigate the bilinear Hilbert transform with oscillatory factors and the truncated bilinear Hilbert transform. The main result is that the $L^{p_1}\times L^{p_2}\to L^q$-boundedness of the two operators is equivalent with $1\le p_1,p_2$ , and $1/q=1/p_1+1/p_2$. In addition, we also discuss the boundedness of a variant operator of bilinear Hilbert transform with a nontrivial polynomial phase.

We investigate the properties of symmetry in two variables related to multiple Euler $q$-$l$-function which interpolates higher-order $q$-Euler polynomials at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to generalized higher-order $q$-Euler polynomials and alternating generalized $q$-power sums.

The “good” Boussinesq equation is transformed into a first order differential system. A fourth order finite difference scheme is derived for this system. The resulting scheme is analyzed for accuracy and stability. Newton’s method and linearization techniques are used to solve the resulting nonlinear system. The exact solution and the conserved quantity are used to assess the accuracy and the efficiency of the derived method. Head-on and overtaking interactions of two solitons are also considered. The numerical results reveal the good performance of the derived method.

We discuss the Laplace transform of the Caputo fractional difference and the fractional discrete Mittag-Leffer functions. On these bases, linear and nonlinear fractional initial value problems are solved by the Laplace transform method.

Codebook is an effective image representation method. By clustering in local image descriptors, a codebook is shown to be a distinctive image feature and widely applied in object classification. In almost all existing works on codebooks, the building of the visual vocabulary follows a basic routine, that is, extracting local image descriptors and clustering with a user-designated number of clusters. The problem with this routine lies in that building a codebook for each single dataset is not efficient. In order to deal with this problem, we investigate the influence of vocabulary sizes on classification performance and vocabulary universality with the kNN classifier. Experimental results indicate that, under the condition that the vocabulary size is large enough, the vocabularies built from different datasets are exchangeable and universal.

The aim of this paper is to give the sufficient conditions on the sequence space $\text{C}\text{e}\text{s}\left(\theta ,p\right)$ defined in Lim (1977) such that the class of all bounded linear operators between any arbitrary Banach spaces with nth approximation numbers of the bounded linear operators in $\text{C}\text{e}\text{s}\left(\theta ,p\right)$ form an operator ideal.

We introduce soft $\alpha$-sets on soft topological spaces and study some of their properties. We also investigate the concepts of soft $\alpha$-continuous and soft $\alpha$-open functions and discuss their relationships with soft continuous and other weaker forms of soft continuous functions. Also counterexamples are given to show the noncoincidence of these functions.

Consideration in this paper is the Cauchy problem of a generalized hyperelastic-rod wave equation. We first derive a wave-breaking mechanism for strong solutions, which occurs in finite time for certain initial profiles. In addition, we determine the existence of some new peaked solitary wave solutions.

The problem of finite-time control for attitude tracking maneuver of a rigid spacecraft is investigated. External disturbance, unknown inertia parameters are addressed. As stepping stone, a sliding mode controller is designed. It requires the upper bound of the lumped uncertainty including disturbance and inertia matrix. However, this upper bound may not be easily obtained. Therefore, an adaptive sliding mode control law is then proposed to release that drawback. Adaptive technique is applied to estimate that bound. It is proved that the closed-loop attitude tracking system is finite-time stable. The tracking errors of the attitude and the angular velocity are asymptotically stabilized. Moreover, the upper bound on the lumped uncertainty can be exactly estimated in finite time. The attitude tracking performance with application of the control scheme is evaluated through a numerical example.

This paper is concerned with the problem of multilevel association rule mining for bridge resource management (BRM) which is announced by IMO in 2010. The goal of this paper is to mine the association rules among the items of BRM and the vessel accidents. However, due to the indirect data that can be collected, which seems useless for the analysis of the relationship between items of BIM and the accidents, the cross level association rules need to be studied, which builds the relation between the indirect data and items of BRM. In this paper, firstly, a cross level coding scheme for mining the multilevel association rules is proposed. Secondly, we execute the immune genetic algorithm with the coding scheme for analyzing BRM. Thirdly, based on the basic maritime investigation reports, some important association rules of the items of BRM are mined and studied. Finally, according to the results of the analysis, we provide the suggestions for the work of seafarer training, assessment, and management.

The fault detection problem is investigated for a class of wireless network control systems which has stochastic uncertainties in the state-space matrices, combined with time delays and nonlinear disturbance. First, the system error observer is proposed. Then, by constructing proper Lyapunov-Krasovskii functional, we acquire sufficient conditions to guarantee the stability of the fault detection observer for the discrete system, and observer gain is also derived by solving linear matrix inequalities. Finally, a simulation example shows that when a fault happens, the observer residual rises rapidly and fault can be quickly detected, which demonstrates the effectiveness of the proposed method.

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions.

We mainly study the exponents of convergence of zeros and poles of difference and divided difference of transcendental meromorphic solutions for certain difference Painlevé III equations.

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear system $\dot{x}(t)=A(t)x(t)+f(t,x)$, $t\ne t_k$, $\mathrm{\Delta }x(t_k)=\tilde{A}(t_k)x(t_k)+\tilde{f}(t_k,x)$, $k\in \mathbb{Z}$, is topologically conjugated to $\dot{x}(t)=A(t)x(t)$, $t\ne t_k$, $\mathrm{\Delta }x(t_k)=\tilde{A}(t_k)x(t_k)$, $k\in \mathbb{Z}$, where $\mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k^{-})$, $x(t_k^{-})=x(t_k)$, represents the jump of the solution $x(t)$ at $t=t_k$. Finally, two examples are given to show the feasibility of our results.

As the core of the effective financial crisis prevention, enterprise finance crisis prediction has been the focal attention of both theorists and businessmen. Financial crisis predictions need to apply a variety of financial and operating indicators for its analysis. Therefore, a new evaluation model based on nonlinear programming is established, the nature of the model is proved, the detailed solution steps of the model are given, and the significance and algorithm of the model are thoroughly discussed in this study. The proposed model can deal with the case of missing data, and has the good isotonic property and profound theoretical background. In the empirical analysis to predict the financial crisis and through the comparison of the analysis of historical data and the real enterprises with financial crisis, we find that the results are in accordance with the real enterprise financial conditions and the proposed model has a good predictive ability.

Some new families of open Newton-Cotes rules which involve the combinations of function values and the evaluation of derivative at uniformly spaced points of the interval are presented. The order of accuracy of these numerical formulas is higher than that of the classical open Newton-Cotes formulas. An extensive comparison of the computational cost, order of accuracy, error terms, coefficients of the error terms, observed order of accuracy, CPU usage time, and results obtained from these formulas is given. The comparisons show that we have been able to define some new open Newton-Cotes rules which are superior to classical open rules for less number of nodes and less computational cost with increased order of accuracy.

Let $f$ be a normalized biholomorphic mapping on the Euclidean unit ball ${𝔹}^n$ in ${ℂ}^n$ and let $\alpha \in \left(\mathrm{0,1}\right)$. In this paper, we will show that if $f$ is strongly starlike of order $\alpha$ in the sense of Liczberski and Starkov, then it is also strongly starlike of order $\alpha$ in the sense of Kohr and Liczberski. We also give an example which shows that the converse of the above result does not hold in dimension $n\ge 2$.

Focusing on the amplifier performance evaluation demand, a novel evaluation strategy based on $\delta$-support vector regression ($\delta$-SVR) is proposed in this paper. Lower computer calculation demand is considered firstly. And this is dealt with by the superiority of $\delta$-SVR which can be significantly improved on the number of support vectors. Moreover, the function of $\delta$-SVR employs the modified RBF kernel function which is constructed from an original kernel by removing the last coordinate and adding the linear term with the last coordinate. Experiment adopted the typical circuit Sallen-Key low pass filter to prove the proposed evaluation strategy via the eight performance indexes. Simulation results reveal that the need of the number of $\delta$-SVR support vectors is the lowest among the other two methods LSSVR and ε-SVR under obtaining nearly the same evaluation result. And this is also suitable for promotion computational speed.

We propose a new method for the multiple attribute decision making problem. In this problem, the decision making information assembles multiple source data. Two main advantages of this proposed approach are that (i) it provides a data fusion technique, which can efficiently deal with the multisource decision making information; (ii) it can produce the degree of credibility of the entire decision making. The proposed method performs very well especially for the scenario that there exists conflict among the multiple source information. Finally, a traffic engineering example is given to illustrate the effect of our method.

To investigate the variability of HFMD in each county of Wenzhou, a spatial-temporal ARMA model is presented, and a general Bayesian framework is given for parameter estimation. The proposed model has two advantages: (i) allowing time series to be correlated, thus it can describe the series both spatially and temporally; (ii) implementing forecast easily. Based on the HFMD data in Wenzhou, we find that HFMD had positive spatial autocorrelation and the incidence seasonal peak was between May and July. In the county-level analysis, we find that after first-order difference the spatial-temporal ARMA $(0,0)\times {(1,0)}_{12}$ model provides an adequate fit to the data.

The humidity sensitive characteristics of the sensor fabricated from 10 mol% La₂O₃ doped CeO₂ nanopowders with particle size 17.26 nm synthesized via hydrothermal method were investigated at different frequencies. It was found that the sensor shows high humidity sensitivity, rapid response-recovery characteristics, and narrow hysteresis loop at 100 Hz in the relative humidity range from 11% to 95%. The impedance of the sensor decreases by about five orders of magnitude as relative humidity increases. The maximum humidity hysteresis is about 6% RH, and the response and recovery time is 12 and 13 s, respectively. These results indicate that the nanosized La₂O₃ doped CeO₂ powder has potential application as high-performance humidity sensor.

We address some forward and inverse problems involving indefinite eigenvalues for discrete $p$-Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues of $p$-Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discrete $p$-Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discrete $p$-Laplacian operators with potential terms involving the smallest indefinite eigenvalues.

This paper is concerned with the numerical simulation for shape reconstruction of the unsteady advection-diffusion problems. The continuous dependence of the solution on variations of the boundary is established, and the explicit representation of domain derivative of corresponding equations is derived. This allows the investigation of iterative method for the ill-posed problem. By the parametric method, a regularized Gauss-Newton scheme is employed to the shape inverse problem. Numerical examples indicate that the proposed algorithm is feasible and effective for the practical purpose.

This paper is concerned with some properties of a periodic two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in $H^s$ ($𝕊$), $s>\mathrm{5}/\mathrm{2}$.

We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz $(\mathrm{\Phi },\rho )$-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.

In solving large scale problems, the quasi-Newton method is known as the most efficient method in solving unconstrained optimization problems. Hence, a new hybrid method, known as the BFGS-CG method, has been created based on these properties, combining the search direction between conjugate gradient methods and quasi-Newton methods. In comparison to standard BFGS methods and conjugate gradient methods, the BFGS-CG method shows significant improvement in the total number of iterations and CPU time required to solve large scale unconstrained optimization problems. We also prove that the hybrid method is globally convergent.

The notion of int-soft subfields, int-soft algebras over int-soft subfields, and int-soft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, int-soft hypervector spaces are discussed.

A class of stochastic set differential equations (SSDEs) with non-Lipschitzian coefficients is investigated. We first give the preliminaries on the stochastic set differential equations. Then the nonexplosion of solutions to the SSDEs is discussed. Moreover, the existence and uniqueness of the solutions to SSDEs are proven. Finally, the continuous dependence of the solutions to SSDEs is studied.

The exponential synchronization issue for stochastic neural networks (SNNs) with mixed time delays and Markovian jump parameters using sampled-data controller is investigated. Based on a novel Lyapunov-Krasovskii functional, stochastic analysis theory, and linear matrix inequality (LMI) approach, we derived some novel sufficient conditions that guarantee that the master systems exponentially synchronize with the slave systems. The design method of the desired sampled-data controller is also proposed. To reflect the most dynamical behaviors of the system, both Markovian jump parameters and stochastic disturbance are considered, where stochastic disturbances are given in the form of a Brownian motion. The results obtained in this paper are a little conservative comparing the previous results in the literature. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.

This paper investigates a problem of how to regulate a firm which has private information about the market capacity, leading to adverse selection, and which can increase the market demand by exerting costly effort, resulting in moral hazard. In such a setting, the regulator offers a regulatory policy to the firm with the objective of maximizing a weighted sum of the consumer surplus and the firm’s profit (i.e., the social total surplus). We firstly find that the regulator will set the firm’s effort level as zero under observable effort regardless of the market capacity being full or private information; that is, the effort has no impact on the optimal regulatory policy. Interestingly, we also show that, it is necessary for regulator to consider the difference between the effort’s impact on the demand and the price’s impact on the demand, which may generate different distortion effects about the regulatory policy.

Let Y be a real separable Banach space and let $\left({𝒦}_C\left(Y\right),d_{\infty }\right)$ be the subspace of all normal fuzzy convex and upper semicontinuous fuzzy sets of Y equipped with the supremum metric $d_{\infty }$. In this paper, we introduce several types of additive fuzzy set-valued functional equations in $\left({𝒦}_C\left(Y\right),d_{\infty }\right)$. Using the fixed point technique, we discuss the Hyers-Ulam-Rassias stability of three types additive fuzzy set-valued functional equations, that is, the generalized Cauchy type, the Jensen type, and the Cauchy-Jensen type additive fuzzy set-valued functional equations. Our results can be regarded as important extensions of stability results corresponding to single-valued functional equations and set-valued functional equations, respectively.

This paper investigates the numerical solution of a class of one-dimensional inverse parabolic problems using the moving least squares approximation; the inverse problem is the determination of an unknown source term depending on time. The collocation method is used for solving the equation; some numerical experiments are presented and discussed to illustrate the stability and high efficiency of the method.

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delay $\tau$ passes through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

We consider some Bernstein-type operators as well as their product and Boolean sum for a function defined on a triangle with all curved sides. Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators.

We study surfaces of revolution with a nonlightlike axis in 3-dimensional Minkowski space and classify such surfaces in terms of the Gauss map $G$ that satisfies the condition ${\mathrm{\Delta }}^{h}G=\mathrm{\Lambda }G$, with Λ being a $3\times 3$ real matrix. Furthermore, this paper completes the classification problem of surfaces of revolution in Minkowski 3-space given by Jin et al. (2013).

A nonlinear impulsive integrodifference equation within the frame of $q_k$-quantum calculus is investigated by applying using fixed point theorems. The conditions for existence and uniqueness of solutions are obtained.