A class of hybrid ratio-dependent three species food chain model with diffusion is investigated. Hurwitz criterion is used to discuss the stability of the non-negative constant equilibrium solution about this system. Then, the Hopf bifurcation is considered. At the same time, the Hopf bifurcation direction and stability of bifurcation periodic solutions are discussed making use of the normal form method and the center manifold theorem. Finally with the help of matlab software, some numerical simulations are shown to support and supply the results of theoretical analysis.

In this article, we adopt the semigroup theory and the spectral analysis method to study the well-posedness and delay-independent stability for a class of delayed ring neural network with two small world connections. Firstly, we formulate the system into an abstract evolution equation and prove the well-posedness by the semigroup approach. Next, we devote to the detailed spectral analysis. It is obtained that with some conditions required on the connection weights between neurons, all eigenvalues are located in the left half complex plane and their real parts go to -∞ as n → ∞. The asymptotic spectral expression is also presented. Moreover, according to the Schur-Cohn criterion, we discuss the stability region of the connection weights, the two small world connections and time delay. The results show that the system remains stable when the weight of the small world connection belongs to a certain interval, for any positive value of the delay τ. Finally, the simulation results of numerical examples are presented to illustrate the validity and feasibility of the conclusion.

In this paper, a Holling-IV type predator-prey model with Michaelis-Menten type harvesting is qualitatively analyzed. At first, by the maximum principle and the linearized stability theory, a priori estimates of the steady-state system and the local asymptotic stability of positive constant solutions are given. Then, with the help of bifurcation theory, the local bifurcation of steady-state system at the positive constant solution U₁ is obtained by treating d₂ as bifurcation parameter; it is shown that under certain conditions, the local bifurcation generated from (d₂^j, U₁) can be extended to global bifurcation.

This paper considers a risk model based on the policy entrance process, in which each customer is allowed to claim more than once within the validity time. The claim sizes caused by each customer are described as extended negatively dependent distributed heavy-tailed random variables, and claims due to different customers are independent and identically distributed heavy-tailed random variables. We derive the large deviations for the loss process of the risk model.

In this paper, based on the property of the l_∞-norm and the method of the singular value threshold, we present an algorithm for Hankel matrix completion. The proposed algorithm ensures each iterative matrix is feasible Hankel matrix, which not only decreases the computation of SVD but also gains more effective approximation to solution in precision. Meanwhile, the convergence of the new algorithm is established. Finally, the numerical examples and inpainted images show that the proposed algorithm is more effective than the APG(an accelerated proximal gradient algorithm), the ALM (augmented Lagrange multiplier) algorithm and the F-NSPTA (structure-preserving thresholding algorithm based on F-norm) algorithm for Hankel matrix completion.

Consider the survival function and hazard rate estimators by the Kaplan-Meier method based on censored data, where the survival and censoring times come from the widely orthant dependent date, respectively. Under some more mild conditions, the uniform strong approximation rates and strong representation for the survival function and hazard rate are investigated, and their uniform strong approximation rates and remainders of strong representation also are obtained with the order O(n^-1/2 log^1/2 n) a.s. Our results established generalize the corresponding ones of negatively associated and negatively superadditive dependent data in the related literatures.

We consider the incompressible limit of 2-d isothermal magnetohydrodynamic equations. If the initial data is well-prepared, we show that the weak solutions of the compressible magnetohydrodynamic equations under the perfectly conducted boundary condition converge to the strong solutions of incompressible system as Mach number goes to zero. The convergence rates are also obtained.

The Hosoya polynomial of a graph encompasses many of its metric properties, for instance the Wiener index and the hyper-Wiener index. In this paper, we first obtain an explicit expression for the expected value of the Hosoya polynomial of a random polyomino chain. Then as a consequence of the result, exact formulas for the expected values of the Wiener and the hyper-Wiener indices are obtained, respectively.

In this paper we establish a modified general modulus-based matrix splitting iteration method for solving the large sparse linear complementarity problems of H₊-matrix and present the convergence analysis. In addition, the optimal parameters are considered under the given methods and we supplement the proof of equivalence of (z, r) from Methods 3.1 and 3.2 in[2] and the solution of the original linear complementary problem LCP (q, A). Finally, we give a numerical example, which illustrates that the modified method is efficient.

To study the solvability of generalized variational inequalities in Hilbert lattices, it is not requiring any kind of continuous and monotone. Using the theory of Banach lattice, this paper investigates the order-preservation of solution correspondence for twoparameter generalized variational inequalities in separable Hilbert lattices which improves the corresponding works on single-parameter generalized variational inequalities in earlier results.

This paper discusses issues of k-person coalitions in the presence of cores in an n-person cooperative game. We take the Nash bargaining solution as our allocation criteria. First, we examine the 2-person coalition and further demonstrate the existence of a stable 2-person coalition in an n-person cooperative game. Then, the concept of a stable k-person coalition is proposed and discussion of the existence of a stable k-person coalition follows. In particular, we present a general approach to realize the search of a stable k-person coalition. Moreover, we additionally show a sufficient condition with which all of the players gain more than the subcoalitions in ak-person coalition. Finally, a numerical illustration is given to verify the correctness of the theory put forward in this paper.

In this paper, we mainly study the existence and fractal dimension of a pullback exponential attractor and a uniform exponential attractor for non-autonomous Schrödinger lattice system. Firstly, we prove the existence of a pullback exponential attractor for the stochastic Schrödinger lattice system with time-dependent coupled coefficients and forces. Then we prove the existence of a uniform exponential attractor for non-autonomous Schrödinger lattice system driven by quasi-periodic external forces.

Let G be a graph with n vertices. A perfect matching of G is a set M of edges such that no two edges are incident with a common vertex and each vertex of G is incident to some edge in M. A near-perfect matching of G is a matching M such that it is incident with all vertices of G except exactly one. If G-v has a perfect matching for every choice of v∈ V(G), then G is said to be factor-critical. In this paper, we present some sufficient conditions for existences of a perfect matching, a near-perfect matching or factor-critical of G in terms of the Laplacian eigenvalues.

The existence of five nontrivial solutions for a class of fourth-order elliptic equations with combined nonlinearities is established by using the minimax method, the Ekeland variational principle and the Morse theory.

A blockwise empirical likelihood approach is developed to construct confidence intervals for nonparametric regression functions when the observations of the response variable and the regressors form a strong mixing sample. The finite-sample performance of the method is evaluated through a simulation study.

In this paper, we study the Green's function of fourth-order difference equation with periodic boundary value problem. We obtain some new results and generalize some main results in A. CabadaandN. Dimitrov's paper.

A vertex cut F of a graph X is called a cyclic vertex cut if at least two components of X-F contain cycles. The cyclic vertex connectivity of X, denoted by κ_c(X), is the minimum cardinality of all cyclic vertex cuts. In this paper, we show that, for the minimal circulant graph X=C(Z_n,S), (1) if |S|≥2, and 2a≡0(mod n) or 3a≡0(mod n) for some a∈S, or (2) if |S|≥3, and 2a≡0(mod n) and 3a≡0(mod n) for any a∈S, then κ_c(G)=g(k-2), where g and k(k>2) are the girth and the regularity of X, respectively.

General Clairaut type differential equations are the generalization of classical Clairaut differential equations. By applying the theories of Legendrian unfolding and transversality, the multi-parameter bifurcations of such differential equations are classified from the geometric point of view. And through the simulation, several typical bifurcation diagrams of the phase portraits are drawn. The results can be used to study the change of the system topology when the parameters are changed.

In this work, we investigate the oscillation of the new even order nonlinear neutral differential equation with distributed delay
(r(t)|z^(n-1)(t)|^α-1z^(n-1)(t))'+F(t,x(g(t)))=0, t ≥ t₀,
where z(t)=x(t)+p(t)x(τ(t)), α>0 is constant and n is even. By using the generalized Riccati transformation and integral averaging technique, we establish some new oscillation criteria. These results extend and improve some existing results in the literature.

Many problems of interest in real world applications can be modeled as a so-called minimax problem, it is of great value to study its theory and solving method. In this paper, a new SQP algorithm with a system of linear equations is proposed to solve minimax problems with equality and inequality constraints. First,based on the ε-active constraint set,a norm-relaxed quadratic programming subproblem and a system of linear equations are established to get the feasible direction of descent,which can overcome the Maratos effect, and greatly reduce the computational complexity of the algorithm; Second, a new curve search step-size strategy without the penalty factors or filters is introduced. The method not only avoids the choice of penalty factors, but also reduces the storage capacity of the computer. Finally, it is proved that,under appropriate assumptions, the new algorithm is globally convergent. Some preliminary numerical experiments are reported to show the effectiveness and competitiveness of the algorithm.

The existence of positive solutions is investigated for a class of integral boundary value problems with fractional derivatives by using the fixed point theorem on cone and the Leggett-Williams fixed point theorem.The existence of at least one positive solution and three positive solutions for these problems is obtained.Meanwhile two examples are delivered to illustrate the approach.

In this note, the exponential stability of the mono-tubular heat exchanger equation with boundary observation possessing a time delay and inner control was investigated. Firstly, the close-loop system was translated into an abstract Cauchy problem in the suitable state space. A uniformly bounded C₀-semigroup generated by the close-loop system, which implies that the unique solution of the system exists, was shown. Secondly, the spectrum configuration of the closed-loop system was analyzed and the eventual differentiability and the eventual compactness of the semigroup were shown by the resolvent estimates on some resolvent sets. This implies that the spectrum determined growth assumption hold. Finally, a sufficient condition, which is related to the physical parameters in the system and independent of the time delay, of the exponential stability of the closed-loop system was given.

In this paper, we study one Kirchhoff equation involving subcritical or critical Sobolev exponents and weigh function. By Nehari manifold and variational methods, we prove that the problem has at least one nontrivial solution under different cases.

We study the existence of positive periodic solutions of second order difference systems.
-Δ²x(n-1)+q(n)x(n)=f(n,x(n)),
where f(n,x):N×R^N\{0}→R^N may be singular at x=0. The proof relies on a nonlinear alternative principle of Leray-Schauder.

The Lasota-Wazewska model is often used to describe the regeneration of red blood cells in animals. Based on the Banach contraction mapping principle, the existence and uniqueness of strictly positive asymptotically almost periodic solution for a class of Lasota-Wazewska models are firstly obtained under some conditions, and then, by constructing a suitable Lyapunov function, the global exponential asymptotic stability of the asymptotically almost periodic solutions is shown. The results obtained in this paper can enrich the characterization of the dynamic behavior of the Lasota-Wazewska models.

In his paper, we study the boundedness, compactness and essential norm of the Volterra-type composition operators from Hardy spaces into the Bloch-type spaces. The boundedness, compactness of these operators from Hardy spaces into little Bloch-type spaces are also investigated.

Case-cohort design provides a cost effective way in large cohort studies. In this article, we consider the proportional mean residual life model under case-cohort design for length-biased data and propose an weighted estimating equation method for the estimation of unknown parameters in the model. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show that the proposed method works well for finite sample situations. A real data example is also provided.

A class of EI Nino/La Nina-Southern Oscillation (ENSO) dynamic system is described. Firstly, the reduced solution to dynamic system was got. Secondly, the time delay function of equation was managed. Then the asymptotic solution to the corresponding problem was obtained by employing the perturbation method. Finally, the parameters of the dynamic system were studied and the curve figures of the solution were obtained. And the physical quantities of the dynamic system were illustrated.

In this paper we investigate the finite time ruin probability in the renewal risk model with dependent random premium rates. Under the assumption of strongly subexponential claim distribution, we obtain the tail behaviour of the ruin probability within finite time t, as initial risk reserve x tends to infinity. The asymptotic formula holds uniformly for t ≥ f(x), where f(x) is an infinitely increasing function.

By using the fixed point theorem and the Gronwall inequality of vector form, the existence and uniqueness of the solution of the coupled system of nonlinear implicit fractional differential equations under the definition of Caputo fractional derivative are obtained. The estimate on solutions, the continuous dependence on initial values, the continuous dependence on parameters and functions, and ε-approximate solutions for coupled systems are also discussed.

Based on the applications in the real world, a new generalized matrix least square problem will be constructed in this paper. A new algorithm, which is the generalization of the iterative algorithm introduced by Allwright, is designed for solving this kind of least square problems. Furthermore, the theoretical analysis and the preliminary numerical result of the new algorithm for the generalized matrix least square problem will be given in this paper.

This paper considers analyzing competing risks data with missing cause of failure under the accelerated failure time model. The missing mechanism is assumed to be missing at random. None parametric model for the probability of missing cause of failure is assumed. The inverse probability weighted and double robust techniques are used to modify the rank based estimating functions. Kernel smoothing technique is used to estimate the probability of missing cause of failure. The algorithm of the estimating equations is developed through transforming the estimating equations into an optimization problem. The asymptotic properties of the proposed estimators are established. A simulation study is carried out to evaluate the performance of the estimators. The proposed estimating method is illustrated by a breast cancer study data.

Volterra integral equation with proportional delay is concerned about in this paper. Firstly, we give some transformation. Then we use the Gauss quadrature formula to get the approximated solution. And then the Chebyshev spectral-collocation method is proposed to solve the equation and a rigorous error analysis is provided which shows that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted L² norm. In the end, numerical example is given to confirm the theoretical result.

In this paper, a new approach of numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional diffusion equations is developed and high accuracy error analysis is provided. Firstly, based on the L1 formula for the approximation of the time distributed order fractional derivative, the fully discrete finite element scheme is derived and the unconditional stability of the scheme is obtained. Secondly, by use of the supercolse estimate between the Ritz projection operator R_h and interpolation operator I_h of the bilinear element and the interpolated post-processing technique, the superclose and superconvergence results for the fully discrete scheme are obtained, which can't be deduced by the interpolation or Ritz projection alone. Furthermore, the proposed method is applied to the equations with variable cofficienet and the unconditional stability and superconvergent eatimates are also proved. Finally, some popular finite elements are investigated.

The existence of meromorphic solutions of Fermat type differential equations and differential-difference equations are investigated by value distribution theory and the complex difference theory in this article. We confirm that there don't exist non-constant meromorphic function f(z) that satisfy the differential equation f'(z)^m+f(z)ⁿ=1 (m, n are positive integers) except when m=2,n=3 or 4 and m=1,n=2. Some examples are given to illustrate the existence of meromorphic solutions of the equation for the particular cases, and we also study the entire function solution of the equation. Meanwhile, we also discuss the existence of non-constant meromorphic solutions of the differential-difference equation f'(z)^m+f(z+c)ⁿ=1.

The existence of periodic solution for a second order differential equation with a singularity of indefinite type x"-(α(t))/(x^μ (t))=h(t) is studied in this paper. Here μ∈ (0,1] is a constant, α(t), h(t) are T-periodic with α,h∈ L¹([0,T],R). The interesting point is that the sign of the function α(t) is allowed to change.

The parameters of the usual epidemic model are determined, but the parameters of the model are difficult to be accurately obtained because of the influence of various uncertain factors. This paper discusses the near-optimal control of a stochastic SIRS model with imprecise parameters. The objective function for the cost of the treatment of disease is as small as possible. The error bounds of the near-optimal control are given. The sufficient conditions for the near-optimal control are established by using the Hamiltonian function, and the effects of the control variables on the disease are verified by a numerical example.

This paper addresses the problem of arbitrary initial states in iterative learning control for high-order nonlinear systems. It presents a new control algorithm. In the process of tracking, this algorithm can rectify the initial errors through a step-by-step rectifying controller. The controller rectifies the error of the state x₂ (t) at first, then the error of the state x₁(t). These rectifying actions are finished in a small interval. After finishing the rectifying actions, the system can achieve complete tracking. Furthermore, the algorithm has shown effective in the improvement of tracking performance through simulations.

In this paper, several difference schemes to the highly nonlinear term ▽·(▽u/√|▽u|²+β) of the total variation-based image denoising problem are proposed. The large nonlinear system is linearized by fixed point iteration method. An algebraic multigrid method with Krylov subspace acceleration is used to solve the corresponding linear equations. Different pictures with Gaussian white noise are processed in the numerical experiments. The numerical experiments demonstrate that our difference schemes are efficient and robust.

This paper studies the regularity criterion of smooth solutions to the Boussinesq equations with fractional dissipation. We obtain an extension of the Beale-Kato-Majda criterion to the Boussinesq equations. As a special case, we recover a previous result of Planchon for the incompressible Euler equations.