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26 April 2020, Volume 40 Issue 2 Previous Issue    Next Issue

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An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds Yue He,Hailong Her Acta mathematica scientia,Series A. 2020, 40 (2):  257-270.  Abstract ( 121 )   RICH HTML PDF (433KB) ( 130 )   Save Let $M$ be an $n$-dimensional compact Riemannian manifold with strictly convex boundary. Suppose that the Ricci curvature of $M$ is bounded below by $(n-1)K$ for some constant $K\geq0$ and the first eigenfunction $f_1$ of Dirichlet (or Robin) eigenvalue problem of a Schrödinger operator on $M$ is log-concave. Then we obtain a lower bound estimate of the gap between the first two Dirichlet (or Robin) eigenvalues of such Schrödinger operator. This generalizes a recent result by Andrews et al. ([4]) for Laplace operator on a bounded convex domain in $\mathbb{R} ^n$. References | Related Articles | Metrics

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An Lp Inhomogeneous Polyharmonic Neumann Problem on Lipschitz Domains in $\mathbb{R} ^{2}$ Zhihua Du,Yumei Li Acta mathematica scientia,Series A. 2020, 40 (2):  271-287.  Abstract ( 81 )   RICH HTML PDF (432KB) ( 103 )   Save In this paper, we study an inhomogeneous polyharmonic Neumann problem with $L^{p}$ boundary data on Lipschitz domains in $\mathbb{R} ^{2}$ by the method of layer potentials. Applying multi-layer ${\cal S}$-potentials, which are higher order analogues of the classical singular layer potential and defined in terms of polyharmonic fundamental solutions, the unique integral representation solution is given for the inhomogeneous polyharmonic Neumann problem on Lipschitz domains in $\mathbb{R} ^{2}$ when the boundary data are in some $L^{p}$ spaces. References | Related Articles | Metrics

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Weighted Composition Operator on the Normal Weight Zygmund Space in the Unit Ball Si Xu,Xuejun Zhang Acta mathematica scientia,Series A. 2020, 40 (2):  288-303.  Abstract ( 83 )   RICH HTML PDF (401KB) ( 103 )   Save Let $\mu$ be a normal function on $[0, 1)$. Suppose that $\varphi$ is a holomorphic self-map on the unit ball $B$ in ${\bf C}^{n}$ and $\psi\in H(B)$. Here the authors characterize the boundedness and compactness of the weighted composition operator $\psi C_{\varphi}$ on the Zygmund-type space ${\cal Z}_{\mu}(B)$ with a normal weight $\mu$ on the unit ball in ${\bf C}^{n}$. References | Related Articles | Metrics

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Products and Commuting of Toeplitz Operators on the Bergman Space of the Polydisk Zhiling Sun Acta mathematica scientia,Series A. 2020, 40 (2):  304-314.  Abstract ( 94 )   RICH HTML PDF (341KB) ( 116 )   Save In this paper, we give some necessary and sufficient conditions for the product of two Toeplitz operators with certain symbols to be a Toeplitz operator and give a formula for the symbol of the product on the Bergman space of polydisk. Next, the corresponding commuting problem of Toeplitz operator is studied. All the results are stated in terms of the Mellin transform of the symbol. References | Related Articles | Metrics

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Dynamical Mechanism and Energy Evolution of a Five-Modes System of the Navier-Stokes Equations For a Two-Dimensional Incompressible Fluid on a Torus Heyuan Wang Acta mathematica scientia,Series A. 2020, 40 (2):  315-327.  Abstract ( 106 )   RICH HTML PDF (2122KB) ( 92 )   Save In this paper we study dynamical mechanism and energy evolution of a five-modes system of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus. The five-modes system is transformed into Kolmogorov type system, which is decomposed into three types of torques:inertial torque, dissipation and external torque. Combining different torques, key factors of chaos generation and the physical meaning of the five-modes system are studied. The evolution of energy is investigated, the relationship between the energies and the Reynolds number is discussed. We conclude that the combination of the three torques is necessary conditions to produce chaos, and only when the dissipative torques match the driving torques (external torque) the system can produce chaos. While any combination of two types of torques cannot produce chaos. The external torque supply the energy for the system, and that leads to bifurcation and chaos. The Casimir function is introduced to analyze the system dynamics, and its derivation is chosen to formulate energy evolution. The bound of chaotic attractor is obtained by the Casimir function and Lagrange multiplier. We find that the Casimir function reflects the energy evolution and the distance between the orbit and the equilibria. Figures and Tables | References | Related Articles | Metrics

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Invariant Measures and Liouville Type Equation for First-Order Lattice System Yongjun Li,Yanmiao Sang,Caidi Zhao Acta mathematica scientia,Series A. 2020, 40 (2):  328-339.  Abstract ( 82 )   RICH HTML PDF (389KB) ( 118 )   Save This article discusses the probability distribution of solutions in the phase space for the first-order lattice system. The authors first prove the the process generated by the solutions operator of the lattice system possesses a pullback attractor. Then they prove that there is a unique family of invariant probability measures contained in pullback attractor and the measures satisfy a Liouville type equation. References | Related Articles | Metrics

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Inequalities Among Eigenvalues of Left-Definite Sturm-Liouville Problems with Periodic Coefficients Longjie Sun,Xiaoling Hao,Tian Niu Acta mathematica scientia,Series A. 2020, 40 (2):  340-351.  Abstract ( 82 )   RICH HTML PDF (297KB) ( 98 )   Save In this paper, by using the relation between left-definite and right-definite problems, the characterization of the periodic and semi-periodic eigenvalues of left-definite Sturm-Liouville problems with periodic coefficients on the interval[a, a+kh] are obtained. For each k >2 we identify explicitly which of the uncountable number of complex conditions generates these periodic eigenvalues. Based on this condition, we give the inequality relation of periodic eigenvalues on the interval[a, a+kh] and 1-1 correspondence between the periodic and semi-periodic eigenvalues on the interval[a, a+kh], k >1 and the corresponding eigenvalues from the interval[a, a+h]. References | Related Articles | Metrics

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Hyers-Ulam-Rassias Stability of a Mixed Type Cubic-Quartic Functional Equation in 2-Banach Spaces Chun Wang,Tianzhou Xu Acta mathematica scientia,Series A. 2020, 40 (2):  352-368.  Abstract ( 78 )   RICH HTML PDF (338KB) ( 94 )   Save In this paper, we investigate the general solution and the Hyers-Ulam-Rassias stability of the mixed type cubic-quartic functional equation $\begin{eqnarray*} &&f(kx+y)+f(kx-y)\\ & = & \frac{k^2+k}{2}[f(x+y)+f(x-y)]+\frac{k^2 -k}{2}[f(-x-y)+f(y-x)]\\ && +(k^4+k^3-k^2 -k)f(x)+(k^4-k^3-k^2+k)f(-x)-(k^2 -1)f(y)-(k^2 -1)f(-y) \end{eqnarray*}$ in 2-Banach spaces, where k >1. The results improve and extend some recent results. References | Related Articles | Metrics

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A Hilbert-Type Integral Inequality with the Mixed Kernel and Its Operator Expressions with Norm Qiong Liu,Yindi Liu Acta mathematica scientia,Series A. 2020, 40 (2):  369-378.  Abstract ( 73 )   RICH HTML PDF (339KB) ( 99 )   Save By using the method of weight function, the technique of real analysis and the theory of special functions, a Hilbert-type integral inequality with the mixed kernel and its equivalent form are established, the optimality for the constant factors are proved and the operator expressions with norm are given. References | Related Articles | Metrics

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Gradient Estimates for Weak Solutions to Non-Homogeneous A-Harmonic Equations Under Natural Growth Yanan Zhang,Shuo Yan,Yuxia Tong Acta mathematica scientia,Series A. 2020, 40 (2):  379-394.  Abstract ( 65 )   RICH HTML PDF (397KB) ( 206 )   Save The gradient estimates for weak solutions to non-homogeneous A-harmonic equations under natural growth is obtained. The Lp-type estimates for such equation is derived under natural growth, and then the Lφ-type estimate in Orlicz space is derived by a new iteration-covering approach. References | Related Articles | Metrics

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Viscosity Implicit Algorithms for a Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces Gang Cai Acta mathematica scientia,Series A. 2020, 40 (2):  395-407.  Abstract ( 101 )   RICH HTML PDF (296KB) ( 92 )   Save In this paper, we study a viscosity implicit algorithm for finding a common element of the set of solutions of a new variational inequality problems for two inverse-strongly monotone operators and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Using modified extragradient method, we obtain strong convergence theorems under some suitable assumptions imposed on the parameters. The results obtained in this paper extend and improve many recent ones. References | Related Articles | Metrics

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Existence of L-Quasi Mild Solutions for Damped Elastic Systems in Banach Spaces Haide Gou Acta mathematica scientia,Series A. 2020, 40 (2):  408-421.  Abstract ( 82 )   RICH HTML PDF (355KB) ( 89 )   Save In this article, we deals with the existence of L-quasi mild solutions for damped elastic systems in Banach spaces. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Under monotone condition and the noncompactness measure condition on nonlinearity, we obtain the existence of extremal mild solutions. In addition, an example is given to illustrate our results. References | Related Articles | Metrics

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An Initial-Boundary Value Problem for the Higher-Order Chen-Lee-Liu Equation on the Half-Line Beibei Hu,Ling Zhang,Ning Zhang Acta mathematica scientia,Series A. 2020, 40 (2):  422-431.  Abstract ( 71 )   RICH HTML PDF (500KB) ( 93 )   Save In this paper, the initial-boundary value problems of the higher-order Chen-Lee-Liu (HOCLL) equation on the half-line has been analysed via the Fokas method. We show that the solution q(x, t) of HOCLL equation can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. Figures and Tables | References | Related Articles | Metrics

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Local Uniqueness of a Single Peak Solution of a Subcritical Kirchhoff Problem in R3 Shimin Xu,Chunhua Wang Acta mathematica scientia,Series A. 2020, 40 (2):  432-440.  Abstract ( 74 )   RICH HTML PDF (312KB) ( 107 )   Save In this paper, we obtain the local uniqueness of a single peak solution to the following Kirchhoff problem $-\Big(\epsilon^2 a+ \epsilon b \int_{\mathbb{R} ^3} |\nabla u|^2{\rm d}x\Big) \Delta u + u=K(x)|u|^{p-1}u, u>0,x \in \mathbb{R} ^3,$ for ε>0 sufficiently small, where a, b>0 and 1 < p < 5 are constants, K: $\mathbb{R}^3$→$\mathbb{R}$ isabounded continuous function. We mainly use a contradiction argument developed by Li G, Luo P, Peng S in[20], applying some local pohozaev identities. Our result is totally new for Kirchhoff equations. References | Related Articles | Metrics

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Reproducing Kernel Shifted Legendre Basis Function Method for Solving the Fractional Differential Equations Quanyi Gong,Huanmin Yao Acta mathematica scientia,Series A. 2020, 40 (2):  441-451.  Abstract ( 80 )   RICH HTML PDF (462KB) ( 86 )   Save Based on the theory of the reproducing kernel, this paper constructed a new reproducing kernel space and the reproducing kernel function of the space is given by using the shifted Legendre polynomials as the basis function. It is different from former that this function is no longer a piecewise function, so when the fractional operator is used on the kernel function, the computation is reduced. Thus the approximate solution is more accurate. Finally, the numerical examples illustrate the validity of the method. Figures and Tables | References | Related Articles | Metrics

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The Fractional Generalized Disturbed Thermal Wave Equation Cheng Ouyang,Weigang Wang,Jiaqi Mo Acta mathematica scientia,Series A. 2020, 40 (2):  452-459.  Abstract ( 95 )   RICH HTML PDF (389KB) ( 89 )   Save A class of generalized fractional nonlinear disturbed thermal wave equation is considered. Firstly, the arbitrary order approximate analytic solutions for the fractional generalized nonlinear disturbed thermal wave equation initial boundary value problem was constructed by using the singular perturbation method. And the uniformly valid asymptotic expansion was proofed by using the fixed point theory of the functional analysis. The physical sense of the solution was stated simply. The approximate analysis solution makes up not enough the simple numerical simulation solution. References | Related Articles | Metrics

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Statistical Inference in Partially Nonlinear Varying-Coefficient Errors-in-Variables Models with Missing Responses Yijia Ma,Liugen Xue,Fei Lu Acta mathematica scientia,Series A. 2020, 40 (2):  460-474.  Abstract ( 99 )   RICH HTML PDF (601KB) ( 127 )   Save This paper considers about the estimation of varying-coefficient partial nonlinear errors-in-variables models with missing responses. Firstly, we develop inverse probability weighted approaches and local bias-corrected restricted profile least squares estimators. Asymptotic normality of estimators is established. Moreover, both simulation results and a real data show that local bias-corrected restricted profile least squares estimated approach are better than the performance ignoring the measurement error. Figures and Tables | References | Related Articles | Metrics

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Asymptotics for the Self-Weighted M-Estimation of Nonlinear Autoregressive Models with Heavy-Tailed Errors Keang Fu,Li Ding,Junqiao Li Acta mathematica scientia,Series A. 2020, 40 (2):  475-483.  Abstract ( 89 )   RICH HTML PDF (393KB) ( 257 )   Save Consider the nonlinear autoregressive model xt=f(xt-1, …, xt-p, θ)+εt, where θ is the q-dimensional unknown parameter and εt's are random errors with possibly infinite variance. In this paper, the self-weighted M-estimator of θ is constructed, and the asymptotic normality of the proposed estimator is also established. Some simulation studies are also given to show that the self-weighted M-estimators have good performances with some heavy-tailed random errors. Figures and Tables | References | Related Articles | Metrics

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Quasi Sure Local Strassen's Law of the Iterated Logarithm for Increments of a Brownian Motion Fengbing Li,Yonghong Liu Acta mathematica scientia,Series A. 2020, 40 (2):  484-491.  Abstract ( 104 )   RICH HTML PDF (287KB) ( 85 )   Save In this paper, we establish the quasi sure local Strassen's law of the iterated logarithm for increments of a Brownian motion. As an application, a quasi sure functional modulus of continuity for a Brownian motion is also derived. References | Related Articles | Metrics

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Averaging Principles for Stochastic Differential Equations Driven by Time-Changed Lévy Noise Lijuan Cheng,Yong Ren,Lu Wang Acta mathematica scientia,Series A. 2020, 40 (2):  492-500.  Abstract ( 77 )   RICH HTML PDF (282KB) ( 119 )   Save This paper concerns averaging principles for a kind of stochastic differential equations driven by time-changed Lévy noise (LSDEs, in short). An averaged LSDE for the original LSDE is proposed. The solution of the averaged LSDE converges to that of the original LSDE in the sense of mean square and probability. Finally, we will give an example to illustrate the obtained results. References | Related Articles | Metrics

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On the Analysis of Ruin-Related Quantities in the Nonhomogeneous Compound Poisson Risk Model Yingchun Deng,Man Li,Ya Huang,Jieming Zhou Acta mathematica scientia,Series A. 2020, 40 (2):  501-514.  Abstract ( 72 )   RICH HTML PDF (467KB) ( 79 )   Save In this paper, the classical risk model is extended to nonhomogeneous compound Poisson risk model. Firstly, both the classical method and the time-varying method are used to calculate the ruin-related quantities for this model, and the analytical expression of the renewal equation is obtained. Secondly, for the time-varying model, the generalized Gerber-Shiu function is defined, which is to verify the effectiveness of the time-varying method for the nonhomogeneous compound Poisson risk model. Finally, when each claim follows an exponentially distribution, the corresponding ruin probability and Gerber-Shiu function are calculated. References | Related Articles | Metrics

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Numerical Method of a Nonlinear Hierarchical Age-Structured Population Model Zerong He,Zhiqiang Zhang,Zheyong Qiu Acta mathematica scientia,Series A. 2020, 40 (2):  515-526.  Abstract ( 74 )   RICH HTML PDF (465KB) ( 90 )   Save Based upon the assumption that the young individuals are more competitive than the old ones within a species, a class of nonlinear hierarchical age-structured population model is established in a form of IBVP of integro-partial differential equations. We propose an algorithm for the solutions to the model, analyze its convergence, and make some numerical experiments. Figures and Tables | References | Related Articles | Metrics

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The Dynamics of an SEIR Epidemic Model with Time-Periodic Latent Period Shuangming Wang,Xingman Fan,Mingjun Zhang,Junrong Liang Acta mathematica scientia,Series A. 2020, 40 (2):  527-539.  Abstract ( 216 )   RICH HTML PDF (608KB) ( 206 )   Save A SEIR ordinary differential epidemic model with time-periodic latent period is studied. Firstly, the model is derived by means of the distribution function of infected ages. Next, the basic reproduction ratio $\mathcal R_0$ is introduced, and it is shown that $\mathcal R_0$ is a threshold index for determining whether the epidemic will go extinction or become endemic using the theory of dissipative dynamic systems. Finally, numerical methods are carried out to validate the analytical results and further to invetigate the effects on evaluating the propagation of disease owning to the neglect of the periodicity of the incubation period. Figures and Tables | References | Related Articles | Metrics

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COVID-19 Transmission Model in an Enclosed Space: A Case Study of Japan Diamond Princess Cruises Xinxin Cheng,Yaqing Rao,Gang Huang Acta mathematica scientia,Series A. 2020, 40 (2):  540-544.  Abstract ( 303 )   RICH HTML PDF (491KB) ( 191 )   Save In this paper, we study the transmission of a novel coronavirus pneumonia (COVID-19) on Japan Diamond Princess Cruises as an example. By establishing a simple susceptible-infected epidemic model, the transmission mechanism of NCP in an enclosed space is established. Dynamic analysis and numerical fitting predict the disease transmission process and the final results, we estimate the infection rate and evaluate the effect of quarantine, and some suggestions for prevention and control strategies are given. Figures and Tables | References | Related Articles | Metrics