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26 December 2020, Volume 40 Issue 6 Previous Issue    Next Issue

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Generalized Uncertainty Principle in Fock-Type Spaces Haigui Wu,Yuxia Liang Acta mathematica scientia,Series A. 2020, 40 (6):  1409-1419.  Abstract ( 103 )   RICH HTML PDF (353KB) ( 169 )   Save The unilateral weighted shift operator is used to construct a general self-adjoint operator pairs in this paper. We obtained the formula of generalized Uncertainty Principle in Fock-type spaces and presented conditions ensuring the equality. This result contains the classical Uncertainty Principle deduced from the derivation and multiplication operators, which can provide some theoretical basis for the solutions to related hot problems in quantum mechanics and other frontal subjects. References | Related Articles | Metrics

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The Quadratic Numerical Range and the Spectrum of Some Unbounded Block Operator Matrices Wenwen Qiu,Yaru Qi Acta mathematica scientia,Series A. 2020, 40 (6):  1420-1430.  Abstract ( 70 )   RICH HTML PDF (470KB) ( 115 )   Save In this paper, we study the operator ${\cal M}=\left[\begin{array}{ccc} 0& I\\ -A& B \end{array} \right]$ which is associated with the second order differential equation $\ddot{z}(t)-B\dot{z}(t)+Az(t)=0$ in a Hilbert space, where $A$ is a self-adjoint and uniformly positive linear operator, $B$ is accretive. We prove that ${\cal M}$ is a boundedly invertible closed operator and $\overline{{\cal M}|_{H_{1}\times H_{1}}}$=${\cal M}$ where $H_{1}={\cal D}(A)$ with the norm $\|x \|_{H_{1}}=\|Ax\|$. And we characterize the spectral distribution of the operator ${\cal M}$ by using the quadratic numerical range of the block operator matrix ${\cal M}|_{H_{1}\times H_{1}}$. Figures and Tables | References | Related Articles | Metrics

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θ Type Marcinkiewicz Integral and Its Commutator on Metric Measure Spaces Guanghui Lu,Shuangping Tao Acta mathematica scientia,Series A. 2020, 40 (6):  1431-1445.  Abstract ( 70 )   RICH HTML PDF (395KB) ( 132 )   Save Let $({\cal X}, d, \mu)$ be a non-homogeneous metric measure space satisfying the so-called geometrically doubling and the upper doubling conditions in the sense of ${\rm Hyt\ddot{o}nen}$. Under the assumption that the dominating function $\lambda$ satisfies the $\epsilon$-weak reverse doubling condition, the authors prove that the $\theta$ type Marcinkiewicz integral ${\cal M}_{\theta}$ and the commutator ${\cal M}_{\theta, b}$ generated by the $b\in\widetilde{{\rm RBMO}}(\mu)$ and the ${\cal M}_{\theta}$ is bounded on homogeneous Herz space $\dot{K}^{\tau, p}_{q}(\mu)$, respectively. Furthermore, the boundeness of the ${\cal M}_{\theta}$ and ${\cal M}_{\theta, b}$ from the $\widetilde{H}\dot{K}^{\tau, p}_{{\rm atb}, q}(\mu)$ into the $\dot{K}^{\tau, p}_{q}(\mu)$ is also obtained. References | Related Articles | Metrics

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ρ-Variation for Singular Integral Operators with Variable Kernels Zhenbing Gong,Yanping Chen,Wenyu Tao Acta mathematica scientia,Series A. 2020, 40 (6):  1446-1460.  Abstract ( 66 )   RICH HTML PDF (377KB) ( 111 )   Save In this paper, we will prove that the variation of singular integral operators with rough variable kernels are bounded on $L^{2}({\mathbb S} ^{n})$ if $\Omega\in L^{\infty}({\mathbb R} ^{n})\times L^{q}({\mathbb S}^{n-1})$ for $q>2(n-1)/n$, and $n\geq2$. Moreover, we can also obtain the weighted variational inequalities for singular integral operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces. References | Related Articles | Metrics

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Lϕ-Type Estimates for Very Weak Solutions of A-Harmonic Equation in Orlicz Spaces Yuxia Tong,Xinru Wang,Jiantao Gu Acta mathematica scientia,Series A. 2020, 40 (6):  1461-1480.  Abstract ( 55 )   RICH HTML PDF (408KB) ( 83 )   Save The paper deals with the gradient estimates in Orlicz spaces for very weak solutions of A-harmonic equations under the assumptions that A satisfies some proper conditions and the given function satisfies some moderate growth condition. References | Related Articles | Metrics

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The Growth and Conditions of Solutions in Some Function Spaces for Complex Linear Differential Equations Weifeng Yang Acta mathematica scientia,Series A. 2020, 40 (6):  1481-1491.  Abstract ( 82 )   RICH HTML PDF (342KB) ( 100 )   Save In this paper, the relationship between the solutions and coefficients of the complex linear differential equation (3.13) $f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_1(z)f'+A_0(z)f=0, z\in {\mathbb D}$ is investigated. A new growth estimate of the solutions is provided. Some sufficient conditions on the coefficients are also obtained such that all solutions belong to the $H_q^\infty$ space, the $F(p, q, s)$ space and the ${\cal N}^p_\alpha$ space. References | Related Articles | Metrics

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The Extensional Koppelman-Leray Type Integral Formulas in the Analytic Varieties of Stein Manifolds Shujin Chen Acta mathematica scientia,Series A. 2020, 40 (6):  1492-1510.  Abstract ( 61 )   RICH HTML PDF (373KB) ( 73 )   Save In this paper, we study how to establish integral formulas for differential forms in the analytic varieties of Stein manifolds. Firstly using different method and technique we derive the corresponding integral representation formulas of differential forms for the complex n-m(0 ≤ m < n) dimensional analytic varieties in two types of bounded domains of Stein manifolds. Secondly we obtain the unified integral representation formulas of differential forms for the complex n-m dimensional analytic varieties in the general bounded domains of Stein manifolds, i.e. Koppelman-Leray type integral formulas for the complex n-m dimensional analytic varieties in the bounded domains of Stein manifolds. In particular, when m=0, the formulas obtained in this paper are the extension of Koppelman-Leray formula in the Stein manifolds. References | Related Articles | Metrics

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Kähler Immersions of Pseudoconvex Hartogs Domains into Complex Space Forms Yihong Hao,An Wang Acta mathematica scientia,Series A. 2020, 40 (6):  1511-1524.  Abstract ( 63 )   RICH HTML PDF (397KB) ( 79 )   Save Let $\Omega$ be a bounded pseudoconvex Hartogs domain in ${\Bbb C}^{n}$. There exists a natural Kähler metric $g^{\Omega}$ in terms of its defining function. In this paper, we study the existence of Kähler immersions of $(\Omega, h g^{\Omega})$ into finite and infinite dimensional complex space forms for $h>0$. References | Related Articles | Metrics

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Existence and Uniqueness of Solutions to the Constant Mean Curvature Equation with Nonzero Neumann Boundary Data in Product Manifold $M^{n}\times{\Bbb R}$ Ya Gao,Jing Mao,Chunlan Song Acta mathematica scientia,Series A. 2020, 40 (6):  1525-1536.  Abstract ( 68 )   RICH HTML PDF (443KB) ( 103 )   Save In this paper, we can prove the existence and uniqueness of solutions to the constant mean curvature (CMC for short) equation with nonzero Neumann boundary data in product manifold $M^{n}\times{\Bbb R}$, where $M^{n}$ is an $n$-dimensional $(n\geq2)$ complete Riemannian manifold with nonnegative Ricci curvature, and ${\Bbb R}$ is the Euclidean 1-space. Equivalently, this conclusion gives the existence of CMC graphic hypersurfaces defined over a compact strictly convex domain $\Omega\subset M^{n}$ and with nonzero Neumann boundary data. References | Related Articles | Metrics

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Generalized Lyapunov Inequalities for a Higher-Order Sequential Fractional Differential Equation with Half-Linear Terms Dexiang Ma,Abdullah Özbekler Acta mathematica scientia,Series A. 2020, 40 (6):  1537-1551.  Abstract ( 65 )   RICH HTML PDF (346KB) ( 123 )   Save In this paper, we establish some new Lyapunov-type inequalities and Hartman-type inequalities for a higher-order sequential Riemann-Liouville fractional boundary value problem with some half-linear terms and a forcing term. We generalize some existing results in literature. References | Related Articles | Metrics

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Cauchy Problem for a Generalized System of Coupled Boussinesq Type Equations Arising from Nonlinear Layered Lattice Model Xiangying Chen,Guowang Chen Acta mathematica scientia,Series A. 2020, 40 (6):  1552-1567.  Abstract ( 68 )   RICH HTML PDF (369KB) ( 94 )   Save In this paper, we prove that the Cauchy problem for a generalized system of the coupled Boussinesq-type equations arising from nonlinear layered lattice model $\begin{eqnarray} &u_{tt}-a_{1}u_{xx}-a_{2}u_{xxtt}+a_{3}(u-w)=f(u_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0, \nonumber\ &w_{tt}-b_{1}w_{xx}-b_{2}w_{xxtt}+b_{3}(w-u)=g(w_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0\nonumber \end{eqnarray}$ has a unique global generalized solution in $C([0, \infty);H^s(\ {\Bbb R})\times H^s(\ {\Bbb R}))(s\geq2$ is a real number) and a unique global classical solution in $C^2([0, \infty);C_B^2(\ {\Bbb R})\times C_B^2(\ {\Bbb R}))(s>\frac{5}{2})$. The sufficient conditions for the blow up of the solution to the Cauchy problem above are given. References | Related Articles | Metrics

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Global Existence and Blowup Phenomena for a Semilinear Wave Equation with Time-Dependent Damping and Mass in Exponentially Weighted Spaces Changwang Xiao,Fei Guo Acta mathematica scientia,Series A. 2020, 40 (6):  1568-1589.  Abstract ( 204 )   RICH HTML PDF (437KB) ( 121 )   Save We consider the global small data solutions and blowup to the Cauchy problem for a semilinear wave equation with time-dependent damping and mass term as well as power nonlinearity. On one hand, if the power of the nonlinearity $p >p_F(N)=1+ \frac 2N$, it is proved that solutions with small initial data exist for all time in exponentially weighted energy spaces. On the other hand, if the power satisfies <p\leq p_F(\alpha, n)=1+\frac{2(1+\alpha)}{N(1+\alpha)-2\alpha}~(0<\alpha<1)$, for some special chosen parameters it is shown that solutions must blow up in finite time provided that the initial data satisfy some integral sign conditions. References | Related Articles | Metrics

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Nontrivial Solution for Schrödinger-Poisson type Systems with Double Critical Terms Xiaojing Feng Acta mathematica scientia,Series A. 2020, 40 (6):  1590-1598.  Abstract ( 108 )   RICH HTML PDF (323KB) ( 106 )   Save The existence of a nontrivial solution to the Schrödinger-Poisson type system with both nonlinear critical growth and nonlocal critical growth is obtained by applying the concentration-compactness principle and mountain pass theorem. The double critical growth in the system presents an obstacle when showing the convergence of the bounded (PS) sequences. At the same time, it is difficult to estimate the critical level of the mountain pass. The key ingredient of the paper is to show that the critical level of the mountain pass is below the non-compactness level of the associated energy functional. References | Related Articles | Metrics

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Weak Solutions to Higher-Order Anisotropic Cahn-Hilliard-Navier-Stokes Systems Jiao Luo,Qian Qi,Hong Luo Acta mathematica scientia,Series A. 2020, 40 (6):  1599-1611.  Abstract ( 52 )   RICH HTML PDF (353KB) ( 133 )   Save In this paper, we are concerned with weak solutions to higher-order anisotropic Cahn-Hilliard-Navier-Stokes systems in a two-dimensional bounded domain. The system consists of an incompressible Navier-Stokes equation coupled with a higher-order anisotropic Cahn-Hilliard equation. First some functional spaces and the definition of weak solutions are introduced. Then energy estimates of the solutions are given, and weak solutions of the system are obtained by Galerkin approximation scheme. Furthermore, the uniqueness of the solution to the system is proved. References | Related Articles | Metrics

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Existence of Multiple Solutions for a Class of Fractional Schrödinger-Kirchhoff Equation Jianli Li,Anran Li,Chongqing Wei,Gang Li Acta mathematica scientia,Series A. 2020, 40 (6):  1612-1621.  Abstract ( 72 )   RICH HTML PDF (348KB) ( 132 )   Save In this article, we use variational method and the critical point theory to study the existence of multiple solutions for a class of Schrödinger-Kirchhoff equation involving the fractional $p$-Laplacian operator $M\bigg(\iint_{ {\ {\Bbb R} }^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\bigg) (-\Delta)_p^{s} u+V(x)|u|^{p-2}u=f(x, u)+\lambda h(x)|u|^{r-2}u, \ x\in {\ {\Bbb R} }^N,$ where $\lambda\in \ {\Bbb R} , 0<s<1<r<p<2, ps<N, (-\Delta)_p^{s}$ is the fractional p-Laplacian operator. Under certain assumptions, we first show the existence of multiple high energy solutions by means of symmetric mountain pass theorem. Secondly, by using dual fountain theorem, we prove that the above equation has a sequence of negative energy solution, whose energy converges to 0. References | Related Articles | Metrics

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Existence and Asymptotic Behavior of Ground State for Fractional p-Laplacian Equations Yaling Han,Jianlin Xiang Acta mathematica scientia,Series A. 2020, 40 (6):  1622-1633.  Abstract ( 76 )   RICH HTML PDF (350KB) ( 165 )   Save In this paper, existence and asymptotic behavior of a fractional p-Laplacian equation with biharmonic potential and critical nonlinear was considered. First, variational method is used to obtain the existence and nonexistence of ground state under different parameter β. Then, the technique of energy estimates was used to get the asymptotic properties of ground state as parameter β tends to critical case. References | Related Articles | Metrics

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Multiple Positive Solutions for a Quasilinear Elliptic System Involving the Caffarelli-Kohn-Nirenberg Inequality and Nonlinear Boundary Conditions Qin Li,Zuodong Yang Acta mathematica scientia,Series A. 2020, 40 (6):  1634-1645.  Abstract ( 72 )   RICH HTML PDF (333KB) ( 79 )   Save This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and nonlinear boundary conditions. By means of variational methods, the existence of at least two positive solutions is obtained. References | Related Articles | Metrics

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Global Existence and Convergence of Solutions to a Chemotactic Model with Logarithmic Sensitivity and Mixed Boundary Conditions Juan Wang,Zixia Yuan Acta mathematica scientia,Series A. 2020, 40 (6):  1646-1669.  Abstract ( 65 )   RICH HTML PDF (438KB) ( 102 )   Save This paper investigates the following chemotactic model with logarithmic sensitivity in a one-dimensional bounded domain: $\begin{eqnarray*}\left\{\begin{array}{lll}\partial_{t}u=Du_{xx}+(u (\ln v)_{x})_{x}, \quad&x\in (0, 1), \quad t>0, \\partial_{t}v=\varepsilon v_{xx}+uv-\mu v, \quad&x\in (0, 1), \quad t>0.\end{array}\right.\end{eqnarray*}$ By using a Cole-Hopf type transformation, we transform the above singular repulsive chemotaxis model into a non-singular system of the form $\begin{eqnarray*}\left\{\begin{array}{lll}p_t = p_{xx} + (pq)_x,\quad & x\in (0,1),\quad t>0,\\q_t = \varepsilon q_{xx} + \varepsilon (q^2)_x + p_x,\quad & x\in (0,1),\quad t>0,\\\end{array}\right.\end{eqnarray*}$ Then under some mixed boundary conditions, we prove the global existence and exponential convergence of solutions to the initial-boundary value problem of the above system with regular initial data. References | Related Articles | Metrics

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On the Complete Moment Convergence for Weighted Sums of Rowwise Asymptotically Almost Negatively Associated Random Variables Bing Meng,Dingcheng Wang,Qunying Wu Acta mathematica scientia,Series A. 2020, 40 (6):  1670-1681.  Abstract ( 78 )   RICH HTML PDF (292KB) ( 119 )   Save In this paper, by applying the moment inequality for AANA random sequence, some results on complete moment convergence for weighted sums of AANA random variables are obtained without assumptions of identical distribution, which generalize and improve the corresponding ones of Baek et al. [1] and Wang et al. [12], respectively. References | Related Articles | Metrics

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Statistical Identification of Reversible Markov Chain on Cyclic Graph Xuyan Xiang,Haiqin Fu,Jieming Zhou,Yingchun Deng,Xiangqun Yang Acta mathematica scientia,Series A. 2020, 40 (6):  1682-1698.  Abstract ( 82 )   RICH HTML PDF (542KB) ( 143 )   Save The statistical identification of Markov chain explores how to identify the transition rate matrix of the underlying Markov chain by partially observable data. SIMC on reversibly cyclic graphs (containing one cycle at least), as the most important and crucial class, is investigated then in this letter. As the differentials of hitting time distribution for a reversible Markov chain are expressed by taboo rates, the necessary condition is developed to identify the transition rate matrix and a general conclusion about sufficiency is provided. The proposed algorithms to exactly calculate all transition rates are developed. A numerical example is included to demonstrate the correctness of the proposed algorithms. Figures and Tables | References | Related Articles | Metrics

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Numerical Methods for the Critical Temperature and Gap Solution of Bogoliubov-Tolmachev-Shirkov Model Zhihao Ge,Ruihua Li Acta mathematica scientia,Series A. 2020, 40 (6):  1699-1711.  Abstract ( 55 )   RICH HTML PDF (344KB) ( 107 )   Save In the paper, the max-mixed scheme and min-mixed scheme are proposed for the critical temperature and gap solution of Bogoliubov-Tolmachev-Shirkov model in superconductivity theory. For the first time, the numerical solutions of critical temperature and gap solution are given for the above model with the alternating kernel function. The convergence analysis of the numerical method is also given. Also, some numerical examples are presented to verify the results of the paper. Figures and Tables | References | Related Articles | Metrics

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Stability of a Class of Nonlinear Hierarchical Age-Dependent Population Model Zerong He,Zhiqiang Zhang,Yang Wang Acta mathematica scientia,Series A. 2020, 40 (6):  1712-1722.  Abstract ( 97 )   RICH HTML PDF (536KB) ( 117 )   Save The article is concerned with the existence of positive equilibria and stability of zero state in a nonlinear hierarchical species. Based on the assumption that young individuals are more competitive than older ones, an integro-partial differential equation is taken to model the revolution process of the population. The net reproductive number is defined and used to show that there are positive steady states in the system. Furthermore, stability results for zero equilibrium are derived via the characteristic equation and a Liapunov function. Finally, some numerical experiments are presented. Figures and Tables | References | Related Articles | Metrics