#### Table of Content

26 June 2020, Volume 40 Issue 3
 On Approximation by Bernstein-Durrmeyer-Type Operators in Movable Compact Disks Zhaojun Pang,Dansheng Yu,Ping Zhou Acta mathematica scientia,Series A. 2020, 40 (3):  545-555.  Abstract ( 14 )   RICH HTML PDF (294KB) ( 26 )   To approximate analytic functions in movable compact disks, we introduce a new kind of Bernstein-Durrmeyer-Type polynomials. The order of simultaneous approximation rate of the new polynomials in the movable compact disks is given.
 The Approximation and Growth of Entire Function Represented by Laplace-Stieltjes Transform with Infinite Order Hongyan Xu,Sanyang Liu Acta mathematica scientia,Series A. 2020, 40 (3):  556-568.  Abstract ( 16 )   RICH HTML PDF (368KB) ( 13 )   The main purpose of this article is to investigate the growth and approximation of Laplace-Stieltjes transform with irregular growth converges in the whole plane, by introducing the concept of the double lower q-type. We obtain some relation theorems concerning the double lower q-type, the error, An* and λn, which are extension and improvement of the previous theorems given by Luo-Kong, Singhal-Srivastava.
 Extended Cesàro Operator from Weighted Bergman Spaces to Ƶμ Type Spaces on the Unit Ball Yanhui Zhao,Xiuyun Wu,Chunyan Liao Acta mathematica scientia,Series A. 2020, 40 (3):  569-578.  Abstract ( 13 )   RICH HTML PDF (314KB) ( 14 )   Some questions of extended Cesàro operator from weighted Bergman space to Ƶμ type spaces in the unit ball were studied in this paper. By the methods of functional analysis and several complex variables, the necessary and sufficient conditions are given for extended Cesàro operator to be bounded and compact from weighted Bergman space to Ƶμ type spaces in the unit ball.
 Some Characterizations of Weakly Quasisymmetric Mappings in Quasi-Metric Spaces Hongjun Liu,Xiaojun Huang Acta mathematica scientia,Series A. 2020, 40 (3):  579-588.  Abstract ( 12 )   RICH HTML PDF (305KB) ( 20 )   This paper is to investigate the properties of weakly quasisymmetric mappings in quasi-metric space. Introduce the concept of a ring and ring properties, and the properties of ring are used to describe the some characterizations of weakly quasisymmetric mappings between two quasi-metric spaces.
 Disjointness of Generalized Frames Wei Zhang,Yunzhang Li Acta mathematica scientia,Series A. 2020, 40 (3):  589-596.  Abstract ( 8 )   RICH HTML PDF (304KB) ( 3 )   The notion of disjointness of frames in Hilbert spaces was firstly introduced by Han and Larson, it is closely related with super frames in Hilbert spaces, and plays an important role in construction of super frames and frames. G-frames is a generalization of frames in Hilbert spaces. In this paper, we establish characterization of disjointness of g-frames, strong disjointness of g-frames and weak disjointness of g-frames in terms of super g-frames; With the results obtained, we give different proof method of the known theorem; We obtain the relation between strongly disjoint and weakly disjoint of dual g-frames; Finally, we use strong disjointness of g-frames to construct (super) dual g-frames, which cover the results obtained by other authors.
 Boundedness of Marcinkiewicz Integral and Its Commutator on Non-Homogeneous Metric Measure Spaces Yaoyao Han,Kai Zhao Acta mathematica scientia,Series A. 2020, 40 (3):  597-610.  Abstract ( 8 )   RICH HTML PDF (367KB) ( 7 )   Let $({\cal X}, d, \mu)$ be a non-homogeneous metric measure space satisfying both the geometrically doubling and the upper doubling conditions. By using the properties of non-homogeneous metric measure space and inequality technique, the authors proved that the Marcinkiewicz integral operator and its commutator are bounded on Herz spaces and Herz type Hardy spaces with non-homogeneous metric measure space.
 Convergence in Lr for Lp-Mixingale Arrays Dehua Qiu,Ju Yang,Yanchun Yi Acta mathematica scientia,Series A. 2020, 40 (3):  611-618.  Abstract ( 7 )   RICH HTML PDF (243KB) ( 13 )   In this paper, the convergence in Lr for Lp-mixingale arrays are discussed by using the properties of Lp-mixingale. These results extend and improve the related known works in the literature.
 Limit Cycles Bifurcations of Liénard System of Degree Four with One Nilpotent Cusp Yi Shao,Chunxiang A Acta mathematica scientia,Series A. 2020, 40 (3):  619-630.  Abstract ( 7 )   RICH HTML PDF (394KB) ( 4 )   In this paper, we study Poincaré bifurcation and Hopf bifurcation of a class of Liénard system of the form ẋ=y, ẏ=f(x)+εg(x)y, where f(x) and g(x) are polynomials of degree 4 and 3, respectively. It is proven that this system can produce at most three limit cycles surrounding the origin.
 Infinite Series Involving Central Binomial Coefficients and Generalized Harmonic Numbers Hongmei Liu Acta mathematica scientia,Series A. 2020, 40 (3):  631-640.  Abstract ( 8 )   RICH HTML PDF (290KB) ( 6 )   In this paper, by appling Gauss's two hypergeometric summation formulas, we derive some infinite series expressions for the central binomial coefficients and the generalized harmonic numbers.
 A Strongly Convergent Generalized Gradient Projection Method for Minimax Optimization with General Constraints Guodong Ma Acta mathematica scientia,Series A. 2020, 40 (3):  641-649.  Abstract ( 5 )   RICH HTML PDF (385KB) ( 5 )   In this paper, minimax optimization problems with inequality and equality constraints is discussed. The original problem is transformed into an associated simple problem with a penalty term and only inequality constraints, then a new generalized gradient projection algorithm is presented. The main characters of the proposed algorithm are as follows:the improved search direction is generated by only one generalized gradient projection explicit formula; the new optimal identification function is introduced; the algorithm is globally and strongly convergent under some mild assumptions. Finally, the numerical results show that the proposed algorithm is promising.
 Spectral Regularization Method for Volterra Integral Equation of the First Kind with Noise Data Lixin Feng,Xiaoxu Yang Acta mathematica scientia,Series A. 2020, 40 (3):  650-661.  Abstract ( 6 )   RICH HTML PDF (377KB) ( 1 )   The main purpose of this work is to solve the Volterra integral equation of the first kind with noise data by using both a Legendre-collocation method and a regularization strategy. We provide a rigorous convergence analysis for the proposed method. Some numerical tests are illustrated to demonstrate the validity and effectiveness of the proposed method.
 The Simple Proof and Generalization of a Conjecture Concerning Generalized Legendre Identity Miaokun Wang,Yuming Chu,Songliang Qiu Acta mathematica scientia,Series A. 2020, 40 (3):  662-666.  Abstract ( 12 )   RICH HTML PDF (268KB) ( 16 )   In this paper, some monotonicity properties of certain combinations of Gaussian hypergeometric function are proved, and using these properties, the simple proof and generalization of a conjecture about generalized Legendre identity are presented, which will be helpful to the study of special functions.
 Spectral Property of Some Self-Affine Measures with N-Element Digits on ${{\mathbb{R}}^{n}}$ Hongguang Li,Pengfei Zhang Acta mathematica scientia,Series A. 2020, 40 (3):  667-675.  Abstract ( 6 )   RICH HTML PDF (355KB) ( 4 )   Let $R \in M_n({\Bbb Z})$ be an expanding matrix and ${\cal D}=\{0, a_1, a_2, \cdots, a_{N-1}\}u \equiv \{0, 1, \cdots, N-1\}u \ ({\rm mod}N)$ be a $N$-element digit set, where $u\in {\Bbb Z}^n\setminus\{0\}$. In this paper, we study the spectral property of the self-affine measures $\mu_{R, {\cal D}}$ which is generated by ${\cal D}$ and $R$, and obtain a sufficient condition such that $\mu_{R, {\cal D}}$ is a spectral measure. Moreover, for a special case, we give a necessary and sufficient condition such that $\mu_{R, {\cal D}}$ is a spectral measure, and the exact spectrum of $\mu_{R, {\cal D}}$ is given.
 Upper Box Dimension of a Class of Homogeneous Moran Sets Jingru Zhang,Yanzhe Li,Manli Lou Acta mathematica scientia,Series A. 2020, 40 (3):  676-683.  Abstract ( 10 )   RICH HTML PDF (316KB) ( 8 )   In this paper, we construct a special homogeneous moran set:{mk}-quasi-homogeneous perfect set by the connected components and the gaps, and prove that the upper box dimension and packing dimension of the set can get the maximum value of the homogeneous moran set under the condition sup{mk} < ∞. We also obtain the range of the upper box dimension of the set under some conditions and find a sufficient condition for getting the exact expression of the upper box dimension.
 A Triangular Prism Finite Element for the Second-Order Elliptic Mixed Problem Zhongjian Zhao,Shaochun Chen Acta mathematica scientia,Series A. 2020, 40 (3):  684-693.  Abstract ( 8 )   RICH HTML PDF (304KB) ( 6 )   There are many researches on the finite element method for second-order elliptic mixed problem, including triangular element, rectangular element, tetrahedral element and cubic element. However, there are few researches on the triangular prism element. The triangular prism element has the advantages of triangular and rectangular elements, and it is more suitable for cylindrical region, especially for the cylindrical region with complex cross-section. In this paper, a lower-order conforming triangular prism element is constructed for the second-order elliptic mixed problem. Its well-posedness and convergence are proved, and the optimal error estimate is given too.
 Nonlinear Integrable Couplings and Bargmann Symmetry Constraint of Super Generalized-Burgers Hierarchy Fang Fang,Beibei Hu,Ling Zhang Acta mathematica scientia,Series A. 2020, 40 (3):  694-704.  Abstract ( 9 )   RICH HTML PDF (302KB) ( 3 )   With the help of the enlarging Lie super algebra, we construct nonlinear integrable couplings for coupled generalized-Burgers hierarchy in this paper. Then, we establish its super-Hamiltonian structures by utilizing super trace identity. Furthermore, we obtain the Bargmann Symmetry Constraint of super generalized-Burgers hierarchy.
 Existence and Multiplicity of Solutions for a Coupled System of Impulsive Differential Equations via Variational Method Wangjin Yao Acta mathematica scientia,Series A. 2020, 40 (3):  705-716.  Abstract ( 13 )   RICH HTML PDF (321KB) ( 20 )   In this paper, we consider a class of coupled system of impulsive differential equations with p-Laplacian operator and obtain the existence and multiplicity of solutions of it with Dirichlet boundary conditions via variational method.
 A Super Order Regularization Method for Determination of an Unknown Source in the Heat Equation Zhenyu Zhao,Riguang Lin,Zhi Li,Duan Mei Acta mathematica scientia,Series A. 2020, 40 (3):  717-724.  Abstract ( 6 )   RICH HTML PDF (345KB) ( 10 )   The problem for determining an unknown source in the heat equation is considered in this paper. We present a Tikhonov regularization method with a super order penalty term to deal with illposedness of the problem. The regularization parameter is chosen by a discrepancy principle and the order optimal error bounds can be obtained for various smooth conditions. The smoothness parameter and the a priori bound of exact solution are not needed in the numerical process. Numerical tests show that the proposed method is effective and stable.
 A Symmetry Result for a Class of p-Laplace Involving Baouendi-Grushin Operators via Constrained Minimization Method Hongli Qian,Xiaotao Huang Acta mathematica scientia,Series A. 2020, 40 (3):  725-734.  Abstract ( 11 )   RICH HTML PDF (341KB) ( 11 )   The purpose of this paper is to investigate a spacial p-Laplace equation involving Baouendi-Grushin operators. Some existence and symmetry results for positive solutions are obtained by rearrangement of its corresponding constrained minimization. These results are in accordance with those for the classical p-Laplace equations and the Baouendi-Grushin type sub-Laplacian equations.
 Blow-Up Analysis for a Weakly Coupled Reaction-Diffusion System with Gradient Sources Terms and Time-Dependent Coefficients Yadong Zheng,Zhongbo Fang Acta mathematica scientia,Series A. 2020, 40 (3):  735-755.  Abstract ( 4 )   RICH HTML PDF (429KB) ( 4 )   This paper investigate the blow-up phenomena for a weakly coupled reaction-diffusion system with gradient sources terms and time-dependent coefficients subject to null Dirichlet boundary condition. By virtue of the differential inequality technique and comparison principle, we derive some sufficient conditions to guarantee that the solutions exist globally or blow up in finite time under several different measure sense. Moreover, the bounds for the blow-up time of the blow-up solution are obtained in higher dimensional space.
 Random Exponential Attractor for Non-Autonomous Stochastic FitzHugh-Nagumo System with Multiplicative Noise in R3 Zongfei Han,Shengfan Zhou Acta mathematica scientia,Series A. 2020, 40 (3):  756-783.  Abstract ( 6 )   RICH HTML PDF (467KB) ( 0 )   The article considers the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for non-autonomous stochastic FitzHugh-Nagumo system in R3, which deduces that the long-term behavior of solutions of system can be characterized by finite independent parameters. The proof is based on the "tail" estimation of solutions of systems and decomposing the difference of two solutions into three parts that one part belongs to finite-dimensional space and both of other two parts become small enough when both time variable and space variable are large enough.
 Oscillation Conditions of Certain Nonlinear Impulsive Neutral Parabolic Distributed Parameter Systems Liping Luo,Zhenguo Luo,Yunhui Zeng Acta mathematica scientia,Series A. 2020, 40 (3):  784-795.  Abstract ( 6 )   RICH HTML PDF (392KB) ( 11 )   The oscillation problems for a class of nonlinear impulse parabolic distributed parameter systems with neutral term and higher order Laplace operator are investigated under first boundary value condition. By using the technique of treating neutral term and higher order Laplace operator and integral averaging method, some new sufficient criteria are established for the oscillation of all solutions of such systems. The conclusions fully indicate that the system oscillation is caused by impulse and delay.
 Smoothness for the Renormalized Self-Intersection Local Time of Bifractional Brownian Motion Liheng Sang,Zhenlong Chen,Xiaozhen Hao Acta mathematica scientia,Series A. 2020, 40 (3):  796-810.  Abstract ( 9 )   RICH HTML PDF (388KB) ( 5 )   Let BH, K={BH, K(t), t ≥ 0 } be a bifractional Brownian motion in Rd with Hurst indexes H ∈ (0, 1) and K ∈ (0, 1]. This process constitutes a natural generalization of fractional Brownian motion(which is obtained for K=1). In this paper, we research the smoothness of the renormalized self-intersection local time of BH, K. By the chaos expansion method of Malliavin analysis, we obtain the smoothness of the renormalized self-intersection local time of BH, K in the sense of Meyer-Watanabe. And our result generalizes that of fractional Brownian motion.
 Research on Signal-to-Noise Ratio in Order Selection of AR Model Zhigang Wang,Yiming Ding Acta mathematica scientia,Series A. 2020, 40 (3):  811-823.  Abstract ( 8 )   RICH HTML PDF (720KB) ( 8 )   There are many methods can be used to determine the order of AR models. For specific time series, different method may provide different results. How to select method adaptively for particular series is an important problem, especially in big data era. In this paper, we introduce a method to estimate the signal-to-noise ratio (SNR) of the AR model in low-order noisy environments. It takes the influence of noise standard deviation, series length and eigenvalue of the model into consideration, which can be used as a criterion to evaluate the accuracy of AIC, BIC and FPE. The experimental results show that when the eigenvalue satisfies|λ1|=|λ2|=…=|λp|=|λmax|, the order determination accuracy reaches the maximum under the condition of maximum eigenvalue. The accuracy is positively correlated with the series length and the distance of eigenvalue from origin, independent of noise standard deviation. Finally, based on the experimental results, we can select the order determination method of AR model according to the SNR of converted reference model, which provides a new perspective on the comparison of the advantages and disadvantages in different order determination methods.
 Stability Analysis for a Class Neural Network with Proportional Delay Based on the Gronwall Integral Inequality Xingshou Huang,Ricai Luo,Wusheng Wang Acta mathematica scientia,Series A. 2020, 40 (3):  824-832.  Abstract ( 9 )   RICH HTML PDF (454KB) ( 8 )   When people study time-delay neural network, structuring Lyapunov function is usually used to analyze the stability of the system. In this paper, we study the stability of a class of neural network with proportional delay by using Gronwall integral inequalities, and obtain the new criterion of global exponential stability of Hopfield neural network and its proportional delay system. Finally, we demonstrate the validity of the results by an numerical example.