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26 February 2024, Volume 44 Issue 1 Previous Issue    Next Issue

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Robust Accessible Hyperbolic Repelling Sets Xiao Jianrong Acta mathematica scientia,Series A. 2024, 44 (1):  1-11.  Abstract ( 221 )   RICH HTML PDF (800KB) ( 333 )   Save By operating Denjoy like surgery on a piecewise linear map, we constructed a family of$C^1$maps$f_\alpha \ (1<\alpha<3 )$admitting the following properties: 1)$f_\alpha$admits a hyperbolic repelling Cantor set$\mathcal{A}_\alpha$with positive Lebesgue measure, and$\mathcal{A}_\alpha$is also a wild attractor of$f_{\alpha}$; 2) The attractor$\mathcal{A}_\alpha$is accessible: the difference set$\mathbb{B}(A_\alpha)\backslash A_\alpha$between the basin of attraction$\mathbb{B}(A_\alpha)$and$A_\alpha$has positive Lebesgue measure; 3) The family is structurally stable:$f_{\alpha}$is topologically conjugate to$f_{\alpha'}$for all$1<\alpha,\ \alpha'<3$. The surgery involves blowing up the discontinuity and its preimages set into open intervals. The$C^1$smoothness of$f_{\alpha}$is ensured by the prescribed lengths of glued intervals and the maps defined on the glued intervals. Figures and Tables | References | Related Articles | Metrics

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The Fine Pseudo-spectra of$2 \times 2$Diagonal Block Operator Matrices Shen Runshuan, Hou Guolin Acta mathematica scientia,Series A. 2024, 44 (1):  12-25.  Abstract ( 127 )   RICH HTML PDF (557KB) ( 207 )   Save Let$A$,$B$be densely closed linear operators in a separable Hilbert space$X$and$M_{0}=\left( \begin{array} {cc}{A} & {0}\\ {0}& {B} \end{array} \right)$be the corresponding$2\times2$block operator matrices. In this paper, we establish the fine pseudo-spectra of$M_{0}$including the pseudo-point spectrum, the pseudo-residual spectrum, and the pseudo-continuous spectrum under diagonal perturbation, which are, respectively, compared with its point spectrum, residual spectrum, and continuous spectrum. And a concrete example is constructed to justify the proved result. Finally, we obtain the pseudo-point spectrum of$M_{0}$under the upper-triangular perturbation by using the technology of space decomposition. References | Related Articles | Metrics

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A Vanishing Theorem for$p$-harmonic$\ell$-forms in Space with Constant Curvature Zhang Youhua Acta mathematica scientia,Series A. 2024, 44 (1):  26-36.  Abstract ( 108 )   RICH HTML PDF (551KB) ( 183 )   Save Let$M^{n}(n \geq 3)$be a complete non-compact submanifold immersed in a space with constant curvature$N^{n+m}(c)$with flat normal bundle. By using Bochner-Weitzenböck formula, Sobolev inequality, Moser iteration and Fatou lemma, we prove that every$L^{\beta}~p$-harmonic forms on$M$is trivial if$M^{n}$satisfies some geometic conditions, where$\beta\geq p\geq 2$. References | Related Articles | Metrics

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An Effective Fourier Spectral Approximation for Fourth-Order Eigenvalue Problems with Periodic Boundary Conditions He Ya, An Jing Acta mathematica scientia,Series A. 2024, 44 (1):  37-49.  Abstract ( 103 )   RICH HTML PDF (624KB) ( 201 )   Save In this paper, we put forward an effective Fourier spectral approximation method for fourth-order eigenvalue problems with periodic boundary conditions. Firstly, we introduce the appropriate Sobolev space and the corresponding approximation space according to the periodic boundary conditions, establish a weak form of the original problem and its discrete form, and derive the equivalent operator form. Then we define an orthogonal projection operator and prove its approximation properties. Combined with the spectral theory of compact operators, we prove the error estimates of approximation eigenvalues. In addition, we construct a set of basis functions of the approximation space, and derive the matrix form based on tensor product associated with the discrete scheme. Finally, we provide some numerical examples, and the numerical results show our algorithm is effective and spectral accuracy. Figures and Tables | References | Related Articles | Metrics

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The Boundedness for Multilinear Commutators of One-Sided Operators on Triebel-Lizorkin Spaces Cheng Xin, Zhang Jing Acta mathematica scientia,Series A. 2024, 44 (1):  50-59.  Abstract ( 79 )   RICH HTML PDF (543KB) ( 183 )   Save In this paper, we study the boundedness of multilinear commutators generated by singular integral operators, discrete square function, Weyl fractional integrals with Lipschitz functions from weighted Lebesgue to weighted Triebel-Lizorkin spaces by extrapolation of weights in the sense of one-side. References | Related Articles | Metrics

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Multiplicity of Positive Solutions to Subcritical Choquard Equation Wen Ruijiang, Liu Fanqin, Xu Ziyi Acta mathematica scientia,Series A. 2024, 44 (1):  60-79.  Abstract ( 80 )   RICH HTML PDF (666KB) ( 171 )   Save In this paper, we are concerned with the multiplicity of solutions for the following subcritical Choquard equation $\begin{equation*} \begin{cases} -{\Delta}{u}+(\lambda V(x)+1)u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^{p_{\varepsilon}}}{|x-y|^\mu}{\rm d}y\Big)|u|^{p_{\varepsilon}-2}u,\quad x\in\mathbb{R}^N,\\{u\in{{H}}^{1}{(\mathbb{R}^N)}}, \end{cases}\end{equation*}$ where$N>3$,$\lambda$is a real parameter,$p_{\varepsilon}=2^\ast_{\mu}-\varepsilon$,$\varepsilon>0$,$\mu\in(0,N)$and$2^\ast_{\mu}=\frac{2N-\mu}{N-2}$is the critical Hardy-Littlewood-Sobolev exponent. Suppose that$\Omega:={\rm int}\,V^{-1}(0)$is a nonempty bounded domain in$\mathbb{R}^N$with smooth boundary, using Lusternik-Schnirelman theory, we prove the problem (0.1) has at least$cat_\Omega(\Omega)$positive solutions for$\lambda$large and$\varepsilon$small enough. References | Related Articles | Metrics

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Multiple Homoclinic Solutions for the Kirchhoff-type Difference Equations with Unbounded Potential Wang Zhenguo, Ding Lianye Acta mathematica scientia,Series A. 2024, 44 (1):  80-92.  Abstract ( 50 )   RICH HTML PDF (597KB) ( 153 )   Save In this paper, we study the existence of multiple homoclinic solutions for the Kirchhoff-type difference equations with unbounded potential by using critical point theory. In our work, the nonlinearity is allowed to grow sublinearly, and some technical methods are used to verify the energy functional satisfying the Palais-Smale conditions. Finally, one example is given to illustrate our main results. References | Related Articles | Metrics

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Existence Result for a Class of Nonlinear Parabolic Equations with$p(\cdot,\cdot)$Structure and an$L^1$Source Li Zhongqing Acta mathematica scientia,Series A. 2024, 44 (1):  93-100.  Abstract ( 64 )   RICH HTML PDF (570KB) ( 124 )   Save This paper is devoted to proving an existence result to a class of nonlinear parabolic equations with variable exponents. The zero order term produces a regularizing effect, which helps to get the$L^\infty$estimate despite the poor summability of the source term. By choosing some appropriate test functions, we obtain the necessary a priori estimates for the approximate solution sequences denoted by$\{u^\epsilon\}_\varepsilon$. Thanks to the Young measure method, the weak convergence of the nonlinear term is identified. References | Related Articles | Metrics

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Multiplicity of High Energy Solutions for a Class of Nonlocal Critical Elliptic System Fu Peiyuan, Xia Aliang Acta mathematica scientia,Series A. 2024, 44 (1):  101-119.  Abstract ( 68 )   RICH HTML PDF (638KB) ( 156 )   Save By using variational methods and topological degree theory, this paper proved a class of coupled nonlocal elliptic system involving the Hardy-Littlewood-Sobolev critical exponents has at least two positive high energy solutions. References | Related Articles | Metrics

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Structural Stability of Temperature Dependent Double Diffusion Model on a Semi-infinite Cylinder Li Yuanfei, Li Dandan, Shi Jincheng Acta mathematica scientia,Series A. 2024, 44 (1):  120-132.  Abstract ( 43 )   RICH HTML PDF (543KB) ( 159 )   Save A simplified temperature dependent double diffusion model defined on a semi-infinite cylinder is studied. By using a prior estimates and weighted energy analysis, it is proved that the solution of the model decays exponentially with the space variable when the boundary conditions satisfy certain constraints. The structural stability of the solution to the interaction coefficient is obtained by using the prior bounds and decay result of the solution. References | Related Articles | Metrics

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Stabilization of Degenerate Wave Equations with Delayed Boundary Feedback Bai Jinyan, Chai Shugen Acta mathematica scientia,Series A. 2024, 44 (1):  133-139.  Abstract ( 60 )   RICH HTML PDF (491KB) ( 156 )   Save In this paper, we study the stabilization of degenerate wave equations with time-delay boundary feedback. Firstly, the well-posedness of the solution is proved by using semigroup theory. And then the exponential stability of the solution is proved by selecting suitable multipliers and constructing suitable Lyapunov functional. References | Related Articles | Metrics

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Periodic Traveling Wave Solutions of Delayed SEIR Systems with Nonlocal Effects Zhang Guangxin, Yang Yunrui, Song Xue Acta mathematica scientia,Series A. 2024, 44 (1):  140-159.  Abstract ( 68 )   RICH HTML PDF (660KB) ( 159 )   Save In this paper, the periodic traveling wave solutions to a class of delayed SEIR systems with nonlocal effects and nonlinear incidence are investigated. Firstly, the existence of periodic traveling waves is transformed into the fixed point problem of an non-monotone operator defined on a closed convex set by defining the basic reproducing number$\Re_{0}$and constructing appropriate upper and lower solutions, and thus the existence of periodic traveling waves of the system is established by using Schauder fixed point theorem and limit theory. Secondly, the non-existence of periodic traveling wave solutions of the system is proved when the basic regeneration number$\Re_{0}<1$by contradictory arguments and comparison principle. References | Related Articles | Metrics

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Multi-scale Image Decomposition Based on ADI Format Luo Xiaoyi, Han Huan, Zhang Yimin Acta mathematica scientia,Series A. 2024, 44 (1):  160-172.  Abstract ( 53 )   RICH HTML PDF (3119KB) ( 118 )   Save For the multi-scale image decomposition model of [Xu R et al., IEEE Trans. Biomed. Eng., 2014], in this paper, a multi-scale image decomposition algorithm based on alternating direction implicit(ADI) scheme is proposed. The convergence and stability of ADI scheme under this model are proved. Further, numerical experiments on different images show that the proposed algorithm has better texture extraction performance. Figures and Tables | References | Related Articles | Metrics

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Global Convergence of a WYL Type Spectral Conjugate Gradient Method Cai Yu, Zhou Guanghui Acta mathematica scientia,Series A. 2024, 44 (1):  173-184.  Abstract ( 55 )   RICH HTML PDF (602KB) ( 155 )   Save In order to solve large scale unconstrained optimization problems, this paper combines the WYL conjugate gradient method with the spectral conjugate gradient method to give a WYL type spectral conjugate gradient method. Without relying on any line search, the search directions generated by the method satisfy the sufficient descent condition. Compared with the convergence of the WYL conjugate gradient method, the spectral WYL conjugate gradient method extends the range of values of the parameter$\sigma$in the line search. Finally, the corresponding numerical results show that the method is effective. Figures and Tables | References | Related Articles | Metrics

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Duality Characterizations for a Class of Two-Stage Adjustable Robust Multiobjective Programming Huang Jiayi, Sun Xiangkai Acta mathematica scientia,Series A. 2024, 44 (1):  185-194.  Abstract ( 54 )   RICH HTML PDF (541KB) ( 144 )   Save In this paper, we deal with a class of two-stage adjustable robust multi-objective programming problems with spectrahedral uncertainty data in both objective and constraints. We first establish a Farkas lemma for a two-stage adjustable robust multiobjective programming with affine adjustable variable. Then, we propose a semidefinite programming dual problem for this multiobjective programming problem. Furthermore, in terms of the obtained Farkas lemma, we explore duality properties between them. References | Related Articles | Metrics

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Research on the Convergence Rate of Bregman ADMM for Nonconvex Multiblock Optimization Chen Jianhua, Peng Jianwen Acta mathematica scientia,Series A. 2024, 44 (1):  195-208.  Abstract ( 120 )   RICH HTML PDF (638KB) ( 169 )   Save Wang et al proposed the alternating direction method of multipliers with Bregman distance (Bregman ADMM) for solving multi-block separable nonconvex optimization problems with linear constraints, and proved its convergence.In this paper, we will further study the convergence rate of Bregman ADMM for solving multi-block separable nonconvex optimization problems with linear constraints, and the sufficient conditions for the boundedness of the iterative point sequence generated by the algorithm.Under the Kurdyka-Łojasiewicz property of benefit function, this paper establish the convergence rates for the values and iterates, and we show that various values of KŁ-exponent associated with the objective function can obtain Bregman ADMM with three different convergence rates. More precisely, this paper proves the following results:if the(KŁ) exponent of the benefit function$\theta=0$, then the sequence generated by Bregman ADMM converges in a finite numbers of iterations; if$\theta \in \left (0, \frac{1}{2}\right ]$, then Bregman ADMM is linearly convergent; if$\theta \in \left ( \frac{1}{2}, 1\right )$, then Bregman ADMM is sublinear convergent. References | Related Articles | Metrics

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Spatio-temporal Dynamics of HIV Infection Model with Periodic Antiviral Therapy and Nonlocal Infection Wu Peng, He Zerong Acta mathematica scientia,Series A. 2024, 44 (1):  209-226.  Abstract ( 82 )   RICH HTML PDF (2753KB) ( 183 )   Save In this paper, a non-autonomous reaction-diffusion HIV cell model is established to study the effects of periodic antiviral therapy, non local infection and spatial heterogeneity on HIV infection spatio-temporal dynamics. Specifically, we derive the functional expression of the basic regeneration number of the model$\mathcal{R}_0$, which is defined by the spectral radius of the next generation regeneration operator$\mathcal{R}$. Then the dynamical behaviors of the model is analyzed, including the global stability of the HIV infection-free steady state, the uniform persistence of HIV infection and the existence of periodic positive steady state. Finally, we conduct some numerical simulations to verify the theoretical results, and study the influence of relevant important factors on the process of HIV infection. Our works can provide some valuable reference suggestions for the clinical treatment of HIV infection. Figures and Tables | References | Related Articles | Metrics

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Global Dynamics Analysis of Anthrax Transmission Model Based on Two Kinds of Vectors Han Mengjie, Liu Junli, Zhang Tailei Acta mathematica scientia,Series A. 2024, 44 (1):  227-245.  Abstract ( 69 )   RICH HTML PDF (840KB) ( 158 )   Save In order to study the influence of vectors on anthrax transmission in animal populations, a deterministic infectious disease model is established based on the transmission mechanism of necrophilic flies and hematophagous flies. The nonnegativity and boundedness of the solutions of the model are proved by using the basic theorem of differential equations, the sufficient conditions for the existence of equilibria are given, and several kinds of reproduction numbers of the model are defined, the stability of the equilibria is analyzed by means of linearization and M-matrix theory, and the persistence of the disease is also studied. The effects of parameters on the basic reproduction number are studied by numerical simulations. The results show that timely cleaning of infected carcasses, eliminating fly breeding sites as much as possible and the use of insecticides on flies can inhibit the spread of anthrax in animal populations. Figures and Tables | References | Related Articles | Metrics

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Entanglement Measure of Two-body Quantum System Based on the Wedge Product of Vectors Yang Zhaodi, He Kan, Duan Zhoubo Acta mathematica scientia,Series A. 2024, 44 (1):  246-256.  Abstract ( 100 )   RICH HTML PDF (570KB) ( 203 )   Save The characterization of quantum entanglement is an unsolved problem. It is interested to measure entanglement based on the geometric properties of vectors because a quantum state is represented as a unit vector in a Hilbert space. Some scholars have defined an entanglement measure on the two-body pure state system$C^{2}\otimes C^{2}$based on the modulus length of the wedge product of two vectors, whose modulus length corresponds geometrically to the area of an oriented parallelogram on a plane. In the work, we give the entanglement measures on the two-body pure state system$C^{3}\otimes C^{3}$and$C^{d}\otimes C^{d}$by using the modular length of the wedge product of vectors. They geometrically correspond to the volume of an oriented parallelepiped and$d\times(d-1)\times\cdots\times4$oriented parallelepipeds. In addition, We propose a geometric criterion for determining separable states. The results show that the entanglement measure$E$defined based on the geometric background of mathematics is a simple and intuitive measurement method. Figures and Tables | References | Related Articles | Metrics