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    Blow-Up of the Smooth Solutions to the Quantum Navier-Stokes-Landau-Lifshitz Equations
    Zhen Qiu,Guangwu Wang
    Acta mathematica scientia,Series A    2022, 42 (4): 1074-1088.  
    Abstract205)      PDF(pc) (346KB)(123)       Save

    In this paper, we investigate the blow-up of the smooth solutions to the quantum Navier-Stokes-Landau-Lifshitz systems(QNSLL) in the domains $\Omega \subseteq \mathbb{R} ^n(n =1, 2)$. We prove that the smooth solutions to the QNSLL will blow up in finite time in the domains half-space $\mathbb{R} _+^n$, whole-space $\mathbb{R} ^n$ and ball. The paper also shows that the blow-up time of the smooth solutions in half-space or whole-space only depends on boundary conditions, while the the blow-up time of the smooth solutions in the ball depends on initial data and boundary conditions. In particular, the above conclusions are also valid for NSLL systems.

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    The General Inverse Bonnesen-Style Inequalities in $\mathbb{R}^n$
    Xu Dong,Yan Zhang,Chunna Zeng,Xingxing Wang
    Acta mathematica scientia,Series A    2022, 42 (3): 641-650.  
    Abstract200)   HTML12)    PDF(pc) (337KB)(210)       Save

    The isoperimetric problem plays an important role in integral geometry. In this paper we mainly investigate the inverse form of the isoperimetric inequality, i.e. the general inverse Bonnesen-type inequalities. The upper bounds of several new general isoperimetric genus are obtained. Futhermore, as corollaries, we get a series of classical inverse Bonnesen-type inequalities in the plane. Finally, the best estimate between the results of three upper bounds is given.

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    Similarity and Unitary Similarity of a Class of Upper Triangular Operator Matrices
    Liqiong Lin,Jiahua Que,Yunnan Zhang
    Acta mathematica scientia,Series A    2022, 42 (5): 1281-1293.  
    Abstract150)   HTML15)    PDF(pc) (268KB)(218)       Save

    This paper introduces a class of upper triangular operator matrices related to Cowen-Douglas operators, and studies its similarity on Banach spaces and its unitary similarity on Hilbert spaces.

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    Complex Symmetry for a Class of Truncated Hankel Operators
    Liling Lai,Jinjin Liang,Yong Chen
    Acta mathematica scientia,Series A    2022, 42 (4): 961-968.  
    Abstract130)   HTML8)    PDF(pc) (293KB)(160)       Save

    The truncated Hankel operator is the compression to the model space of Hankel operator on the Hardy space. In this paper, the complex symmetry for a class of truncated Hankel operators is studied and the complete characterization is given. The obtained results show that, the complex symmetry of truncated Hankel operator may be related to the model space only, or to the model space and the symbol function of the operator both.

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    Approximate Optimality Conditions and Mixed Type Duality for a Class of Non-Convex Optimization Problems
    Jiaolang Wang,Donghui Fang
    Acta mathematica scientia,Series A    2022, 42 (3): 651-660.  
    Abstract117)   HTML4)    PDF(pc) (298KB)(134)       Save

    By using the properties of the Fréchet subdifferentials, we first introduce a new constraint qualification and then establish some approximate optimality conditions for the non-convex constrained optimization problem with objective function and/or constraint function being α-convex function. Moreover, some results for the weak duality, strong duality and converse-like duality theorems between this non-convex optimization problem and its mixed type dual problem are also given.

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    Perturbations of Canonical Unitary Involutions Associated with Quantum Bernoulli Noises
    Nan Fan,Caishi Wang,Hong Ji
    Acta mathematica scientia,Series A    2022, 42 (4): 969-977.  
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    Quantum Bernoulli noises (QBN) are annihilation and creation operators acting on the space of square integrable Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time and can play an important role in describing the environment of an open quantum system. In this paper, we address a type of perturbations of the canonical unitary involutions associated with QBN. We analyze these perturbations from a perspective of spectral theory and obtain exactly their spectra, which coincide with their point spectra. We also discuss eigenvectors of these perturbations from an algebraic point of view and unveil the structures of the subspaces consisting of their eigenvectors. Finally, as application, we consider the abstract quantum walks driven by these perturbations and obtain infinitely many invariant probability distributions of these walks.

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    The Self-Adjointness and Dependence of Eigenvalues of Fourth-Order Differential Operator with Eigenparameters in the Boundary Conditions
    Wenwen Yan,Meizhen Xu
    Acta mathematica scientia,Series A    2022, 42 (3): 671-693.  
    Abstract83)   HTML1)    PDF(pc) (450KB)(108)       Save

    In this paper we consider the self-adjointness and the dependence of eigenvalues of a class of discontinuous fourth-order differential operator with eigenparameters in the boundary conditions of one endpoint. By constructing a linear operator T associated with problem in a suitable Hilbert space, the study of the above problem is transformed into the research of the operator in this space, and the self-adjointness of this operator T is proved. In addition, on the basis of the self-adjointness of the operator T, we show that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, and give the corresponding differential expressions. In particular, giving the Fréchet derivative of the eigenvalue with respect to the eigenparameter-dependent boundary condition coefficient matrix, and the first-order derivatives of the eigenvalue with respect to the left and right sides of the inner discontinuity point c.

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    The Maximal Operator of Vilenkin-like System on Hardy Spaces
    Chuanzhou Zhang,Chaoyue Wang,Xueying Zhang
    Acta mathematica scientia,Series A    2022, 42 (5): 1294-1305.  
    Abstract82)   HTML2)    PDF(pc) (311KB)(92)       Save

    In this paper, we discuss the boundedness of maximal operator with respect to bounded Vilenkin-like system (or $\psi\alpha$ system) which is generalization of bounded Vilenkin system. We prove that when $0 < p <1/2$ the maximal operator $\tilde{\sigma}_p^*f=\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{(n+1)^{1/p-2}}$ is bounded from the martingale Hardy space $H_p$ to the space $L_p$, where $\sigma_nf$ is $n$-th Fej\'er mean with respect to bounded Vilenkin-like system. By a counterexample, we also prove that the maximal operator $\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{\varphi(n)}$ is not bounded from the martingale Hardy space $H_{p}$ to the space $L_{p,\infty}$ when $0 < p <1/2$ and $\mathop{\overline{\lim}}\limits_{n\rightarrow \infty}\frac{(n+1)^{1/p-2}}{\varphi(n)}=+\infty$.

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    Ground State Travelling Waves in Infinite Lattices with Superquadratic Potentials
    Chunhui Shao,Jijiang Sun,Shiwang Ma
    Acta mathematica scientia,Series A    2022, 42 (5): 1451-1461.  
    Abstract79)   HTML6)    PDF(pc) (396KB)(37)       Save

    In this paper, we consider one dimensional FPU type lattices with particles of unit mass. The dynamics of the system is described by the followingwhere U is the potential of interaction between two adjacent particles and qn denotes the displacement. By directly using the usual variational method, we study the existence of ground state travelling waves, i.e., non-trivial travelling waves with least possible energy, for the above system with more general superquadratic potentials than the previous work of Pankov[10] and Zhang and Ma [20]. Moreover, we also concern the monotonicity of the solitary ground state travelling waves.

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    Classification of Calabi Hypersurfaces in ${\mathbb R}^5$ with Parallel Fubini-Pick Form
    Ruiwei Xu,Miaoxin Lei
    Acta mathematica scientia,Series A    2022, 42 (2): 321-337.  
    Abstract78)   HTML2)    PDF(pc) (391KB)(109)       Save

    The classifications of locally strongly convex equiaffine hypersurfaces (centroaffine hypersurfaces) with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Blaschke-Berwald metric (centroaffine metric) have been completed in the last decades. In [20], the authors studied Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric and classified 2 and 3-dimensional cases. In this paper, we extend such calssification results to 4-dimensional Calabi hypersurfaces in the affine space ${\mathbb R}^5$.

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    A Class of Differential Operators with Eigenparameter Dependent Boundary Conditions
    Kang Sun,Yunlan Gao
    Acta mathematica scientia,Series A    2022, 42 (3): 661-670.  
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    In this paper, A class of third-order differential operators with transition conditions and two boundary conditions containing spectral parameters is studied, and the analytical method is used to do two aspects of work. First, by constructing a new space and a new operator, the eigenvalues of the problem and the operator are connected so that the eigenvalues of the original problem are consistent with the eigenvalues of the new operator. Second, the properties of the eigenvalues of the original problem are studied, and the conclusion that the spectrum of the original problem has only point spectrum is given.

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    Breather Wave Solutions, Lump Solutions and Semi-Rational Solutions of a Reduced (3+1)Dimensional Hirota Equation
    Chunmei Fang,Shoufu Tian
    Acta mathematica scientia,Series A    2022, 42 (3): 775-783.  
    Abstract75)   HTML0)    PDF(pc) (1085KB)(45)       Save

    In this paper, the long wave limit method is used to study the exact solutions of the (3+1)dimensional Hirota equation under dimensional reduction $z$=$x$. First, the bilinear form is constructed by using Bell polynomials. Based on the bilinear form, the $n$-order breather wave solutions are obtained under some parameter constraints on the $N$-order soliton solution. Secondly, by using the long wave limit method, high order lump wave solutions are obtained. Finally, the combined solutions of the first-order, second-order lump wave solutions and single solitary wave solutions are derived, i.e. semi-rational solutions. All the obtained solutions were analyzed with Maple software for physical characteristics.

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    Q-(Approximate) Dual of g-Frames in Hilbert Spaces
    Wei Zhang,Yanling Fu,Shuanbao Li
    Acta mathematica scientia,Series A    2022, 42 (3): 694-704.  
    Abstract68)   HTML2)    PDF(pc) (350KB)(84)       Save

    In this paper, fusing the ideas of dual Fusion frames and approximate dual g-frames, the definitions of Q-(approximate) dual g-frames are given. The relationship between Q-approximate dual g-frames and Q-dual g-frames is discussed. The characterizations of Q-(approximate) dual g-frames are obtained. Finally, by means of Q-approximate dual g-frames, some equivalent conditions for a g-frame to be close to another g-frame are given.

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    Existence of Positive Ground State Solutions for the Choquard Equation
    Xudong Shang,Jihui Zhang
    Acta mathematica scientia,Series A    2022, 42 (3): 749-759.  
    Abstract68)   HTML1)    PDF(pc) (355KB)(67)       Save

    In this paper we study the following nonlinear Choquard equation where $N \geq 3$, $\alpha \in (0, N)$, $I_{\alpha}$ is the Riesz potential, $V(x):\mathbb{R} ^{N} \rightarrow \mathbb{R} $ is a given potential function, and $F\in {\cal C}^{1}(\mathbb{R}, \mathbb{R})$ with $F'(s)=f(s)$. Under assumptions on $V$ and $f$, we do not require the $(AR)$ condition of $f$, the existence of ground state solutions are obtained via variational methods.

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    On a Nonlinear Non-Autonomous Ratio-Dependent Food Chain Model with Delays and Feedback Controls
    Changyou Wang,Nan Li,Tao Jiang,Qiang Yang
    Acta mathematica scientia,Series A    2022, 42 (1): 245-268.  
    Abstract66)   HTML0)    PDF(pc) (1166KB)(42)       Save

    In this paper, we study a 3-species nonlinear non-autonomous ratio-dependent food chain system with delays and feedback controls. Firstly, based on the theory of delay differential inequality, some new analytical methods are developed and a suitable Lyapunov function is constructed. Secondly, sufficient conditions for the permanence and global attractivity of positive solutions for the system are obtained. Thirdly, by using the theoretical analysis and fixed point theory, the corresponding periodic systems are discussed, and the conditions for the existence, uniqueness and stability of positive periodic solutions of periodic systems are established. Moreover, we give some numerical simulations to prove that our theoretical analysis are correct. Finally, we still give an numerical example for the corresponding stochastic food chain model with multiplicative noise sources, and achieve new interesting change process of the solution for the model.

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    Stability Analysis of an HIV Infection Dynamic Model with CTL Immune Response and Immune Impairment
    Meng Deng,Rui Xu
    Acta mathematica scientia,Series A    2022, 42 (5): 1592-1600.  
    Abstract64)   HTML2)    PDF(pc) (393KB)(62)       Save

    In this paper, we study an HIV infection model with saturation incidence rate, CTL immune response, immune impairment, and intracellular delay. Firstly, the basic reproduction ratio $ \Re_{0} $ of virus infection is obtained by using the next generation matrix method. Secondly, the local stability of feasible equilibria is proved by analyzing the distribution of the root of the corresponding characteristic equations. By constructing appropriate Lyapunov functionals and using LaSalle's invariance principle, we prove that when $ \Re_{0}<1 $, the virus infection-free equilibrium is globally asymptotically stable; when $ \Re_ {0}>1 $, the immunity-inactivated equilibrium is globally asymptotically stable. Finally, the parameter with critical influence on $ \Re_{0} $ is determined by the parameter sensitivity analysis.

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    A Class of Weakly Nonlinear Critical Singularly Perturbed Integral Boundary Problems
    Hao Zhang,Na Wang
    Acta mathematica scientia,Series A    2022, 42 (4): 1060-1073.  
    Abstract64)   HTML2)    PDF(pc) (438KB)(97)       Save

    Based on the boundary layer function method, a class of singularly perturbed problems with integral boundary conditions in weakly nonlinear critical cases are studied. In the framework of this paper, we not only construct the asymptotic expansion of the solution of the original equation, but also prove the uniformly effective asymptotic expansion. At the same time, we give an example to illustrate our results, The comparison images of approximate solution and exact solution under different small parameters are drawn.

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    Ground State Solutions for Quasilinear Schrödinger Equation of Choquard Type
    Yanan Wang,Kaimin Teng
    Acta mathematica scientia,Series A    2022, 42 (3): 730-748.  
    Abstract63)   HTML1)    PDF(pc) (399KB)(82)       Save

    In this paper, we consider the following quasilinear Schrödinger equations of Choquard type where $N\geq3$, 0 < $\alpha$ < $N$, $<p<\frac{N+\alpha}{N-2}$, $I_{\alpha}$ is the Riesz potential, $V(x)$ is a positive continuous potential and $k$ is a non-negative parameter. The existence of ground state solutions is established via Pohožaev manifold approach.

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    Dimension Theory of Uniform Diophantine Approximation Related to Beta-Transformations
    Wanlou Wu,Lixuan Zheng
    Acta mathematica scientia,Series A    2022, 42 (4): 978-1002.  
    Abstract63)   HTML0)    PDF(pc) (485KB)(59)       Save

    For $\beta>1$, let $T_\beta$ be the $\beta$-transformation defined on $[0, 1)$. We study the sets of points whose orbits of $T_\beta$ have uniform Diophantine approximation properties. Precisely, for two given positive functions $\psi_1, \ \psi_2:{\Bbb N}\rightarrow{\Bbb R}^+$, define where $\gg$ means large enough. We calculate the Hausdorff dimension of the set ${\cal L}(\psi_1)\cap{\cal U}(\psi_2)$. As a corollary, we obtain the Hausdorff dimension of the set ${\cal U}(\psi_2)$. Our work generalizes the results of [4] where only exponential functions $\psi_1, \ \psi_2$ were taken into consideration.

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    Two-Dimensional Infinite Square Well in Fractional Quantum Mechanics
    Yunjie Tan,Xiaohui Han,Jianping Dong
    Acta mathematica scientia,Series A    2022, 42 (4): 1018-1026.  
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    Fractional quantum mechanics is a generalization of standard quantum mechanics, which is described by fractional Schrödinger equation with fractional Riesz derivative operator. In this paper, we consider a free particle moving in a two-dimensional infinite square well, By using Lévy path integral method, the wave function and energy eigenvalue of the two-dimensional infinite square well are obtained. Then the perturbation expansion method is used to study the two-dimensional infinite square well with $\delta$ function, and the corresponding energy-dependent Green's function is obtained.

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