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    The Schödinger Uncertainty Relation in the Fock-Type Spaces
    Li Wenxin,Lian Pan,Liang Yuxia
    Acta mathematica scientia,Series A    2023, 43 (5): 1321-1332.  
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    In this paper, the Schödinger uncertainty relation for the unilateral weighted shift operators on Fock space is established, and the explicit expression when the equality attained is given, which further extends the Heisenberg uncertainty relation on Fock space established in [4] and overcomes the difficulty in [16]. In addition, we generalize the uncertainty relation to the multiple operators case. A new uncertainty inequality in the form of non-self adjoint operators is obtained as well.

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    An Extension of Minkowski Formulae for Free Boundary Hypersurfaces in a Ball
    Sheng Weimin, Wang Yinhang
    Acta mathematica scientia,Series A    2023, 43 (6): 1641-1648.  
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    In this article, we prove a generalization of Hsiung-Minkowski formula for free boundary hypersurfaces in a ball in space forms. As corollaries, we obtain some Alexandrov-type results.

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    Robust Accessible Hyperbolic Repelling Sets
    Xiao Jianrong
    Acta mathematica scientia,Series A    2024, 44 (1): 1-11.  
    Abstract173)   HTML10)    PDF(pc) (800KB)(279)       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.

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    The Reproducing Kernel of Bergman Space and the Eigenvectors of Toeplitz Operator
    Ding Xuanhao,Hou Lin,Li Yongning
    Acta mathematica scientia,Series A    2023, 43 (5): 1333-1340.  
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    In the Bergman space, it is well-known that $ T_{\varphi}K_{z}=\varphi(z)K_{z} $ for $ \varphi\in \overline{H^{\infty}} $, that is, $ K_{z} $ is the eigenvector of $ T_{\varphi} $ corresponding the eigenvalue $ \varphi(z) $, where $ K_{z} $ is the reproducing kernel of Bergman space. Conversely, if $ \varphi $ is a bounded harmonic function and if there is $ z\in \mathbb{D} $ (or for every $ z\in\mathbb{D} $), $ K_{z} $ is a eigenvector of $ T_{\varphi} $, whether there must be $ \varphi\in \overline{H^{\infty}} $ ? In view of the above questions, in this paper we give a complete characterization of the Toeplitz operator with the bounded harmonic symbol which have the reproducing kernels $ K_{z} $ as their eigenvectors. Moreover, we partially describe the Toeplitz operators with the bounded harmonic symbol whose eigenvalues are all $ \varphi(z) (z\in \mathbb{D}) $.

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    On Second Order Complex Differential Equations with Coefficients of Period $ {2\pi{\rm i}}$
    Zhang Jie,Zhao Donghai
    Acta mathematica scientia,Series A    2023, 43 (5): 1382-1390.  
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    This paper mainly learned classic book `Nevanlinna theory and complex differential equations' due to Laine and considered the second order complex differential equation $f ''(z) + A (z) f(z)=0, \lambda(f)<\infty$ with coefficient $A (z) $ whose period is $2 \pi{\rm i}$. It found a possible error in the original proof and gave its partial correction, and also it gave a slightly weaker conclusion than its possibly controversial result in the original literature.

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    On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement
    Jian Hui, Gong Min, Wang Li
    Acta mathematica scientia,Series A    2023, 43 (5): 1350-1372.  
    Abstract99)   HTML5)    PDF(pc) (797KB)(340)       Save

    This paper is devoted to the Cauchy problem of inhomogeneous nonlinear Schrödinger equation in the presence of a partial confinement, which is an important model in Bose-Einstein condensates. Combining the variational characterization of the ground state of a nonlinear elliptic equation and the conservations of mass and energy, we first obtain a global solution and show the existence of blow-up solutions for some special initial data by scaling techniques. Then, we study the $L^2$-concentration phenomenon for the blow-up solutions. Finally, we apply the variational arguments connected to the above ground state to investigate the dynamics of $L^2$-minimal blow-up solutions, i.e., the limiting profile, mass-concentration and blow-up rate of the blow-up solutions with minimal mass. We extend the global existence and blow-up results of Zhang[34] to the case of inhomogeneous nonlinearities and improve partial results of Pan and Zhang[23] to space dimensions $N\geq2$ in the inhomogeneous case.

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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.  
    Abstract95)   HTML1)    PDF(pc) (557KB)(159)       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.

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    The Discrete Series of Affine Symmetric Space ${SO^\ast(6)/SO(3,\mathbb{C})}$
    Lan Chao, Fan Xingya
    Acta mathematica scientia,Series A    2023, 43 (6): 1649-1658.  
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    In this paper, the partial discrete sequence of $SO^\ast(6)/SO(3,\mathbb{C})$ is obtained by local isomorphism of Hermite-type affine symmetric space, and the specific form of the holomorphic discrete sequence generated by the cyclic vector is given.

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    Time Decay Rate for Large-Solution About 3D Compressible MHD Equations
    Chen Fei,Wang Shuai,Zhao Yongye,Wang Chuanbao
    Acta mathematica scientia,Series A    2023, 43 (5): 1397-1408.  
    Abstract81)   HTML4)    PDF(pc) (697KB)(342)       Save

    This paper focus on time decay rate for large-solution about compressible magnetohydrodynamic equations in $\mathbb{R}^3$. Provided that $(\sigma_{0}-1,u_{0},M_{0})\in L^1\cap H^2$, based on the work of Chen et al.[1], $\|\nabla(\sigma-1,u,M)\|_{H^1}\leqslant C(1+t)^{-\frac{5}{4}}$ is obtained in reference [2], obviously, time decay rate of the 2nd-order derivative of the solution in [2] is not ideal. Here, we improve that of $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ to be $(1+t)^{-\frac{7}{4}}$ by the frequency decomposition method[3].

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    Existence of Positive Solutions for a Class of Schrödinger-Newton Systems with Critical Exponent
    Cheng Qingfang,Liao Jiafeng,Yuan Yanxiang
    Acta mathematica scientia,Series A    2023, 43 (5): 1373-1381.  
    Abstract77)   HTML5)    PDF(pc) (629KB)(311)       Save

    In this paper, we study the existence of positive solutions for a class of Schrödinger-Newton system with critical exponents on bounded domain, and obtain two positive solutions by the variational method.

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    $q$-Ramanujan Asymptotic Formula and $q$-Ramanujan $R$-function
    Bao Qi, Wang Miaokun, Chu Yuming
    Acta mathematica scientia,Series A    2023, 43 (6): 1659-1666.  
    Abstract77)   HTML2)    PDF(pc) (530KB)(194)       Save

    In this paper, the Ramanujan asymptotic formula of the Gaussian hypergeometric function $_{2}F_{1}$ and its related Ramanujan $R$-function will be generalized to the case of basic hypergeometric series $_{2}\phi_{1}$. On the one hand, we shall present the $q$-Ramanujan asymptotic formula of $_{2}\phi_{1}$ and introduce the $q$-Ramanujan $R$-function; on the other hand, we shall mainly study the $q$-Ramanujan $R$-function, and prove some analytical properties of the $q$-Ramanujan $R$-function including series expansions, complete monotonicity property and monotonicity property with respect to the parameter $q$. As applications, several sharp inequalities for the $q$-Ramanujan $R$-function will be derived.

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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.  
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    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$\beta\geq p\geq 2$.

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    Some Reverse Bonnesen-style Inequalities in $n$-Dimensional Euclidean Space $\mathbb{R} ^n$
    Wang Hejun
    Acta mathematica scientia,Series A    2023, 43 (4): 985-993.  
    Abstract71)   HTML3)    PDF(pc) (320KB)(89)       Save

    This paper mainly studies reverse Bonnesen-style inequalities in $n$-dimensional Euclidean space $\mathbb{R} ^n$. By the Urysohn inequality, the dual isoperimetric inequality, mean width and mean intersection area, some new reverse Bonnesen-style inequalities for general convex bodies are obtained in $\mathbb{R} ^n$.

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    Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity
    Li Xiaodong, Gao Hongliang, Xu Jing
    Acta mathematica scientia,Series A    2023, 43 (5): 1341-1349.  
    Abstract70)   HTML8)    PDF(pc) (770KB)(321)       Save

    In this paper, we study the exact multiplicity and bifurcation diagrams of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of

    $ \left\{\begin{array}{ll} -\left(\frac{u'}{\sqrt{1-u'^{2}}}\right)'=\lambda f(u), x\in(-L,L),\\ u(-L)=0=u(L), \end{array} \right. $

    where $\lambda>0$ is a bifurcation parameter and $L>0$ is an evolution parameters, $f\in C^{2}([0,\infty), \mathbb{R})$ satisfies $f(0)<0$ and $f$ is concave for $0. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of $\lambda$. The arguments are based upon a detailed analysis of the time map.

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    Multi-Scale Approach for Diffeomorphic Multi-Modality Image Registration
    Ding Zijuan,Han Huan
    Acta mathematica scientia,Series A    2023, 43 (5): 1620-1640.  
    Abstract70)   HTML6)    PDF(pc) (2647KB)(236)       Save

    Multi-modality image registration is widely used in remote sensing, clinical medicine and other fields. Many models for multi-modality image registration have been proposed in the past few decades. Concerning this problem, there are two major challenges: (1) the existence of physical mesh folding; (2) the ill-posedness of similarity measure minimization/maximization problem. In order to address those problems, a multi-scale approach for diffeomorphic image registration based on Rényi's statistical dependence measure is proposed, which can avoid estimating joint probability density function, and obtain a smooth minimizer of the energy functional without mesh folding and prior regularization. In addition, the existence of solution for the proposed model and the convergence of the multi-scale approach are proved. And numerical experiments are performed to show the efficiency of the proposed algorithm in the monomodality image registration and multi-modality image registration.

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    The Completely Regular Growth of Solutions of Higher Order Linear Differential Equations
    Chen Li,Liu Huifang
    Acta mathematica scientia,Series A    2023, 43 (3): 733-742.  
    Abstract68)   HTML3)    PDF(pc) (349KB)(74)       Save

    In this paper, the existence of completely regular growth solutions of higher order linear differential equations is studied, where its dominant coefficient is an exponential polynomial. By using the Nevanlinna characteristic of exponential polynomials, some conditions which guarantee the non-existence of such solutions are obtained. At the same time, for the higher order linear differential equation with exponential polynomial solutions, the relationship between the expression of its solutions and dominant coefficient is given.

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    Structure Expression Form of Isotropic Growth Surface
    Qian Jinhua,Bian Jinxin,Fu Xueshan
    Acta mathematica scientia,Series A    2023, 43 (3): 657-668.  
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    The isotropic growth surface in complex 3-space is investigated by evolving an isotropic curve as dictated growth velocity. The structure expression form of the isotropic growth surface is explored by the aid of the structure function of its generating isotropic curve. As an important application, the isotropic growth surface initiated by the isotropic helix is discussed deeply and explicitly. At the same time, several typical examples are constructed to characterize the generating process of such growth surfaces.

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    Uniqueness and Asymptotic Stability of Time-Periodic Solutions for the Fractional Burgers Equation
    Xu Fei, Zhang Yong
    Acta mathematica scientia,Series A    2023, 43 (6): 1710-1722.  
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    The paper is concerned with the time-periodic (T-periodic) problem of fractional Burgers equation on the real line. Based on the Galerkin approximates and Fourier expansion, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution for the nonlinear equation are established by constructing a suitable contraction mapping. Furthermore, we show that the unique T-periodic solution is asymptotically stable. In addition, our method can be extended to the classical forced Burgers equation in a bounded region, which improves the known result.

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    Global Bifurcation for the Yamabe Equation on the Unit Sphere
    Dai Guowei,Gao Siyu,Ma Ruyun
    Acta mathematica scientia,Series A    2023, 43 (5): 1391-1396.  
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    We study the Yamabe equation on the $N$-dimensional unit sphere $\mathbb{S}^N$

    $\begin{equation} -\Delta_{\mathbb{S}^N} v+\lambda v=v^{\frac{N+2}{N-2}}.\nonumber \end{equation}$

    By bifurcation technique, for each $k\geq1$, we prove that this equation has at least one non-constant solution $v_k$ for any $\lambda>\lambda_k:=(k+N-1)(N-2)/4$ such that $v_k-\lambda^{1/(N^{*}-1)}$ has exactly $k$ zeroes, all of them are in $(-1, 1)$ and are simple, where $N^{*}$ is the sobolev critical exponent. As application, we obtain the existence of non-radial solutions of a nonlinear elliptic equation on $\mathbb{R}^N$ with $n\geq4$. Moreover, we also obtain the global bifurcation results of the Yamabe problem in product manifolds with one of the manifold is the unit sphere.

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    Survival Analysis of an SVIR Epidemic Model with Media Coverage
    Li Dan,Wei Fengying,Mao Xuerong
    Acta mathematica scientia,Series A    2023, 43 (5): 1595-1606.  
    Abstract64)   HTML2)    PDF(pc) (1602KB)(301)       Save

    We consider the long-term properties of a stochastic SVIR epidemic model with media coverage and the logistic growth in this paper. We firstly derive the fitness of a unique global positive solution. Then we construct appropriate Lyapunov functions and obtain the existence of ergodic stationary distribution when ${R}_{0}^{s}>1$ is valid, and also derive sufficient conditions for persistence in the mean. Moreover, the exponential extinction to the density of the infected is figured out when ${R}_{0}^{e}<1$ holds.

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