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当期目录

    2022年, 第42卷, 第3期 刊出日期:2022-06-26 上一期   
    本期栏目: 论文 
    论文
    $\mathbb{R}^n$中的广义逆Bonnesen型不等式
    董旭,张燕,曾春娜,王星星
    数学物理学报. 2022 (3):  641-650. 
    摘要 ( 124 )   RICH HTML PDF(337KB) ( 122 )   收藏

    等周问题在积分几何中具有举足轻重的地位. 该文主要研究$\mathbb{R}^n$中等周不等式的逆形式, 即广义逆Bonnesen型不等式. 该文获得了$\mathbb{R}^n$中几个新广义等周亏格上界的结果, 作为推论, 得到了更一般的平面上的逆Bonnesen型不等式; 最后给出其中三个上界结果之间的最佳估计.

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    一类非凸约束优化问题的近似最优性条件及其混合型对偶
    王娇浪,方东辉
    数学物理学报. 2022 (3):  651-660. 
    摘要 ( 64 )   RICH HTML PDF(298KB) ( 67 )   收藏

    利用函数Fréchet次微分性质, 引入新的约束规范条件, 建立了目标函数和(或)约束函数为α-凸函数的非凸约束优化问题的近似最优性条件以及该问题及其混合型对偶问题之间的弱对偶、强对偶和逆对偶定理.

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    一类边界条件含谱参数的微分算子
    孙康,高云兰
    数学物理学报. 2022 (3):  661-670. 
    摘要 ( 45 )   RICH HTML PDF(355KB) ( 46 )   收藏

    该文考虑了一类具有转移条件且两个边界条件含谱参数的三阶微分算子, 利用分析法做了两方面的工作, 一是通过构造新空间和新算子将所考虑问题的特征值与新算子的特征值建立联系, 使原问题的特征值和新算子的特征值一致. 二是研究了原问题的特征值的性质, 且给出了原问题的谱只有点谱的结论.

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    边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性
    闫文文,许美珍
    数学物理学报. 2022 (3):  671-693. 
    摘要 ( 27 )   RICH HTML PDF(450KB) ( 40 )   收藏

    该文考虑了一类边界条件一端含有特征参数且具有转移条件的四阶微分算子的自共轭性及特征值的依赖性. 通过在适当的Hilbert空间上定义一个与问题相关的线性算子T, 将上述问题的研究转化为对此空间中算子的研究, 并证明算子T是自共轭的. 另外, 在算子T自伴的基础上证明特征值不仅连续依赖而且可微依赖于问题的各个参数, 并给出相应的微分表达式. 特别地, 给出特征值关于特征参数依赖的边界条件系数矩阵的Fréchet导数及关于内部不连续点c左右两侧的一阶导数.

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    Hilbert空间中广义框架的Q-(近似)对偶
    张伟,付艳玲,李拴保
    数学物理学报. 2022 (3):  694-704. 
    摘要 ( 34 )   RICH HTML PDF(350KB) ( 55 )   收藏

    该文融合Fusion对偶框架与近似对偶广义框架的思想, 给出了Q-(近似)对偶广义框架的概念, 讨论了Q-近似对偶广义框架与Q-对偶之间的关系, 得到了Q-(近似)对偶广义框架的刻画. 最后, 借助Q-近似对偶广义框架, 给出了一广义框架接近另一广义框架的若干等价条件.

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    三个数字集生成的自相似测度的乘积谱
    李海雄,吴新林,丁道新
    数学物理学报. 2022 (3):  705-715. 
    摘要 ( 18 )   RICH HTML PDF(390KB) ( 18 )   收藏

    该文研究了自相似测度$\mu_{k, a, b}(\cdot)=\frac{1}{3}\sum\limits_{i=0}^{2}\mu_{k, a, b}(3k(\cdot)-ki)$的谱特征值问题, 利用整数同余及有限乘法群元素阶的相关性质, 得到了集合为自相似测度$\mu_{k, a, b}$乘积谱的几个充分条件. 最后通过一个具体的例子来应用文中的结果.

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    Chern-Simons-Schrödinger方程能量泛函的L2约束极小化问题
    杨迎,沈烈军
    数学物理学报. 2022 (3):  716-729. 
    摘要 ( 20 )   RICH HTML PDF(394KB) ( 25 )   收藏

    该文主要研究$\mathbb{R}^2$上一类Chern-Simons-Schrödinger (CSS)方程在给定$L^{2}$范数下解的存在性.这类问题可转化为该方程对应能量泛函$E^\beta_{p}(u)$在约束条件$\|u\|_{L^{2}(\mathbb{R}^2)}=1$下的变分求极小问题.对于质量次临界的情形,即$p\in (0,2)$,该文应用简洁的方法证明了无论位势函数$V (x)$是否为$0$,这类约束变分极小化问题都是可达的;对于质量临界的情形,即$p=2$,该文找到了两个可显式给出的正常数$\beta^{*}>\beta_{*}$,使得$V (x)\equiv0$时的约束变分极小化问题对于$\beta>\beta^{*}$$\beta\in (0,\beta_{*}]$均不可达,而对于$V (x)\not\equiv 0$时的约束变分极小化问题则在$\beta\in (0,\beta_{*}]$可达,$\beta>\beta^{*}$不可达.此外,该文还讨论了质量次临界的约束极小能量在$p\rightarrow2$时的极限行为.

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    带Choquard项的拟线性薛定谔方程的基态解
    王亚男,滕凯民
    数学物理学报. 2022 (3):  730-748. 
    摘要 ( 25 )   RICH HTML PDF(399KB) ( 40 )   收藏

    该文研究如下Choquard型拟线性薛定谔方程 其中$N\geq3$, 0 < $\alpha$ < $N$, $< p<\frac{N+\alpha}{N-2}$, $I_{\alpha}$是Riesz位势, $V(x)$是一个正连续位势, $k$是一个非负参数. 采用Pohožaev流形方法, 证明了基态解的存在性.

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    一类非线性Choquard方程基态解的存在性
    尚旭东,张吉慧
    数学物理学报. 2022 (3):  749-759. 
    摘要 ( 30 )   RICH HTML PDF(355KB) ( 41 )   收藏

    该文研究如下一类非线性Choquard方程 其中$N \geq 3$$\alpha \in (0, N)$$I_{\alpha}$是Riesz势, 位势函数$V:\mathbb{R} ^{N} \rightarrow \mathbb{R} $为连续函数,$F\in {\cal C}^{1}(\mathbb{R},\mathbb{R})$, 且$F'(s)=f(s)$. 在函数$V$$f$满足合适的条件下(但$f$不必满足(AR)条件), 利用变分方法,该文证明了上述方程存在一个正的基态解.

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    依赖参数的2$n$阶差分方程边值问题多个非平凡解的存在性
    王振国
    数学物理学报. 2022 (3):  760-766. 
    摘要 ( 22 )   RICH HTML PDF(313KB) ( 24 )   收藏

    该文研究了一类具有参数的$2n$阶差分方程边值问题多个非平凡解的存在性问题.当$\lambda\in\left (\frac{p (T)}{2B}, \frac{1}{2A}\right)$时, 运用临界点理论得到这类差分方程边值问题无穷多个非平凡解存在的充分条件.最后, 举例验证文中的主要结论.

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    一类非局部拟线性椭圆方程组无穷多解的存在性
    王倩,陈林,汤楠
    数学物理学报. 2022 (3):  767-774. 
    摘要 ( 19 )   RICH HTML PDF(281KB) ( 24 )   收藏

    该文运用对称山路引理研究一类拟线性椭圆方程组无穷多解的存在性,其中$M(s)=s^{k},k>0,N\geq3,p>1,p(k+1)<q\leq d<p^{\ast}\leq N$ (当$p<N$时,$p^{\ast}=\frac{Np}{N-p}$; 当$p=N$时,$p^{\ast}=\infty$),$\lambda, \mu>0, \sigma\in\mathbb{R} ^{N}$,权函数$V(x)\in C(\mathbb{R} ^N)$满足某些给定的条件.

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    约化的(3+1)维Hirota方程的呼吸波解、lump解和半有理解
    房春梅,田守富
    数学物理学报. 2022 (3):  775-783. 
    摘要 ( 21 )   RICH HTML PDF(1085KB) ( 20 )   收藏

    该文利用长波极限方法研究了(3+1)维Hirota方程在维数约化$z$=$x$下的精确解.首先利用贝尔多项式构造了其双线性形式.基于双线性形式,对$N$-孤子解做某些参数约束,获得了$n$-阶呼吸波解.其次,利用长波极限方法获得了高阶lump波解.最后导出了一阶,二阶lump波解分别与单孤子解的混合解,即半有理解.所有得到的解都通过Maple软件进行物理特征分析.

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    脉冲离散Ginzburg-Landau方程组的统计解及其极限行为
    赵才地,姜慧特,李春秋,TomásCaraballo
    数学物理学报. 2022 (3):  784-806. 
    摘要 ( 11 )   RICH HTML PDF(497KB) ( 16 )   收藏

    该文研究脉冲离散Ginzburg-Landau方程组的统计解及其极限行为.文章首先证明该脉冲离散方程组的全局适定性,接着证明由解算子生成的过程存在拉回吸引子和一族Borel不变概率测度,然后给出该脉冲离散方程组统计解的定义并证明其存在性.该文的结果揭示了脉冲系统的统计解只分段地满足Liouville型定理.最后,文章证明了脉冲离散Ginzburg-Landau方程组的统计解收敛于脉冲离散Schrödinger方程组的统计解.

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    具结构化的细菌种群增生中迁移方程解的渐近稳定性分析
    吴红星,袁邓彬,王胜华
    数学物理学报. 2022 (3):  807-817. 
    摘要 ( 19 )   RICH HTML PDF(377KB) ( 17 )   收藏

    该文借助于线性算子理论,探讨了以细菌种群为背景的具结构化的更一般的边界条件的迁移方程, 运用预解算子和比较算子等方法证明了该迁移方程相应迁移算子谱在带域$\Gamma_{\alpha, \beta}$中仅由有限个具有限代数重数的离散本征值构成, 证明了该迁移方程解在$\psi_0 \in D (A_{H_{\alpha, \beta}})$时是渐近稳定的.

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    求解一类新的广义绝对值方程组的动力学模型
    郑雯丽,唐嘉,陈彩荣
    数学物理学报. 2022 (3):  818-825. 
    摘要 ( 25 )   RICH HTML PDF(352KB) ( 30 )   收藏

    该文提出了求解一类新的广义绝对值方程组(GAVE)的动力学模型.在适当的条件下,证明了GAVE的解等价于动力学模型的平衡点,并证明了平衡点的渐近稳定性.数值实验验证了所提出的模型是可行且有效的.

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    BOUNDEDNESS AND EXPONENTIAL STABILIZATION IN A PARABOLIC-ELLIPTIC KELLER–SEGEL MODEL WITH SIGNAL-DEPENDENT MOTILITIES FOR LOCAL SENSING CHEMOTAXIS
    江杰
    数学物理学报. 2022 (3):  825-846.  DOI: 10.1007/s10473-022-0301-y
    In this paper we consider the initial Neumann boundary value problem for a degenerate Keller—Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in the boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos—Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in [1] and the exponential stabilization is obtained for the first time.
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    一类自变量分段连续系统的振动性分析
    刘莹,高建芳
    数学物理学报. 2022 (3):  826-838. 
    摘要 ( 13 )   RICH HTML PDF(378KB) ( 20 )   收藏

    该文主要运用$\theta$-方法对一类滞后型自变量分段连续系统的振动性进行分析,讨论了解析解和数值解的振动性和非振动性,得到了数值方法在解析解振动条件下保持方程振动性的充分条件,并给出了数值算例.

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    时间轴上二阶非线性非自治延迟动力系统的振动性
    张萍,杨甲山,覃桂茳
    数学物理学报. 2022 (3):  839-850. 
    摘要 ( 17 )   RICH HTML PDF(418KB) ( 13 )   收藏

    研究了时间轴T上一类二阶非线性非自治变延迟的阻尼动态方程的振动性,考虑的是方程为非正则情形,通过引入广义Riccati变换,借助时间轴上的理论和一些经典不等式,建立了该方程振动的一些新准则,这些振动准则充分反应了延迟函数和阻尼项在系统振动中的影响作用.所举例子说明,所得准则不仅推广、改进并丰富了一些已知结果,而且具有较好的有效性和实用性.

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    ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS
    杨镇杭, 田景峰
    数学物理学报. 2022 (3):  847-864.  DOI: 10.1007/s10473-022-0302-x
    Let $\mathcal{K}\left( r\right) $ be the complete elliptic integrals of the first kind for $r\in \left( 0,1\right) $ and $f_{p}\left( x\right) =\left[ \left( 1-x\right) ^{p}\mathcal{K}\left( \sqrt{x}\right) \right] $. Using the recurrence method, we find the necessary and sufficient conditions for the functions $-f_{p}^{\prime }$, $\ln f_{p}$, $-\left( \ln f_{p}\right) ^{\left( i\right) }$ ($i=1,2,3$) to be absolutely monotonic on $\left( 0,1\right) $. As applications, we establish some new bounds for the ratios and the product of two complete integrals of the first kind, including the double inequalities \begin{eqnarray*} \frac{\exp \left[ r^{2}\left( 1-r^{2}\right) /64\right] }{\left( 1+r\right) ^{1/4}} &<&\frac{\mathcal{K}\left( r\right) }{\mathcal{K}\left( \sqrt{r}% \right) }<\exp \left[ -\frac{r\left( 1-r\right) }{4}\right] , \\ \frac{\pi }{2}\exp \left[ \theta _{0}\left( 1-2r^{2}\right) \right] &<&\frac{% \pi }{2}\frac{\mathcal{K}\left( r^{\prime }\right) }{\mathcal{K}\left( r\right) }<\frac{\pi }{2}\left( \frac{r^{\prime }}{r}\right) ^{p}\exp \left[ \theta _{p}\left( 1-2r^{2}\right) \right] , \\ \mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) &\leq &\mathcal{K}\left( r\right) \mathcal{K}\left( r^{\prime }\right) \leq \frac{1}{\sqrt{% 2rr^{\prime }}}\mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) \end{eqnarray*}% for $r\in \left( 0,1\right) $ and $p\geq 13/32$, where $r^{\prime }=% \sqrt{1-r^{2}}$ and $\theta _{p}=2\Gamma \left( 3/4\right) ^{4}/\pi ^{2}-p$.
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    具有时滞和季节性的炭疽模型的动力学分析
    张太雷,刘俊利,韩梦洁
    数学物理学报. 2022 (3):  851-866. 
    摘要 ( 26 )   RICH HTML PDF(480KB) ( 38 )   收藏

    该文建立了一个关于炭疽的时滞传染病模型,该模型考虑了炭疽传播的季节性和潜伏期.给出了模型的基本再生数$R_{0}$,研究表明模型的动力学行为完全由$R_{0}$来决定.当$R_{0}$ < 1时无病周期解全局吸引,疾病绝灭;当$R_{0}$>1时,系统存在一个正的周期解,疾病持久生存.对相应的自治系统,根据基本再生数得到了无病平衡点和地方病平衡点的全局渐近稳定性.最后数值模拟研究了$R_{0}$关于参数的敏感性及接种和尸体处理策略对炭疽传播的影响.

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    THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS
    孟令旭
    数学物理学报. 2022 (3):  865-875.  DOI: 10.1007/s10473-022-0303-9
    We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if all of its components do as well.
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    具有尺度等级和时滞的种群系统的最优边界控制
    何泽荣,窦艺萌,韩梦杰
    数学物理学报. 2022 (3):  867-880. 
    摘要 ( 13 )   RICH HTML PDF(438KB) ( 26 )   收藏

    该文基于个体尺度确定的等级差异并考虑孕育期时滞和幼体投放行为,建立了一类种群系统演化控制模型.针对某种预定的理想种群分布,研究最优投放控制问题:适当选取投放策略,使得经过一段指定时间后,种群实际状态与预定分布的差异及执行成本之和最小化.运用特征线方法确立了模型的适定性,借助Ekeland近似极值原理证明了最优策略的存在唯一性,构造恰当的共轭系统并利用法锥结构精细刻画了最优策略,为实际应用奠定理论基础.

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    PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS
    刘美琪, 乔会杰
    数学物理学报. 2022 (3):  876-886.  DOI: 10.1007/s10473-022-0304-8
    This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Finally, we estimate the errors between solutions of these equations and that of their numerical equations.
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    状态空间模型的正规核方法
    王超,李波,王磊
    数学物理学报. 2022 (3):  881-890. 
    摘要 ( 19 )   RICH HTML PDF(465KB) ( 13 )   收藏

    状态空间模型(SSMs)为随机过程研究提供了一个通用框架,在被应用到诸如揭示单一经济现象背后潜藏的经济过程、手机信号识别、雷达屏幕中干扰源条件下的飞机位置检测等等具体实践中表现出了较高的应用价值.其中,马尔可夫状态空间模型在具备较高应用价值的同时,还具备良好的理论性质.再生核希尔伯特空间由于具备良好的学习性,能够适应更加广泛的规律拟合包括线性和非线性拟合,因此尝试从再生核空间中寻找一个能够拟合状态空间中状态转移规律的函数具备很好的研究意义.为了保证解的唯一性,正则化方法被添加到最优化问题中.其中,解的存在性和唯一性是探讨预测误差的前提条件,在此基础上,误差上界被给出.最后,通过机场能见度数据(全国研究生数学建模竞赛—中国学位研究生教育发展中心提供),将正规化核方法的状态空间模型和传统ARMA模型以及Kalman Filter(Kalman)模型进行进行对比,用以验证正规化核方法下状态空间模型的有效性.

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    GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW
    赖宁安, 向伟, 周忆
    数学物理学报. 2022 (3):  887-902.  DOI: 10.1007/s10473-022-0305-7
    In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. A non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity which is a perturbation of plane waves. Therefore, unlike in the one-dimensional case, the multi-dimensional plane shocks are not stable globally. Moreover, a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.
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    一种维持Saint-Venant方程组移动稳态解的中心格式
    罗一鸣,李订芳,刘敏,董建
    数学物理学报. 2022 (3):  891-903. 
    摘要 ( 21 )   RICH HTML PDF(5011KB) ( 18 )   收藏

    针对Saint-Venant方程组提出了一种具有二阶精度的非交错中心有限体积格式.相较于经典中心格式为维持静稳态解选择重构守恒变量和水位值,但在求解移动稳态问题时会产生巨大误差,格式通过重构守恒变量和能量值,以及一种新的源项离散方法能够精确维持移动稳态解并捕捉其小扰动.最后,通过一些经典数值算例验证了格式的收敛性,谐性以及稳健性.

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    ESTIMATES FOR EXTREMAL VALUES FOR A CRITICAL FRACTIONAL EQUATION WITH CONCAVE-CONVEX NONLINEARITIES
    郝江浩, 张亚静
    数学物理学报. 2022 (3):  903-918.  DOI: 10.1007/s10473-022-0306-6
    In this paper we study the critical fractional equation with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the critical fractional equation.
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    关于伪单调变分不等式与不动点问题的新投影算法
    杨静,龙宪军
    数学物理学报. 2022 (3):  904-919. 
    摘要 ( 30 )   RICH HTML PDF(694KB) ( 28 )   收藏

    该文在Hilbert空间中给出了一个新投影算法,找到了伪单调变分不等式问题的解集与半压缩映射的不动点集的公共元.在映射是伪单调和一致连续的条件下,证明了强收敛定理.数值实验结果表明了新算法的有效性和优越性.

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    THE SYSTEMS WITH ALMOST BANACH-MEAN EQUICONTINUITY FOR ABELIAN GROUP ACTIONS
    朱斌, 黄小军, 连媛
    数学物理学报. 2022 (3):  919-940.  DOI: 10.1007/s10473-022-0307-5
    In this paper, we present the concept of Banach-mean equicontinuity and prove that the Banach-, Weyl- and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system of Abelian group action is zero. As an application of our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.
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    双重稀疏约束优化问题的一种贪婪单纯形算法
    潘庭葳,贺素香
    数学物理学报. 2022 (3):  920-933. 
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    鉴于交替最小化方法在求解双重稀疏约束优化问题时需要计算目标函数梯度的Lipschitz常数和构建该问题的L-稳定点时需要借助于Lipschitz条件等方面的不足,该文提出了一种求解该问题的贪婪单纯形算法.刻画了双重稀疏约束优化问题的CW最优性条件.基于CW最优性条件,具体设计了该算法的迭代步骤,并在较弱的假设条件下,证明了由算法产生的迭代点列全局收敛到问题的CW最优解.

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    二维回归相依风险模型的精细大偏差
    陈振龙,刘扬,傅可昂
    数学物理学报. 2022 (3):  934-942. 
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    该文讨论了一类非标准二维风险模型, 其中索赔额向量$ \{\vec{X}_k=(X_{1k}, X_{2k})^T, $ $k\ge 1\}$是独立同分布的非负随机向量序列, 向量内部两分量之间相依, 且向量与索赔发生时间间隔之间还存在回归相依关系.在索赔额向量边际分布具有一致变化尾的条件下, 得到该二维回归相依风险模型损失和的精细大偏差.

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    A NEW SUFFICIENT CONDITION FOR SPARSE RECOVERY WITH MULTIPLE ORTHOGONAL LEAST SQUARES
    李海锋, 张静
    数学物理学报. 2022 (3):  941-956.  DOI: 10.1007/s10473-022-0308-4
    A greedy algorithm used for the recovery of sparse signals, multiple orthogonal least squares (MOLS) have recently attracted quite a big of attention. In this paper, we consider the number of iterations required for the MOLS algorithm for recovery of a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$. We show that MOLS provides stable reconstruction of all $K$-sparse signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{w}$ in $\lceil\frac{6K}{M}\rceil$ iterations when the matrix $\mathbf{A}$ satisfies the restricted isometry property (RIP) with isometry constant $\delta_{7K}\leq0.094$. Compared with the existing results, our sufficient condition is not related to the sparsity level $K$.
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    违约风险下目标收益型养老金计划的α-鲁棒最优投资策略
    石媛,赵永霞
    数学物理学报. 2022 (3):  943-960. 
    摘要 ( 25 )   RICH HTML PDF(560KB) ( 28 )   收藏

    该文研究具有违约风险和模型不确定性的目标收益养老金计划的最优投资和收益支付问题.假定养老基金投资于无风险资产、可违约债券和股票,其中股票价格服从常弹性方差(CEV)模型.养老金的支付取决于计划的财务状况,且风险由不同代人分担.同时为保障退休之前发生死亡的养老金持有者权益,在模型中加入保费返还条款.此外,该文的模型允许养老金管理者有不同程度的模糊厌恶,而不是只考虑极端的模糊厌恶.应用随机控制方法,分别建立违约后和违约前两种情况下的Hamilton-Jacobi-Bellman方程,推导出α-鲁棒的最优投资策略和最优福利调整策略的闭型解.最后数值分析说明了金融市场参数对最优控制问题的影响.

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    CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT
    李应求, 黄绪兰, 彭朝晖
    数学物理学报. 2022 (3):  957-974.  DOI: 10.1007/s10473-022-0309-3
    We are interested in the convergence rates of the submartingale ${W}_{n} =\frac{Z_{n}}{\Pi_{n}}$ to its limit ${W},$ where $(\Pi_{n})$ is the usually used norming sequence and $(Z_{n})$ is a supercritical branching process with immigration $(Y_{n})$ in a stationary and ergodic environment $\xi$. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law: (i) $W-W_{n}$ with suitable normalization converges to the normal law $N(0,1)$, and similar results also hold for $W_{n+k}-W_{n}$ for each fixed $k\in \mathbb{N}^{\ast};$ (ii) for a branching process with immigration in a finite state random environment, if $W_{1}$ has a finite exponential moment, then so does $W,$ and the decay rate of $\mathbb{P}(|W-W_{n}|>\varepsilon)$ is supergeometric; (iii) there are normalizing constants $a_{n}(\xi)$ (that we calculate explicitly) such that $a_{n}(\xi)(W-W_{n})$ converges in law to a mixture of the Gaussian law.
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    QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES
    Edcarlos D. SILVA, Jefferson S. SILVA
    数学物理学报. 2022 (3):  975-1002.  DOI: 10.1007/s10473-022-0310-x
    It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.
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    SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS
    Nacira AGRAM, Saloua LABED, Bernt ØKSENDAL, Samia YAKHLEF
    数学物理学报. 2022 (3):  1003-1017.  DOI: 10.1007/s10473-022-0311-9
    This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution $X^{u,\xi}(t)=X(t)$ is given by $$X(t) =\phi(t)+{ \int_{0}^{t}}b\left( t,s,X(s),u(s)\right){\rm d}s+% { \int_{0}^{t}} \sigma\left( t,s,X(s),u(s)\right) {\rm d}B(s) + { \int_{0}^{t}} h\left( t,s\right) {\rm d}\xi(s). $$ Here ${\rm d}B(s)$ denotes the Brownian motion Itô type differential, $\xi$ denotes the singular control (singular in time $t$ with respect to Lebesgue measure) and $u$ denotes the regular control (absolutely continuous with respect to Lebesgue measure).
    Such systems may for example be used to model harvesting of populations with memory, where $X(t)$ represents the population density at time $t$, and the singular control process $\xi$ represents the harvesting effort rate. The total income from the harvesting is represented by $$ J(u,\xi) =\mathbb{E}\bigg[ \int_{0}^{T} f_{0}(t,X(t),u(t)){\rm d}t+ \int_{0}^{T} f_{1}(t,X(t)){\rm d}\xi(t)+g(X(T))\bigg] $$ for the given functions $f_{0},f_{1}$ and $g$, where $T>0$ is a constant denoting the terminal time of the harvesting. Note that it is important to allow the controls to be singular, because in some cases the optimal controls are of this type.
    Using Hida-Malliavin calculus, we prove sufficient conditions and necessary conditions of optimality of controls. As a consequence, we obtain a new type of backward stochastic Volterra integral equations with singular drift.
    Finally, to illustrate our results, we apply them to discuss optimal harvesting problems with possibly density dependent prices.
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    HE EXPONENTIAL OF QUASI BLOCK-TOEPLITZ MATRICES
    Elahe BOLOURCHIAN, Bijan Ahmadi KAKAVANDI
    数学物理学报. 2022 (3):  1018-1034.  DOI: 10.1007/s10473-022-0312-8
    The matrix Wiener algebra, $\mathcal{W}_N:=\mathrm{M}_{N} (\mathcal{W})$ of order $N>0$, is the matrix algebra formed by $N \times N$ matrices whose entries belong to the classical Wiener algebra $\mathcal{W}$ of functions with absolutely convergent Fourier series. A block-Toeplitz matrix $T(a)=[A_{i,j}]_{i,j \geq 0}$ is a block semi-infinite matrix such that its blocks $A_{i,j}$ are finite matrices of order $N$, $A_{i,j}=A_{r,s}$ whenever $i-j=r-s$ and its entries are the coefficients of the Fourier expansion of the generator $a:\mathbb{T} \rightarrow \mathrm{M}_{N} (\mathbb{C})$. Such a matrix can be regarded as a bounded linear operator acting on the direct sum of $N$ copies of $L^2(\mathbb{T})$. We show that $\exp(T(a))$ differes from $T(\exp(a))$ only in a compact operator with a known bound on its norm. In fact, we prove a slightly more general result: for every entire function $f$ and for every compact operator $E$, there exists a compact operator $F$ such that $f(T(a)+E)=T(f(a))+F$. We call these $T(a)+E's$ matrices, the quasi block-Toeplitz matrices, and we show that via a computation-friendly norm, they form a Banach algebra. Our results generalize and are motivated by some recent results of Dario Andrea Bini, Stefano Massei and Beatrice Meini.
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    BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS
    邱蜀燕, 穆春来, 易红
    数学物理学报. 2022 (3):  1035-1057.  DOI: 10.1007/s10473-022-0313-7
    This work explores the predator-prey chemotaxis system with two chemicals \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta u+\chi\nabla\cdot(u\nabla v)+\mu_1u(1-u-a_1w),\quad &x\in \Omega, t>0,\\ v_t=\Delta v-\alpha_1 v+\beta_1w,\quad &x\in \Omega, t>0,\\ w_t=\Delta w-\xi\nabla \cdot(w\nabla z)+\mu_2 w(1+a_2u-w),\quad &x\in\Omega, t>0,\\ z_t=\Delta z-\alpha_2 z+\beta_2u,\quad &x\in \Omega, t>0,\\ \end{array} \right. \end{eqnarray*} in an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^n$ under homogeneous Neumann boundary conditions. The parameters in the system are positive.
    We first prove that if $n\leq3$, the corresponding initial-boundary value problem admits a unique global bounded classical solution, under the assumption that $\chi, \xi$, $\mu_i, a_i, \alpha_i$ and $\beta_i(i=1,2)$ satisfy some suitable conditions. Subsequently, we also analyse the asymptotic behavior of solutions to the above system and show that
    $\bullet$ when $a_1<1$ and both $\frac{\mu_1}{\chi^2}$ and $\frac{\mu_2}{\xi^2}$ are sufficiently large, the global solution $(u, v, w, z)$ of this system exponentially converges to $(\frac{1-a_1}{1+a_1a_2}, \frac{\beta_1(1+a_2)}{\alpha_1(1+a_1a_2)}, \frac{1+a_2}{1+a_1a_2}, \frac{\beta_2(1-a_1)}{\alpha_2(1+a_1a_2)})$ as $t\rightarrow \infty$;
    $\bullet$ when $a_1>1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently large, the global bounded classical solution $(u, v, w, z)$ of this system exponentially converges to $(0, \frac{\alpha_1}{\beta_1}, 1, 0)$ as $t\rightarrow \infty$;
    $\bullet$ when $a_1=1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently large, the global bounded classical solution $(u, v, w, z)$ of this system polynomially converges to $(0, \frac{\alpha_1}{\beta_1}, 1, 0)$ as $t\rightarrow \infty$.
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    THE GLOBAL EXISTENCE AND A DECAY ESTIMATE OF SOLUTIONS TO THE PHAN-THEIN-TANNER MODEL
    位瑞英, 李银, 姚正安
    数学物理学报. 2022 (3):  1058-1080.  DOI: 10.1007/s10473-022-0314-6
    In this paper, we study the global existence and decay rates of strong solutions to the three dimensional compressible Phan-Thein-Tanner model. By a refined energy method, we prove the global existence under the assumption that the $H^3$ norm of the initial data is small, but that the higher order derivatives can be large. If the initial data belong to homogeneous Sobolev spaces or homogeneous Besov spaces, we obtain the time decay rates of the solution and its higher order spatial derivatives. Moreover, we also obtain the usual $L^p-L^2(1\leq p\leq2)$ type of the decay rate without requiring that the $L^p$ norm of initial data is small.
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    THE TIME DECAY RATES OF THE CLASSICAL SOLUTION TO THE POISSON-NERNST-PLANCK-FOURIER EQUATIONS IN $\mathbb{R}^3$
    童雷雷, 谭忠, 张旭
    数学物理学报. 2022 (3):  1081-1102.  DOI: 10.1007/s10473-022-0315-5
    In this work, the Poisson-Nernst-Planck-Fourier system in three dimensions is considered. For when the initial data regards a small perturbation around the constant equilibrium state in a $H^3\cap\dot{H}^{-s} (0\leq s\leq 1/2)$ norm, we obtain the time convergence rate of the global solution by a regularity interpolation trick and an energy method.
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    DK SPACES AND CARLESON MEASURES
    李东行, 乌兰哈斯, 赵如汉
    数学物理学报. 2022 (3):  1103-1112.  DOI: 10.1007/s10473-022-0316-4
    We give some characterizations of Carleson measures for Dirichlet type spaces by using Hadamard products. We also give a one-box condition for such Carleson measures.
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    A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION
    李奇, 向长林
    数学物理学报. 2022 (3):  1113-1124.  DOI: 10.1007/s10473-022-0317-3
    In this paper, we consider a class of fractional problem with subcritical perturbation on a bounded domain as follows: \begin{equation*} (P_{k})\quad \left\{ \begin{array}{ll} \displaystyle (-\Delta)^s u=g(x)[(u-k)^+]^{q-1}+u^{2^{*}_{s}-1},\ \ &x\in \Omega,\\ \displaystyle u>0,\ \ &x\in \Omega,\\ \displaystyle u=0,\ \ &x\in \mathbb{R}^N\backslash \Omega. \end{array} \right. \end{equation*} We prove the existence of nontrivial solutions $u_{k}$ of $(P_{k})$ for each $k\in (0,\infty)$. We also investigate the concentration behavior of the solutions $u_{k}$ as $k\to \infty$.
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    THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL
    谭金岚, 李勇勇, 唐春雷
    数学物理学报. 2022 (3):  1125-1140.  DOI: 10.1007/s10473-022-0318-2
    In this paper, we investigate a class of nonlinear Chern-Simons-Schrödinger systems with a steep well potential. By using variational methods, the mountain pass theorem and Nehari manifold methods, we prove the existence of a ground state solution for λ > 0 large enough. Furthermore, we verify the asymptotic behavior of ground state solutions as λ → +∞.
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    CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT
    黄明湛, 刘守宗, 宋新宇, 邹秀芬
    数学物理学报. 2022 (3):  1141-1159.  DOI: 10.1007/s10473-022-0319-1
    This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy. First, a stochastic impulsive model describing the interaction and competition among normal cells, tumor cells and immune cells under periodically pulsed chemotherapy is established. Then, sufficient conditions for the extinction, non-persistence in the mean, weak and strong persistence in the mean of tumor cells are obtained. Finally, numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features. The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries. In clinical practice, doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.
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    A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS
    黄腾
    数学物理学报. 2022 (3):  1160-1172.  DOI: 10.1007/s10473-022-0320-8
    Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a result from [7], we prove that the moduli space of the stable flat $SL(2,\mathbb{C})$ connections is disconnected under certain technical assumptions.
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    LEARNING RATES OF KERNEL-BASED ROBUST CLASSIFICATION
    王淑华, 盛宝怀
    数学物理学报. 2022 (3):  1173-1190.  DOI: 10.1007/s10473-022-0321-7
    This paper considers a robust kernel regularized classification algorithm with a non-convex loss function which is proposed to alleviate the performance deterioration caused by the outliers. A comparison relationship between the excess misclassification error and the excess generalization error is provided; from this, along with the convex analysis theory, a kind of learning rate is derived. The results show that the performance of the classifier is effected by the outliers, and the extent of impact can be controlled by choosing the homotopy parameters properly.
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    BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES
    何骞君, 吴新峰, 燕敦验
    数学物理学报. 2022 (3):  1191-1208.  DOI: 10.1007/s10473-022-0322-6
    In this paper, we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear $L^{r^{\prime},\alpha}$-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space. Moreover, we give an endpoint estimate for multilinear fractional integral operators. As an application, we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.
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    EXISTENCE RESULTS FOR SINGULAR FRACTIONAL p-KIRCHHOFF PROBLEMS
    向明启, Vicenţiu D. Rădulescu, 张彬林
    数学物理学报. 2022 (3):  1209-1224.  DOI: 10.1007/s10473-022-0323-5
    This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional $p$-Laplacian operator. More precisely, we study the following nonlocal problem: \begin{align*} \begin{cases} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|x|^{\alpha_1p}|y|^{\alpha_2p}|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right) \mathcal{L}^{s}_pu= |x|^{\beta} f(u)\,\, \ &{\rm in}\ \Omega,\\ u=0\ \ \ \ &{\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*} where $\mathcal{L}^{s}_p$ is the generalized fractional $p$-Laplacian operator, $N\geq1$, $s\in(0,1)$, $\alpha_1,\alpha_2,\beta\in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, and $M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0$, $f:\Omega\rightarrow\mathbb{R}$ are continuous functions. Firstly, we introduce a variational framework for the above problem. Then, the existence of least energy solutions is obtained by using variational methods, provided that the nonlinear term $f$ has $(\theta p-1)$-sublinear growth at infinity. Moreover, the existence of infinitely many solutions is obtained by using Krasnoselskii's genus theory. Finally, we obtain the existence and multiplicity of solutions if $f$ has $(\theta p-1)$-superlinear growth at infinity. The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.
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    CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES
    商绍强
    数学物理学报. 2022 (3):  1225-1237.  DOI: 10.1007/s10473-022-0324-4
    In this paper, we prove that if $X$ is an almost convex and 2-strictly convex space, linear operator $T: X \to Y$ is bounded, $N(T)$ is an approximative compact Chebyshev subspace of $X$ and $R(T)$ is a 3-Chebyshev hyperplane, then there exists a homogeneous selection ${T^\sigma }$ of ${T^\partial }$ such that continuous points of ${T^\sigma }$ and ${T^\partial }$ are dense on $Y$.
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    A ROBUST COLOR EDGE DETECTION ALGORITHM BASED ON THE QUATERNION HARDY FILTER
    毕文姗, 程冬, 刘万凯, 高洁欣
    数学物理学报. 2022 (3):  1238-1260.  DOI: 10.1007/s10473-022-0325-3
    This paper presents a robust filter called the quaternion Hardy filter (QHF) for color image edge detection. The QHF can be capable of color edge feature enhancement and noise resistance. QHF can be used flexibly by selecting suitable parameters to handle different levels of noise. In particular, the quaternion analytic signal, which is an effective tool in color image processing, can also be produced by quaternion Hardy filtering with specific parameters. Based on the QHF and the improved Di Zenzo gradient operator, a novel color edge detection algorithm is proposed; importantly, it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique. From the experimental results, we conclude that the minimum PSNR improvement rate is 2.3% and the minimum SSIM improvement rate is 30.2% on the CSEE database. The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.
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