中国科学院数学与系统科学研究院期刊网

15 March 2025, Volume 68 Issue 2
    

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  • Shengxiang Lü, Yuxi Wang, Licheng Zhang
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 197-210. https://doi.org/10.12386/A20240074
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    A $1$-plane graph $D$ of a graph $G$ is a drawing of $G$ in the plane such that each edge is crossed at most once. The crossing number of $G$ is the minimum number of edge crossings in any drawing of $G$ in the plane. Determining the crossing number of a graph is NP-hard, and determining whether a graph is $1$-planar is NP-complete. In this paper, we establish the lower bound on the number of non-crossed edges in $2$-connected locally maximal $1$-plane graphs and locally crossing-optimal maximal $1$-plane graphs, respectively. Consequently, we also determine the upper bound of their crossing numbers in relation to the number of edges.
  • Xiaojie Wang, Fuyi Xu
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 211-223. https://doi.org/10.12386/B20230674
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    The present paper is dedicated to the study of the Cauchy problem for 3D incompressible inhomogeneous asymmetric fluids with only rough density. By exploiting some extra time-weighted energy estimates, and employing the interpolation argument and Lorentz norms for the time variable, we first construct the Lipschitz regularity of the velocity. Based on it, following the duality approach, we finally settle the uniqueness issue of the global weak solution constructed by [Qian,Chen and Zhang,Math.Ann.,2023,386:1555-1593].
  • Gang Yu, Wei Gao, Ningzhong Shi
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 224-239. https://doi.org/10.12386/A20230178
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    In this paper, we propose a new estimation approach for binary panel data model with error cross-sectional dependence. The estimation approach does not need to estimate the interactive effects in model. The asymptotic property of this proposed estimator is established as long as $N$ is fixed and $T$ goes to infinity. Finally, we present some Monte Carlo studies on the small sample properties of the proposed estimator for binary panel data model with error cross-sectional dependence, showing that our proposed estimator performs well.
  • Haiqiang Ma, Zhiyan Sheng, Xuan Liu, Jianbao Chen
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 240-267. https://doi.org/10.12386/A20230170
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    With the development of big data technology, the dimensionality of spatial data is becoming higher and higher, and the endogeneity and heterogeneity of data often exist simultaneously. In this paper, we propose a quantile regression model of high-dimensional spatial dependent data with endogenous spatial weight matrix so as to analyze high-dimensional spatial dependent data robustly. We then develop a three-step penalized quantile estimation procedure through combining the instrumental variable method, variable selection method with robust statistic method, and establish the consistency and the asymptotic normality of the corresponding estimators. In addition, the oracle theoretical properties of variable selection are derived under some mild conditions. At last, we investigate the effectiveness and robustness of the proposed model and method through simulations and an application to housing prices in 284 prefecture-level cities across the country.
  • Tong Wei, Zhishan Yang
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 268-277. https://doi.org/10.12386/A20230144
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    In this paper, we use Dudek's [Bull.Aust.Math.Soc., 2019, 99(1): 1-9] method to study the mean of divisor function defined on polynomial rings over finite fields over a quadratic polynomial sequence, and comparing the mean of divisor function defined on integer ring over quadratic arithmetic sequence, we find that the two are the same from the perspective of order.
  • Zezhen Sun
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 278-289. https://doi.org/10.12386/A20230128
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    This paper deals with a non-local curvature flow which preserves convexity and the modified elastic energy $\int^{L}_{0}\kappa^{2}ds-\epsilon L (\epsilon\ge0)$ of the evolving curve. We show that the flow exists globally, the length of the evolving curve is non-increasing, and the evolving curve converges to a finite circle in $C^{\infty}$ topology as time goes to infinity. As an application of this flow, we prove two new geometric inequalities.
  • Haiyang He, Xiao Li
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 290-303. https://doi.org/10.12386/B20230427
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    In this paper,our purpose is to study the following Hénon type Choquard system $$ \left\{\begin{array}{l} -\Delta u=\int_{\mathbb{R}^3} \frac{|x|^\alpha|y|^\alpha v^p(y)}{|x-y|^{3-\mu}} d y \cdot v^{p-1} \text { in } \mathbb{R}^3, \\ -\Delta v=\int_{\mathbb{R}^3} \frac{|x|^\alpha|y|^\alpha u^q(y)}{|x-y|^{3-\mu}} d y \cdot u^{q-1} \text { in } \mathbb{R}^3, \end{array}\right. $$ where $0<\mu<3, \alpha>0$. We will show that there are no positive classical solutions in three dimension-space $\mathbb{R}^3$ for $p, q>2$ and $$ \frac{1}{p}+\frac{1}{q}>\frac{2}{3+2 \alpha+\mu}. $$
  • Jiangfu Zhao, Jun Jiang
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 304-324. https://doi.org/10.12386/A20230158
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    Using the chord power integral and its inequalities of a convex body, we establish inequalities about moments for $\mu$-random chord length, $\nu$-random chord length, and $\lambda$-random chord length in $\mathbb{R}^n$. Based on the relationship between the chord power integral and containment function of a convex body, we obtain a new expression for moments of three kinds of random chord length mentioned above. By utilizing the properties of the distribution function and probability density function of $\mu$-random chord length, we get the calculation formulas for the distribution function and probability density function of $\nu$-random chord length, and the distribution function and probability density function of $\lambda$-random chord length, respectively. Further, we establish the relationships among three kinds of distribution functions. On this basis, taking a rhombus, regular pentagon, and regular hexagon as examples in $\mathbb{R}^2$, we give the expressions of their 1-order moment for three kinds of random chord length and the distribution function of $\nu$-random chord length.
  • Hujun Yang, Xiaoling Han, Caidi Zhao
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 325-349. https://doi.org/10.12386/A20230094
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    This article studies the trajectory statistical solution and its properties of the 3D tropical climate model. Firstly, the authors establish that the 3D tropical climate model with damping terms possesses a trajectory attractor, and use this trajectory attractor and generalized Banach limit to construct the trajectory statistical solution. Then they prove that the trajectory statistical solution has degenerate regularity provided that the associated generalized Grashof number is small enough. Finally they verify that the trajectory statistical solution converges to that of the 3D tropical climate model without damping term when the damping coefficients tends to zero.
  • Jinlong Wei, Guangying LÜ
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 350-368. https://doi.org/10.12386/A20230095
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    We extend Davie's trick (it Int. Math. Res. Not., 2007, 2007(1): 1-26) from stochastic differential equations with bounded measurable drifts to the ones in which the drifts are square integrable in time variable and Hölder continuous in space variable, and obtain the gradient estimates as well as the uniformly local quasi-Lipschitz estimates for strong solutions. As applications, we prove the unique strong solvability for stochastic transport equations driven by Wiener noise with square integrable drift as well as the uniformly local quasi-Lipschitz estimates for stochastic strong solutions, which partially solves the open problem posed by Fedrizzi and Flandoli (J. Funct. Anal., 2013, 264(6): 1329-1354).
  • Longfa Sun, Yipeng Zhang, Jingfeng Tian
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 369-378. https://doi.org/10.12386/A20230051
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    Let $X$ and $Y$ be real finite-dimensional Banach spaces with the same dimension and $f:X\rightarrow Y$ be a mapping. In this note, we show that if $X$ is smooth, then $f$ satisfies $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\},\; x, y\in X$, if and only if $f$ is phase equivalent to a linear surjective isometry.
  • Fucai Lin, Qiyun Wu, Chuan Liu
    Acta Mathematica Sinica, Chinese Series. 2025, 68(2): 379-396. https://doi.org/10.12386/A20230026
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    A topological space is called dense-separable if each dense subset of it is separable. Therefore, each dense-separable space is separable. This paper is devoted to establishing some basic properties of dense-separable topological groups. We prove that each separable space with a countable tightness is dense-separable, and give a dense-separable topological group which is not hereditarily separable. We also prove that, for a Hausdorff locally compact group, it is locally dense-separable iff it is metrizable. Moreover, we study dense-subgroup-separable topological groups. We prove that, for each locally compact abelian group, it is dense-subgroup-separable iff it is dense-separable iff it is metrizable. Finally, we discuss some applications in $d$-independent topological groups and related structures. We prove that each regular dense-subgroup-separable abelian semitopological group with $r_{0}(G)\geq\mathfrak{c}$ is $d$-independent. We also prove that, for each regular dense-subgroup-separable bounded paratopological abelian group $G$ with $|G|>1$, it is $d$-independent iff it is a nontrivial $M$-group iff each nontrivial primary component $G_{p}$ of $G$ is $d$-independent. Applying this result, we prove that a separable metrizable almost torsion-free paratopological abelian group $G$ with $|G|=\mathfrak{c}$ is $d$-independent. Further, we prove that each dense-subgroup-separable MAP abelian group with a nontrivial connected component is also $d$-independent.