中国科学院数学与系统科学研究院期刊网

15 July 2024, Volume 67 Issue 4
    

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  • Statistical Analysis Based on the Application and Funding Data of the National Natural Science Foundation of China for Youth Science Fund Projects in Mathematics
    Gui Ping ZHAO, De Cai LIANG, Qian Yu SHU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 611-623. https://doi.org/10.12386/A20240023
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    The National Natural Science Foundation of China (NSFC) is one of the fundamental departments funding mathematical research. Since its establishment, it has supported over 22,000 mathematical projects in China, with more than 8,600 being Youth Science Fund projects. Examining the application and funding status of the Youth program in mathematical disciplines is beneficial for understanding the scale of young mathematical talent development in China, assessing the quality of fund reviews, and planning fund support strategies. Utilizing data science techniques, the paper initially conducts an overall analysis of the application and funding status of Youth Science Fund projects. We then conduct hierarchical statistics analysis on project funding rates based on factors such as discipline, age, institution, and gender. Addressing the recent phenomenon of an overall “decreasing growth rate" in the Youth program, we further conduct statistical analysis from the applicants' perspective, including analyzing the changing patterns of funding proportions and comparing the funding rates of different applicants. Regarding the situation where recipients of Youth projects continue to apply for General projects, the paper utilizes the censored data mechanism and Kaplan—Meier estimation to infer the distribution of the time intervals between the approval of Youth and General programs. This approach provides more accurate estimates of the mean and quantiles of the time interval, effectively addressing the “sampling bias" problem that arises when considering only those who have obtained general project approvals.
  • Normal Elements and Irreducible Polynomials in Finite Fields
    Wei CAO, Wei Hua LI, Bi Yun XU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 624-633. https://doi.org/10.12386/A20220014
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    Let $\mathbb{F}_{q}$ be the finite field of $q$ elements, and $\mathbb{F}_{q^{n}}$ be its extension of degree $n$. An element $\alpha\in \mathbb{F}_{q^{n}}$ is called a normal element of $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ if $\{\alpha,\alpha^{q},\ldots, \alpha^{q^{n-1}}\}$ constitutes a basis of $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$. Normal elements over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. The minimal polynomial of a normal element is certainly an irreducible polynomial with nonzero trace, while the converse does not hold in general. Using linearized polynomials, we give some necessary and sufficient conditions for this problem, which extend the known results.
  • A Note on $k$-hyperbolicity of Hermitian Manifolds
    Kai TANG, Shuo YAN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 634-640. https://doi.org/10.12386/A20220107
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    Motivated by the recent work of Ni on $k$-hyperbolicity of K\"{a}hler manifolds, we investigate $k$-hyperbolicity of Hermitian manifolds. We prove a new Schwarz-lemma-type estimate for holomorphic map between complete Hermitian manfolds. As a corollary, we show that a compact Hermitian manifold with negative first $k$-Ricci curvature is $k$-hyperbolic.
  • A Class of Inverse Mean Curvature Flow in the Eculid Space
    A Huan GONG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 641-651. https://doi.org/10.12386/A20220147
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    We study a class of inverse mean curvature flows in the Euclidean space. If the hypersurface is closed, star-shaped and mean convex, after rescaling, they will converge to a sphere.
  • On Poisson Algebra Structure of the Schrödinger Algebra in One and Two Dimensional Space-time
    Lin Lu YAN, Yu Feng YAO
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 652-665. https://doi.org/10.12386/A20220148
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    Poisson algebras are a class of algebras that admit both associative algebra structure and Lie algebra structure satisfying the Leibniz rule. In this paper, the Poisson algebra structures of the Schrödinger algebras in one and two dimensional space-time are determined.
  • 2-Local Derivations on the Extended Schrödinger—Virasoro Lie Algebra
    Fei WANG, Xiao Min TANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 666-674. https://doi.org/10.12386/A20220162
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    Let $\widetilde{\mathfrak{sv}}$ be the extended Schrödinger—Virasoro Lie Algebra. A map $\Delta: \widetilde{\mathfrak{sv}}\rightarrow \widetilde{\mathfrak{sv}}$ is called a 2-local derivation if for every $x,y\in \widetilde{\mathfrak{sv}},$ there exists a derivation $D_{x,y}$ of $\widetilde{\mathfrak{sv}}$ such that $\Delta(x)=D_{x,y}(x)$ and $\Delta(y)=D_{x,y}(y)$. We prove that any 2-local derivation on $\widetilde{\mathfrak{sv}}$ is a derivation.
  • Functional Panel Quantile Regression Models with Group Structured Fixed Effect Functions
    Hua LIU, Zi Jian WANG, Jia Qi MEN, Jin Hong YOU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 675-703. https://doi.org/10.12386/A20220167
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    Panel data analysis and functional data analysis are both popular research in many fields, such as statistics and econometrics. As practical applications become more complex, the types of data become more and more diverse, and the homogeneity and heterogeneity of data often exist simultaneously. Thus, it is very necessary to identify the homogeneity and heterogeneity of data before making correct and effective statistical inferences. In this paper, we propose functional panel quantile regression models with unknown group fixed effect functions. The computation of quantile regression can be extremely challenging due to the fact that panel data are a combination of time series data and cross-sectional data, as well as the infinite-dimensional feature of functional data. We adopt the newly proposed convolution method to smooth the objective function to deal with the computational complexity associated with such large samples and high-dimensional data. We combine the Bayesian information criterion and the hierarchical agglomerative clustering algorithm to identify potential group structures. Based on the identified group structure information, we further propose an efficient method to estimate the bivariate time-varying coefficient function. In this paper, we also prove the consistency of the group identification methods and the asymptotic normality of the corresponding estimators. And through several numerical simulation results and analyses of the household electricity consumption data and Gini coefficient data of 87 counties in Jiangxi Province, we illustrate the rationality of the proposed model and the validity of the estimation procedure.
  • An Algorithm to Solve the Variational Inequality Problem Based on the Common Solutions of Two Classes of Problems
    Yuan Heng WANG, Tian Tian XU, Jen-Chih YAO, Bing Nan JIANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 704-718. https://doi.org/10.12386/A20220171
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    We study a new algorithm to solve a common solution of the split feasibility problem and the fixed point problem involving quasi-nonexpansive mappings in Hilbert spaces. Based on the common solutions of these two classes of problems, we solve the variational inequality problem. Compared with the predecessors, the self-adaptive technique and the inertial iteration method are added, which can speed up the convergence rate of the iterative sequence generated by our algorithms. At the same time, we extend the involving previous nonexpansive mappings to extensive quasi-nonexpansive mappings. In addition, a strong positive bounded operator is added to the algorithm, which extends the original viscous iterative algorithm to a more general viscous iterative algorithm. The effectiveness of the algorithm is verified by numerical examples.
  • The Set-Valued Metric Generalized Inverses and Its Single-Valued Selection of Multi-Valued Linear Operator in Banach Spaces
    Hong Li LIU, Guan Qi LIU, Yu Wen WANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 719-731. https://doi.org/10.12386/A20220184
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    In this paper, the exact representation of domain, range and expression of the set valued metric generalized inverse from the multi-valued closed linear operators in the inverse Banach spaces are given by means of the Banach spaces geometry method, especially the generalized orthogonal decomposition theorem. At the same time, the construction method and characterization of the single valued homogeneity selection of the metric generalized inverse are given, which makes some rencent related achievements can be deduced directly.
  • The Relative Curvature Integral Inequalities of Two Mixed Convex Bodies
    Xu DONG, Ya Ling WANG, Yu Qi ZHOU, Chun Na ZENG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 732-742. https://doi.org/10.12386/A20220190
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    The curvature integral inequalities are important in geometric inequalities. In this paper, by using the nonsymmetric Green—Osher inequality, we obtain the bounds of the relative curvature integral of the higher power of two mixed convex bodies. On the basis of Green and Osher's work, these results remove the symmetric condition and are the famous strengthened form of Green—Osher inequality. In particular, when a convex body is a unit circle, we obtain the Ros inequality and the inequalities of curvature entropy in $\mathbb{R}^2$. And the new simplified proofs of the inequalities of curvature entropy are provided.
  • CFI Operators and Property $UW_\Pi$
    Jiong DONG, Xiao Hong CAO, Nan LI
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 743-755. https://doi.org/10.12386/B20220738
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    In this paper, using the new spectrum set defined by the property of consistency in Fredholm index, we characterize the property $UW_\Pi$ for bounded linear operators and their functions. Moreover, the sufficient and necessary conditions that an operator, its adjoint operator and their functions satisfy simultaneously the property $UW_\Pi$ are investigated. As an application, the property $UW_\Pi$ for $\ast$-paranormal operators is considered.
  • Quantum Correlation of Quantum Networks Based on Their Covariance Matrices
    Yi ZHAO, Zhi Hua GUO, Huai Xin CAO
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 756-780. https://doi.org/10.12386/A20230074
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    The characteristics of a quantum network are determined by the topology of the network and quantum states shared between nodes. In this paper, we establish a mathematical framework for quantum networks based on directed acyclic graphs. We provide a method for constructing the covariance matrix of a quantum network based on probability tensors, and obtain relevant properties of the covariance matrix. We further constructed a covariance matrix of a quantum network with respect to some quantum measurement. By defining a block coherence measure of the covariance matrix, we obtain a correlation measure of quantum networks. We proved that quantum correlation of quantum networks is 0 if and only if all shared states of the network are product states.
  • Rectifiable Surfaces on Self-affine Sierpinski Sponges
    Jia Qi FAN, Li Feng XI
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 781-793. https://doi.org/10.12386/A20230081
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    This paper discusses the tangent planes and the areas of rectifiable surfaces on self-affine Sierpinski sponges by using Rademacher's theorem, the coarea formula and recurrence of classical points on Euclidean spaces.
  • The Generalized Green Ring of Category $\widetilde{\mathcal{O}}$ of $\frak{sl}_2$
    Cai E LV, Cai Xia SHEN, Li Meng XIA
    Acta Mathematica Sinica, Chinese Series. 2024, 67(4): 794-806. https://doi.org/10.12386/B20230357
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    Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbb{C}$. The category $\widetilde{\mathcal{O}}$ is a tensor category which contains the well-known BGG category. In this paper, we first introduce the generalized Green ring which is a natural generalization of the well-known Green ring. Then we study the explicit structure of $\widetilde{\mathcal{O}}$ for the case $\mathfrak{g}=\frak{sl}_2$. Finally, we give an abelian tensor category containing $\mathcal{O}$ for $\frak{sl}_2$ and it is minimal in such categories.
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