中国科学院数学与系统科学研究院期刊网

15 May 2024, Volume 67 Issue 3
    

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  • Yun GAO, Fang Wei FU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 413-427. https://doi.org/10.12386/A20220016
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    Let $\mathbb{F}_q$ be a finite field of cardinality $q$, where $q$ is a power of a prime $p$, $t \ge 2$ is an even number satisfying $t\not \equiv 1\ (\bmod \,p)$ and $\mathbb{F}_{{q^t}}$ is an extension field of $\mathbb{F}_q$ with degree $t$. Firstly, a trace bilinear form on $\mathbb{F}_{{q^t}}^n$ which is called $\Delta$-bilinear form is given, where $n$ is a positive integer coprime to $q$. Then according to this trace bilinear form, bases and enumeration of cyclic $\Delta$-self-orthogonal and cyclic $\Delta$-self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{{q^t}}$-codes are investigated when $t=2$. Furthermore, some good $\mathbb{F}_q$-linear $\mathbb{F}_{{q^2}}$-codes are obtained.
  • Yi XUAN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 428-443. https://doi.org/10.12386/B20220154
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    We study weighted fractional Sobolev-Poincaré inequalities in irregular domains. The weights considered here are distances to the boundary to certain powers, and the domains are the so-called $s$-John domains and $ beta$-Hölder domains. Our main results extend that of Hajlasz-Koskela [J. Lond. Math. Soc., 1998, 58(2): 425-450] from the classical weighted Sobolev-Poincaré inequality to its fractional counter-part and Guo [Chin. Ann. Math., 2017, 38B(3): 839-856] from the fractional Sobolev-Poincaré inequality to its weighted case.
  • Yong Xin BAI, Man Ling QIAN, Mao Zai TIAN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 444-467. https://doi.org/10.12386/A20220026
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    We propose an effective iterative screening method for the ultra-high dimensional additive quantile regression with missing data. Specifically, the canonical correlation analysis is introduced into the maximum correlation coefficient based on the optimal transformation, and the marginal contribution of important variables is sorted by the maximum correlation coefficient after the optimal transformation of covariates and model residuals. On the basis of variable screening, the sparse smooth penalty is used to make further variable selection. The proposed variable selection method has three advantages: (1) The maximum correlation based on optimal transformation can reflect the nonlinear dependent structure of response variable to covariable more comprehensively; (2) In the iteration process, the residual can be used to obtain the relevant information of the model so as to improve the accuracy of variable screening; (3) The variable screening process can be separated from model estimation to avoid regression of redundant covariables. Under appropriate conditions, the sure independent screening property of the variable screening method and the sparsity and consistency of the estimator under the sparse-smooth penalty are proved. Finally, the performance of the proposed method is given by Monte Carlo simulation and the rat genome data is used to illustrate the effectiveness of the proposed method.
  • You Qi LIU, Jin XIA, Xiao Feng WANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 468-481. https://doi.org/10.12386/A20220087
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    We study Toeplitz operators with positive symbols on $A_2$ weighted harmonic Bergman spaces in bounded smooth domain of $n$-dimensional real space. We characterize sufficient and necessary conditions for bounded or compact Toeplitz operators via average function and Berezin transform. The Schatten class Toeplitz operators are obtained as well.
  • Yi SHI
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 482-488. https://doi.org/10.12386/A20220115
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    Let $\rho$ be an orthogonal representation on a Euclidean space $V$, and $SV$ be the unit sphere of $V$. Let $\bar{d}_{\mathcal{H}}$ and $d_{\mathcal{H}}$ be the horizontal metrics on $V$ and $SV$ induced by $\rho$, respectively. Our main result is to show that the following conditions are equivalent: (1) The representation $\rho$ is polar. (2) $(V, \bar{d}_{\mathcal{H}})$ is a CAT$(0)$ space. (3) $(SV, d_{\mathcal{H}})$ is a CAT$(1)$ space.
  • Gang ZHANG, Long JIANG, Wei ZHANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 489-499. https://doi.org/10.12386/A20220129
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    This paper establishes the existence and uniqueness results for solutions to multidimensional backward stochastic differential equation ($G$-BSDE) driven by $G$-Brownian motion, in which the generators $f$ and $g$ of $G$-BSDE satisfy the $\beta$-order Mao's condition in $y$ and the Lipschitz conditon in $z$.
  • Yan Hong SONG, Zhi Cheng WANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 500-510. https://doi.org/10.12386/A20220127
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    In the paper, Viterbi algorithms for hidden Markov models are studied. When partial states, initial probability distributions, transition probability matrices and observation probability matrices are given, the optimal state sequences are estimated by the Viterbi algorithms. Compared with existing algorithms, the algorithms presented in the paper have not only considered the influence of partially visible states on the initial conditions and recursion formulas, but also ensured that the predicted state sequences are overall optimal. Finally, fault recognition is investigated to verify the feasibility of the algorithms.
  • Jin Lin GUAN, Yan TANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 511-520. https://doi.org/10.12386/A20220139
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    We introduce a modified iterative algorithm for solving the split common fixed point problem for multi-valued demicontractive mapping in real Hilbert spaces. Under very mild conditions, we prove a strong convergence theorem for our algorithm by using inertial and hybrid projection technique. Further, we give the application and example of the algorithm to illustrate the effectiveness of the algorithm. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.
  • Lang Mei BU, Guo Xing JI
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 521-530. https://doi.org/10.12386/A20220149
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    Let $\mathcal M$ be a factor von Neumann algebra on a complex Hilbert space $\mathcal{H}$ with $\dim\mathcal M>1$ and $\mathcal M_{+}$ the positive cone of $\mathcal M$. We consider automorphisms of $\mathcal M_{+}$ with respect to convex sequential product $\circ_{\lambda}$ on $\mathcal M_{+}$ for some $\lambda \in [0,1]$ defined by $A\circ_{\lambda}B=\lambda A^{\frac12}BA^{\frac12}+(1-\lambda)B^{\frac12}AB^{\frac12}$ for any $ A,B\in \mathcal M_{+}$. We show that an automorphism of $\mathcal M_{+}$ with respect to convex sequential product is implemented by a $*$-isomorphism or an anti-$*$-isomorphism of $\mathcal M$.
  • Chu QIN, Yi Chao CHEN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 531-538. https://doi.org/10.12386/A20220161
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    The partial duality $G^A$ of an embedded graph $G$ can be seen as geometric duality over a partial edge set $A$ of $G$. It is a generalization of the classic Poincaré duality $G^*$. Unlike the classic Poincaré duality, the genus of $G^A$ is often not equal to the genus of $G.$ Similar to the Huang-Liu's characterization theorem of non upper-embedability of graphs, we first prove a structure theorem for nonmaximal partial-dual planar graphs. Then, we determine the maximum partial-dual genus for a planar triangulated graph $G$, that is, if $G$ is $3$-cycle, the maximum partial-dual genus of $G$ is $1$; Otherwise the maximum partial-dual genus of $G$ is equal to the number of vertices minus 1.
  • Ya GAO, Yan Ling GAO, Jing MAO
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 539-564. https://doi.org/10.12386/B20220651
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    In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen as a prescribed curvature problem and a continuous work of [Mathematische Nachrichten, 2024, 297: 833-860].
  • Yu FAN, Ying Ying HU, Yi SUN, Pei HENG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 565-581. https://doi.org/10.12386/A20220174
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    Bayesian networks utilize directed acyclic graphs (DAGs) to constrain conditional independencies in multivariate joint probability distribution, so as to realize its modular decomposition in uncertainty reasoning and reduce the computational complexity of probabilistic reasoning. They are widely used in probabilistic reasoning, machine learning and causal inference. In practice, if structure learning or statistical inference was performed by adopting the idea of dividing and conquering or model collapsing, we have to establish the marginal models by finding their minimal Markov subgraphs (or minimal independence maps). Therefore, this paper details minimal Markov subgraphs for marginal models of Bayesian networks, and provides the refined characterization on them from the perspectives of statistics and graph theory. For the collapsibility of DAG, this paper gives more intuitive equivalent conditions based on the properties of directed inducing paths, and also proposes some sufficient conditions, which provides more theoretical tools for judging whether the considered models can be collapsible onto local sub-models.
  • Jing ZHANG, Yan Yan LIU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 582-598. https://doi.org/10.12386/A20220179
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    Linear regression models are often used to study the relationship between variables in various fields of scientific research, such as medicine, genetics, economics. However, main effects may not be sufficient to characterize the relationship between the response and predictors in complex situations, the interaction effects between variables will also have an important influence on the response variable in many practical problems. Interaction model that considers both the main effect and the interaction effect can describe the relationship between variables more comprehensively. For high-dimensional data, the number of variables $p$ is relatively large, and the number of second-order interaction terms $\frac{p(p+1)}{2}$ is much larger, the statistical analysis of the interaction model faces many difficulties and challenges. How to select the important interaction effects that have a significant impact on the event of interest from huge number of interaction effects is a very important problem. The existing research on this problem mainly focuses on the complete data under the framework of the linear model. In this paper, we will consider this problem for ultrahigh-dimensional right-censored survival data. Based on distance correlation and the two-step analysis method, we propose a model-free screening method for interaction effects which does not depend on any model assumptions. This method can select the important main effects and important interaction effects at the same time, and can handle ultrahigh-dimensional data with large $p$. Extensive simulation studies are carried out to evaluate the finite sample performance of the proposed procedure, and the results show that this method can effectively select the important interaction effects for ultrahigh-dimensional right-censored survival data. As an illustration, we apply the proposed method to analyze the diffuse large-B-cell lymphoma (DLBCL) data.
  • Yue Lu ZHANG, Gang CAI
    Acta Mathematica Sinica, Chinese Series. 2024, 67(3): 599-610. https://doi.org/10.12386/A20230043
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    This paper introduces a Bregman extragradient method and applies it to solve pseudo-monotone variational inequality problems in Hilbert spaces. Under some reasonable assumptions imposed on the parameters, a weak convergence theorem for the suggested method is achieved. The results obtained in this paper generalize and improve many recent ones in the literature.