中国科学院数学与系统科学研究院期刊网

15 July 2022, Volume 65 Issue 4
    

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  • Xiang Yu ZHOU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 581-598. https://doi.org/10.12386/A20220050
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    In this article, we elaborate and reveal the contribution of Chinese ancient mathematics to the Chinese civilization, not only to the material civilization, but also to Sinology, Chinese language and Chinese culture; elaborate the respect and influence of Chinese culture on Mathematics and discover mathematics in Chinese Classics; and describe the achievements of Chinese ancient mathematics and disclose its influence and contribution to modern mathematics.
  • Wei ZHANG, Deng Feng LI
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 599-606. https://doi.org/10.12386/A20200154
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    In this paper, we investigate the characterization of g-Riesz bases in term of g-biorthogonal sequences. We obtain that a sequence of operators is a g-Riesz basis if and only if it is a g-complete g-Bessel sequence with g-biorthogonal sequence which is also a g-complete g-Bessel sequence, and further prove that the condition for gcompleteness of one (any one) of two g-Bessel sequences can be removed from the characterization. Examples are given to illustrate the relations for g-biorthogonality, g-completeness and g-Bessel condition.
  • Gui Ping SUN, Mu ZHAO, Yong ZHOU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 607-624. https://doi.org/10.12386/A20210019
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    Residual life is an important quantity to characterize the individual life expectancy. Early studies on the residual life mainly focus on the mean residual life. However, when the potential survival function of population is skewed or heavy-tailed, the mean residual life does not exist. So the statisticians suggest the quantile residual life to characterize the individual life expectancy. With the complete data and the right censored data, the modeling and theoretical properties of the quantile residual life have been established well. However, biased sampling data are often encountered in practical investigations. For example, left truncated data are often encountered in clinical trial, case cohort sampling data frequently occur in epidemiology, length biased sampling data also frequently occur in large medical cohort studies. Ignoring sampling biases will lead to biased estimators and unreasonable inferences. We consider a quantile residual life regression model under the general biased and right censored data. First, we propose a quantile regression model of residual life with the general biased and right censored data. Estimation procedures by using general estimation equation method are proposed. Comparing to the independent assumption of the censored variables and the covariates, which is commonly used in the existing literatures, this paper mainly considers that censored variables depend on the covariates. Second, because the asymptotic variance of the estimator involves the non-parametric density functions, so a bootstrap method is adopted. Finally, the simulation results show that the proposed estimators have good finite sample properties.
  • Xiao Li WANG, Alatancang
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 625-638. https://doi.org/10.12386/A20210049
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    We consider the local spectral properties for the upper triangular operator matrix $M_{C}=\binom{A \ C}{ 0 \ B}\in L(X\oplus Y)$, where $A\in L(X)$, $B\in L(Y)$, $C\in L(Y,X)$ and $X,Y$ are infinite-dimensional complex Banach spaces. Firstly we investigate the single-valued extension property for $M_{C}$ by considering the set $\mathcal{S}(T)=\{\lambda \in \mathbb{C}: T$ does not have the single-valued extension property at $\lambda\}.$ By means of vector valued analytic functions and analytic kernels, we obtain the characterization of $\mathcal{S}(M_{C})$ and in particular we develop some conditions on $A$ and $B$ under which the equality $\mathcal{S}(M_{C})=\mathcal{S}(A)\cup\mathcal{S}(B)$ holds for arbitrary $C$. Further, we show that for certain operator $C$ the equality $\mathcal{S}(M_{C})=\mathcal{S}(A)\cup\mathcal{S}(B)$ holds for arbitrary $A,B$. Also, we give some examples to illustrate our results. At the end, we apply the obtained results to spectral perturbation and local spectral perturbation for upper triangular operator matrices, obtain conditions for which equalities $\sigma(M_{C})=\sigma(A)\cup\sigma(B)$ and $\sigma_{M_{C}}(x\oplus 0)=\sigma_{A}(x)$ hold true, and give a characterization for the local spectral subspaces of $M_{C}$.
  • Ming Chu GAO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 639-656. https://doi.org/10.12386/A20210069
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    We defining bi-circular element pairs of random variables, which provide examples of $R$-diagonal pairs of random variables. Formulae are given for calculating the distributions of the product pairs of two $*$-bi-free $R$-diagonal pairs. When focusing on pairs of left acting operators and right acting operators from finite von Neumann algebras in the standard form, we characterize $R$-diagonal pairs of random variables in terms of the $*$-moments of the random variables. Finally, we define $\eta$-diagonal pairs of random variables, and give a characterization of $\eta$-diagonal distributions in terms of the $*$-moments of the distribution.
  • Qi HAN, Ya Xin KOU, Ya Nan HAN, Zi Qiang LU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 657-664. https://doi.org/10.12386/A20210004
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    We give the quantum decomposition of general Bernoulli random variable $X$, the integral representation of the number of return routes in $m$ steps of a three point path graph, and the asymptotic spectral distribution of two kinds of growing graphs—cycle graph and complete bipartite graph. This makes it possible for us to study a classical variable or a probability distribution within the framework of quantum probability. To some extent, this also shows that the quantum probabilistic techniques in the spectral analysis of graphs.
  • Hua Ning LIU, Xi LIU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 665-678. https://doi.org/10.12386/A20200128
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    Let $p$ be a prime and let $n$ be an integer coprime to $p$. The Fermat quotient $q_p(n)$ is defined as the unique integer with $$q_p(n)\equiv \dfrac{n^{p-1}-1}{p} \ (\bmod\ p),\quad 0\leq q_p(n)\leq p-1. $$ We also define $q_p(kp)=0$ for $k\in \mathbb{Z}$. In this paper we constructed large families of binary sequences of length $p^2$ by using the estimates for character sums of Fermat quotients, and studies the pseudorandomness: well-distribution, correlation, linear complexity, collision and avalanche effect.
  • Peng Cheng TANG, Xue Jun ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 679-690. https://doi.org/10.12386/A20200213
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    It is well known that the problem of composition operator is a basic problem in function spaces. In general, the problem of composition operator for holomorphic function spaces characterized by derivative is much more complicated than that of one complex variable case. In this paper, the authors give the necessary and sufficient conditions for the boundedness or compactness of composition operators from the boundary general function spaces $F^{p,q,s}(B)$ to the Bloch type spaces $\mathscr{B}^{\frac{q+n}{p}}(B)$. In particular, the authors give the simple necessary and sufficient conditions for the compactness of the composition operator when $p\neq q+n$.
  • Ya Di WU, Xiao Qing YUE
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 691-698. https://doi.org/10.12386/A20210045
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    Let ${\mathbb L}$ be a super Heisenberg-Virasoro algebra with the ${\mathbb C}$-basis {$L_{n},I_{n},G_{n}|$ $ n\in {\mathbb Z}$}, which satisfies the relations $[L_{m},L_{n}]=(m-n)L_{m+n}$, $[L_{m},I_{n}]=\!-nI_{m+n}$, $[L_{m},G_{n}]=-nG_{m+n}$ and $ [G_{m},G_{n}]=I_{m+n}.$ In this paper, we prove that all super-skewsymmetric super-biderivations of ${\mathbb L}$ are inner. Furthermore, we prove that every linear super-commuting map on ${\mathbb L}$ has the form $\Psi(x)=f(x)I_{0}$ for all $x\in{\mathbb L}$, where $f(x)$ is a linear map from ${\mathbb L}$ to ${\mathbb C}$.
  • Li Fang ZHOU, Jin LU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 699-708. https://doi.org/10.12386/A20210083
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    In this paper, for $1<p<\infty$, we characterize that the essential norm of a noncompact Toeplitz operator $T_\mu$ with $\mu$ being a positive $p$-Carleson measure equals its distance to the set of compact Toeplitz operators on the Bergman space with regular weight $A^p_\omega$. Moreover, the distance is realized by infinitely many compact Toeplitz operators.
  • Dong Xu LV
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 709-722. https://doi.org/10.12386/A20200184
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    Suppose $M^{m}$ is an immersed submanifold in the unit sphere space $S^{n}$, on which the Blaschke tensor $A$ is the fundamental Möbius invariant. In this paper, we study of which the Möbius rigidity of the conformal invariants $\|A\|^{2}- (\operatorname{{\rm tr}}A)^{2}$ is constant submanifolds. We obtain an inequality about the integral of the cubic function of the norm of isotropic Blaschke tensor, and classify this class of submanifolds when the equality holds in the inequality.
  • Guang Ming HU, Jun Ming LIU, Yi QI, Shu An TANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 723-732. https://doi.org/10.12386/A20210085
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    We provide the characterizations of distances from Bloch functions to BMOA by the high order derivatives. These results generalize the distance formula from Bloch functions to BMOA by Peter Jones and Ruhan Zhao. As applications, we give some equivalent characterizations of the small Teichmüller space.
  • Sheng Fan ZHOU, Xu Ying JIANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 733-750. https://doi.org/10.12386/A20200229
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    We first present the conditions for the existence of a random attractor of continuous cocycle defined by the solutions of locally coupled stochastic retarded lattice dynamical systems in a space of continuous functions from finite closed interval into the space of infinite sequences. Then we consider the existence and upper semicontinuity of the random attractor for the nonautonomous retarded Schrödinger lattice system with multiplicative white noise.
  • Yong Xiu CAO, Ji Chang YU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(4): 751-762. https://doi.org/10.12386/A20200143
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    The case–cohort design has been widely used in epidemiological and biomedical studies due to its cost-effectiveness. Existing statistical methods are primarily focused on obtaining consistent and efficient estimators for the regression parameters. However, there is little work to estimate the causal effect of a nonrandomized treatment on the outcome. In this article, we develop an efficient inference procedure to estimate the average treatment effect based on data from the case–cohort design. We also establish the consistency and asymptotic normality of the proposed estimator. The finite sample performance of the proposed estimator is evaluated through simulation studies. Finally, we use the proposed method to analyze a real data set.