中国科学院数学与系统科学研究院期刊网

15 March 2020, Volume 63 Issue 2
    

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  • Yan Hong SONG
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 97-104. https://doi.org/10.12386/A2020sxxb0008
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    We study geometric and algebraic transience for discrete-time Markov chains on countable state spaces. Criteria are presented based on the moment of the last exit time for some state and the existence of solution for some equation. Moreover, we apply the results to investigating the stochastic stability of Geom/G/1 queueing models.

  • Yu Tao LIU, Jing PAN, Yong ZHOU
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 105-122. https://doi.org/10.12386/A2020sxxb0009
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    We propose a novel nonparametric estimator of the quantile difference based on the length-biased data subject to potential right censoring. In order to improve efficiency, the new estimator incorporates the auxiliary information inherent in the prevalent sampling design. And it has a simple expression, which is easy to compute. Moreover, the consistency and asymptotic normality of this quantile difference estimator are established using the empirical process theory and the asymptotic variance can be obtained consistently via a resampling method. We also demonstrate that the proposed estimator exhibits satisfactory performance with finite sample size through the Monte-Carlo studies and an analysis of a real data example about the Alzheimer's disease.

  • Rui Pu BAI, Shuai HOU, Chuang Chuang KANG
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 123-136. https://doi.org/10.12386/A2020sxxb0010
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    We studied the structure of 3-Lie algebras with involutive derivations, and proved that if A is an m-dimensional 3-Lie algebra with an involutive derivation, then there exists a compatible 3-pre-Lie algebra and a local cocycle 3-Lie bialgebraic structure on the 2m-dimensional semi-direct product 3-Lie algebra Aad* A*. By means of involutive derivations, we constructed the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra Aad* A*, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras.

  • Ji Chang YU, Yong Xiu CAO
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 137-148. https://doi.org/10.12386/A2020sxxb0011
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    Case-cohort design is a well-known cost-effective design and has been widely used in survival analysis. Many statistical methods have been developed to estimate the covariates effects on the survival time based on case-cohort data. However, little work has focused on checking the proportional hazards model assumptions with case-cohort data. In this article, we propose a class of test statistics through the asymptotically mean-zero processes for testing the proportional hazards assumption with case-cohort data. Re-sampling scheme is proposed to approximate the asymptotic distribution of the test statistics. Simulation studies are conducted to evaluate the finite sample performances of the proposed method and a data set from the National Wilm's Tumor Study Group is analyzed to illustrate the proposed method.

  • Lun Chuan ZHANG
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 149-156. https://doi.org/10.12386/A2020sxxb0012
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    We prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space. Our results include the relevant conclusions of predecessors as special cases, and refine B. Biane's work as a corollary.

  • Xun Gui GUAN
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 157-170. https://doi.org/10.12386/A2020sxxb0013
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    Let p1, p2, p3 be diverse odd primes, and c > 1 be integer. We obtain all nonnegative integer solutions(x, y, z) on the Pell equations x2-(c2-1)y2=y2-2p1p2p3z2=1. It generalizes the previous work of Keskin (2017) and Cipu (2018).

  • Pin Ling WANG, Ming Liang FANG
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 171-180. https://doi.org/10.12386/A2020sxxb0014
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    Let f, g be two nonconstant meromorphic functions, let a be a nonzero finite complex number, and let n ≥ 5 be a positive integer. If[f(z)]n and[g(z)]n share a CM, f(z) and g(z) share ∞ CM, and N1)(r, f)=S(r, f), then either f(z) ≡ tg(z), where tn=1, or f(z)g(z) ≡ t, where tn=a2. This improves some unicity results concerning derivatives and differences of meromorphic functions.

  • Xian Yuan WU, Rui ZHU
    Acta Mathematica Sinica, Chinese Series. 2020, 63(2): 181-192. https://doi.org/10.12386/A2020sxxb0015
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    It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently,[Internet Mathematics, DOI:10.1080/15427951, 2015.101208] indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the d (d ≥ 1)-dimensional lattice torus (with size nd) according to[Internet Mathematics, DOI:10.1080/15427951, 2015.101208], and show that both the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in log n.