中国科学院数学与系统科学研究院期刊网

15 December 2020, Volume 36 Issue 12
    

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  • Jun LIAO, He Guo LIU, Xing Zhong XU, Ji Ping ZHANG
    Acta Mathematica Sinica. 2020, 36(12): 1315-1340. https://doi.org/10.1007/s10114-020-9132-8
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    The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial Z-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups, we finally obtain the structure and invariants of the group G.

  • Ricardo ABREU-BLAYA, Juan BORY-REYES
    Acta Mathematica Sinica. 2020, 36(12): 1341-1356. https://doi.org/10.1007/s10114-020-9332-2
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    The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé-Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel-Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.

  • Aristides V. DOUMAS, Vassilis G. PAPANICOLAOU
    Acta Mathematica Sinica. 2020, 36(12): 1357-1383. https://doi.org/10.1007/s10114-020-9425-y
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    A collector samples coupons with replacement from a pool containing g uniform groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur (while coupons of different groups have different probabilities to occur). For each j=1,..., g, let Tj be the number of trials needed to detect Group j, namely to collect all Mj coupons belonging to it at least once. We first derive formulas for the probabilities P {T1 < … < Tg} and P {T1j=1g Tj}. After that, without severe loss of generality, we restrict ourselves to the case g=2 and compute the asymptotics of P {T1 < T2} as the number of coupons grows to infinity in a certain manner. Then, we focus on T:=T1T2, i.e. the number of trials needed to collect all coupons of the pool (at least once), and determine the asymptotics of E[T] and V[T], as well as the limiting distribution of T (appropriately normalized) as the number of coupons becomes large.

  • Xiao Fang LUO, Xiao Xiao NIE, Jian Dong YIN
    Acta Mathematica Sinica. 2020, 36(12): 1384-1394. https://doi.org/10.1007/s10114-020-9331-3
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    In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property.

  • Gui Li LIAO, Qi Meng LIU, Rong Mao ZHANG
    Acta Mathematica Sinica. 2020, 36(12): 1395-1416. https://doi.org/10.1007/s10114-020-9428-8
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    A self-weighted quantile procedure is proposed to study the inference for a spatial unilateral autoregressive model with independent and identically distributed innovations belonging to the domain of attraction of a stable law with index of stability α, α ∈ (0, 2]. It is shown that when the model is stationary, the self-weighted quantile estimate of the parameter has a closed form and converges to a normal limiting distribution, which avoids the difficulty of Roknossadati and Zarepour (2010) in deriving their limiting distribution for an M-estimate. On the contrary, we show that when the model is not stationary, the proposed estimates have the same limiting distributions as those of Roknossadati and Zarepour. Furthermore, a Wald test statistic is proposed to consider the test for a linear restriction on the parameter, and it is shown that under a local alternative, the Wald statistic has a non-central chisquared distribution. Simulations and a real data example are also reported to assess the performance of the proposed method.

  • Dong Han ZHANG, You LU, Sheng Gui ZHANG
    Acta Mathematica Sinica. 2020, 36(12): 1417-1428. https://doi.org/10.1007/s10114-020-0144-1
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    Pil?niak and Wo?niak put forward the concept of neighbor sum distinguishing (NSD) total coloring and conjectured that any graph with maximum degree Δ admits an NSD total (Δ+3)-coloring in 2015. In 2016, Qu et al. showed that the list version of the conjecture holds for any planar graph with Δ ≥ 13. In this paper, we prove that any planar graph with Δ ≥ 7 but without 6-cycles satisfies the list version of the conjecture.

  • Hong Yan YIN, Da ZHOU, Xing An ZHANG
    Acta Mathematica Sinica. 2020, 36(12): 1429-1440. https://doi.org/10.1007/s10114-020-6350-z
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    In this paper, we investigate the isolated closed orbits of two types of cubic vector fields in R3 by using the idea of central projection transformation, which sets up a bridge connecting the vector field X(x) in R3 with the planar vector fields. We have proved that the cubic vector field in R3 can have two isolated closed orbits or one closed orbit on the invariant cone. As an application of this result, we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones, and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.

  • Kuldeep Singh CHARAK, Ilpo LAINE
    Acta Mathematica Sinica. 2020, 36(12): 1441-1444. https://doi.org/10.1007/s10114-020-9516-9
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    This is a corrigendum to our paper, "On a class of prime entire functions", published in Acta Math. Sin., Engl. Ser., 25, 1647-1652 (2009).