中国科学院数学与系统科学研究院期刊网

15 November 2022, Volume 65 Issue 6
    

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  • Li Jie MA, Xiao Chuan XU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 959-966. https://doi.org/10.12386/A20210064
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    In this paper, we study the transmission eigenvalue problem with the Robin boundary condition. Applying the related properties of entire function of exponential type, we show the relationship between the density of eigenvalues and the length of the support interval of the potential function. Meanwhile, we prove that the transmission eigenvalue problem is equivalent to a kind of Sturm-Liouville problem with spectral parameter in the boundary condition.
  • Zheng Jun ZHAO, Xiang CHEN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 967-978. https://doi.org/10.12386/A20210073
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    Let $\mathbb{C}_v$ be an algebraically closed non-archimedean field, complete with respect to a valuation $v$. Let $\varphi:\mathbb{P}^N\rightarrow \mathbb{P}^N$ be a morphism of degree greater than one defined over $\mathbb{C}_v$, $\Phi$ a lift of $\varphi$. Let $\mathcal{G}_\Phi$ be the Green function of $\Phi$ and $\rho$ the chordal metric on $\mathbb{P}^N(\mathbb{C}_v)$. In this paper, we first study the properties of reduction of points in high dimensional projective space and reduction of automorphisms of $\mathbb{P}^N$ with degree one. With the help of Green function $\mathcal{G}_\Phi$ of $\Phi$, we introduce the arithmetic distance of morphisms and investigate its property. The necessary and sufficient condition which $\Phi$ has good reduction is obtained in this paper. We also describe explicitly the Filled Julia set of $\Phi$ by its Green function.
  • Shuang Jian GUO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 979-988. https://doi.org/10.12386/A20200223
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    We study Leibniz algebras equipped with higher derivations. We call such a tuple of a Leibniz algebra and a higher derivation by LeibHDer pair. First, we define representations of LeibHDer pairs and construct the semi-direct product. Finally, we define a suitable cohomology for a LeibHDer pair with coefficients in a representation, and study central extensions and deformations of LeibHDer pairs.
  • Zhen Dong GU, Li Ying SUN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 989-1002. https://doi.org/10.12386/A20210034
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    Spectral collocation method is investigated for the nonlinear Caputo fractional multi-point value problems. The main idea of the presented method is to solve the corresponding nonlinear weakly singular Volterra-Fredholm integral equations obtained from the nonlinear Caputo fractional multi-point value problems. In order to carry out convergence analysis for the presented method, we investigate the Gronwall type inequality with Volterra-Fredholm integral terms. The provided convergence analysis shows that the presented method has spectral convergence, which is confirmed by the provided numerical experiments. At present, numerical methods for fractional multi-point value problems are rarely studied. The method and convergence analysis in this paper are useful references for the researches of related subjects.
  • Mei Ying LÜ, Jing XIE
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1003-1008. https://doi.org/10.12386/A20210058
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    For any real number $x\in(0,1)$, there exists a unique Engel continued fractions of $x$. In this paper, we mainly discuss the exceptional set which the logarithms of the partial quotients grow with non-linear rate. We completely characterize the Hausdorff dimension of the relevant exceptional set.
  • Guo Wang CHEN, Fang DA
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1009-1022. https://doi.org/10.12386/A20200216
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    In this paper, the existence and uniqueness of the global generalized solution and the global classical solution of the initial boundary value problem for the generalized nonlinear telegraph equation with nonlinear damping \begin{align*} &v_{tt}-\alpha v_{xxtt}-v_{xx}+ \beta v_{xxt}=\beta f(v)_{xxt}, \ \ x\in(0,1),\ \ t>0,\\ &v(0,t)=0, \ \ v(1,t)=0, \ \ t>0,\\ &v(x,0)=v_{0}(x),\ \ v_{t}(x,0)=v_{1}(x), \ \ x\in(0,1), \end{align*} are proved. When $f(v)$ is a linear function, the asymptotic behavior of the solution for the problem is studied. The existence and uniqueness of the global solution for the following initial boundary value problem \begin{align} &v_{tt}-v_{xx}=\alpha v_{xxtt}+\beta\Big(\frac{v^{3}}{3}-v\Big)_{xxt},\nonumber\\ &v(0,t)=0, \ \ v(1,t)=0,\nonumber\\ &v(x,0)=v_{0}(x),\ \ v_{t}(x,0)=v_{1}(x)\nonumber \end{align} are also proved.
  • Hang Long ZHANG, Xi Wang CAO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1023-1032. https://doi.org/10.12386/A20200131
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    Let $q=p^k$ and $\mathbb F_{q^{n}}$ be the extension field of $\mathbb F_{q}$ of degree $n$, where $p$ is an odd prime and $n, k$ are positive integers. The main contribution of this paper is as follows:If $n\,|\,(q-1),\, k\geq11,\, n\geq14$ or $n\nmid(q-1),\, k\geq10,\, n\geq8$, then there exists a primitive element $\alpha$ in $\mathbb F_{q^n}$ such that $\alpha+\alpha^{-1}$ is a normal element, and $1+\alpha^2$ is a square element, and there exists a normal element $\beta$, such that $\beta+\beta^{-1}$ is a primitive element, and $1+\beta^2$ is a square element.
  • Ke Wei ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1033-1044. https://doi.org/10.12386/B20210292
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    In this article we study Chow stability and K-stability on algebraic manifolds. Using the language of filtrations, we relate the Chow weight to the slope at infinity of the quantized K-energy along Bergman geodesics, which implies an inequality between $\delta$- and $\delta_m$-invariants. We also introduce a series of new invariants, which can characterize Chow Stability and K-stability.
  • Rui Chang PEI
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1045-1056. https://doi.org/10.12386/B20210187
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    The main purpose of this paper is to investigate the existence of one positive solution and the existence of infinitely many nontrivial solutions for a class of ($p$, 2)-Laplacian equation with subcritical polynomial growth and subcritical (critical) exponential growth. Some existence results for nontrivial solutions are established by using the mountain pass theorem and the fountain theorem.
  • Xiao Xia SUN, Xuan Ming NI
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1057-1066. https://doi.org/10.12386/A20190028
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    The relationship between a fractional diffusion process and its integration by parts formula is studied. By constructing a pull back formula, the integration by parts formula for fractional diffusion process is established. Conversely, a fractional diffusion process can be characterized through its integration by parts formula.
  • Lin XIAO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1067-1082. https://doi.org/10.12386/A20190054
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    In this paper, I consider that the actuarial model is affected by the environmental process $\Theta$, premium income counting process $\eta$, claim counting process I and the claim process B, and establish a compound binomial risk model with random income in Markov chain environment, which is called MRICM, for short. The characteristic five-tuple set is given. It is proved that there exists a probabilistic space $(\Omega,\mathscr{F},P)$, and MRICM$(\Theta,\eta,I,B)$ defined on it, and its characteristic five-tuple set coincides with the given one. The recursive equations of conditional ruin probability for finite time and infinite time are obtained.
  • Yong Hong LIU, Qing Qing ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1083-1092. https://doi.org/10.12386/A20190074
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    In this paper, using large deviations for a Brownian sheet and increments of a Brownian sheet, we obtain local functional law of the iterated logarithm for a Brownian sheet and increments of a Brownian sheet.
  • Shu Jin WU, Nan HUA
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1093-1104. https://doi.org/10.12386/A20190057
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    The advantage of time series with matrix cross-section data is that multiple attributes of multiple objects can be characterized simultaneously. This paper focuses on autoregression model of time series with matrix cross-section data and presents the methods of parameter estimation, model identification and white noise test. Finally, the daily yield series and daily volume change rate series of two bank stocks are analyzed by this model.
  • Yu Ting LAN, Xin Wei FENG, Ning ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1105-1122. https://doi.org/10.12386/A20190091
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    In this paper, a strong law of large numbers for arrays of rowwise negatively associated random variables is obtained under nonlinear probabilities, from which Kolmogorov type and Marcinkiewicz-Zygmund type strong laws of large numbers are derived. And the notion of negative association is weaker than some existing notions of dependence in nonlinear probabilities. Furthermore, an extension of strong law of large numbers for arrays of rowwise independent random variables under nonlinear probabilities is obtained. As a special case, a Kolmogorov type strong law indicates that not only the cluster points of empirical averages lie in the interval between the lower expectation and upper expectation quasi-surely, but such an interval is also the smallest one that covers the empirical averages quasi-surely. Furthermore, the strong law also states that the upper and lower limits of the empirical averages will converge to the upper and lower expectations with upper probabilities one, respectively.
  • Jian Zhang ZHU, Zhang LIU, Xin Yan WANG, Yi Jun HU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1123-1136. https://doi.org/10.12386/A20180105
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    We follow Cheung and Lo[Scandinavian Actuarial Journal] and Chi et al.[Insurance:Mathematics and Economics] to investigate the optimal reinsurance problem where risks of the insurer is measured by distortion risk measures, and premiums are calculated under the generalized distortion premium principle. Our novelty is the inclusion of constraints on the maximum level of risk the reinsurer can tolerate. Our objective is to seek for all the optimal reinsurance strategies which minimize the insurer's risk measurement of its total loss under the stipulated constraints.
  • Zhi Feng ZHU, Shao Yi ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1137-1142. https://doi.org/10.12386/A20190048
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    We first study the basic coupling and obtain an equation between total variation norm and the basic coupling. Then by use this equation we investigate the ergodicity property of continuous time Markov processes in general state space. For an ergodic continuous-time Markov processes, adding condition $\pi(f)<\infty$, by using the coupling method, there exists the full absorption set, such that the continuous time Markov processes are $f$-ergodic on it.
  • Xu YANG, Yu Rong WEI
    Acta Mathematica Sinica, Chinese Series. 2022, 65(6): 1143-1152. https://doi.org/10.12386/A20200098
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    In this paper, we establish the discrete approximation of continuous-state nonlinear branching processes in Lévy random environments by using tightness and convergence sequence in infinite dimensional product space via stochastic differential equations. Taking α-stable branching as an example, the conditions which are given to discretize continuous-state nonlinear branching processes in Lévy random environments are verified.