中国科学院数学与系统科学研究院期刊网

15 July 2024, Volume 40 Issue 3
    

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    ARTICLES
  • Global Phase Portraits of Uniform Isochronous Centers System of Degree Six with Polynomial Commutator
    Li-na GUO, Ai-yong CHEN, Shuai-feng ZHAO
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 577-599. https://doi.org/10.1007/s10255-024-1081-z
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    This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form $\dot{x}=-y+xf(x,y),\ \dot{y}=x+yf(x,y)$, where $f(x,y)=a_{1}x+a_{2}xy+a_{3}xy^{2}+a_{4}xy^{3}+a_{5}xy^4=x\sigma(y)$, and any zero of $1+a_{1}y+a_{2}y^2+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}$, $y=\overline{y}$ is an invariant straight line. At last, all global phase portraits are drawn on the Poincarédisk.
  • Ramsey Numbers of Trees Versus Multiple Copies of Books
    Xiao-bing GUO, Si-nan HU, Yue-jian PENG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 600-612. https://doi.org/10.1007/s10255-024-1117-4
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    Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any two-coloring of the edges of $K_{N}$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the number of vertices in a minimum color class among all proper $\chi(G)$-colorings of $G$. Burr showed that $R(G,H)\geq (v(G)-1)(\chi(H)-1)+s(H)$ if $G$ is connected and $v(G)\geq s(H)$. A connected graph $G$ is $H$-good if $R(G,H)=(v(G)-1)(\chi(H)-1)+s(H)$. %Ramsey goodness is a special property of graph. Let $tH$ denote the disjoint union of $t$ copies of graph $H$, and let $G\vee H$ denote the join of $G$ and $H$. Denote a complete graph on $n$ vertices by $K_n$, and a tree on $n$ vertices by $T_n$. Denote a book with $n$ pages by $B_n$, i.e., the join $K_2\vee \overline{K_n}$. Erdös, Faudree, Rousseau and Schelp proved that $T_n$ is $B_m$-good if $n\geq 3m-3$. In this paper, we obtain the exact Ramsey number of $T_n$ versus $2B_2$. Our result implies that $T_n$ is $2B_2$-good if $n\geq5$.
  • Clustering for Bivariate Functional Data
    Shi-yun CAO, Yan-qiu ZHOU, Yan-ling WAN, Tao ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 613-629. https://doi.org/10.1007/s10255-024-1116-5
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    In this paper, we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject. The $k$-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data, and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered. In addition, we also consider two other clustering methods, $k$-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis. Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index. The approaches are further illustrated through empirical analysis of human mortality data.
  • Sliced Average Variance Estimation for Tensor Data
    Chuan-quan LI, Pei-wen XIAO, Chao YING, Xiao-hui LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 630-655. https://doi.org/10.1007/s10255-024-1024-8
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    Tensor data have been widely used in many fields, e.g., modern biomedical imaging, chemometrics, and economics, but often suffer from some common issues as in high dimensional statistics. How to find their low-dimensional latent structure has been of great interest for statisticians. To this end, we develop two efficient tensor sufficient dimension reduction methods based on the sliced average variance estimation (SAVE) to estimate the corresponding dimension reduction subspaces. The first one, entitled tensor sliced average variance estimation (TSAVE), works well when the response is discrete or takes finite values, but is not $\sqrt{n}$ consistent for continuous response; the second one, named bias-correction tensor sliced average variance estimation (CTSAVE), is a de-biased version of the TSAVE method. The asymptotic properties of both methods are derived under mild conditions. Simulations and real data examples are also provided to show the superiority of the efficiency of the developed methods.
  • A Degree Condition for Graphs Having All (a,b)-parity Factors
    Hao-dong LIU, Hong-liang LU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 656-664. https://doi.org/10.1007/s10255-024-1090-y
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    Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\cdots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 2(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\delta(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.
  • Dynamic Analysis of the Multi-state Reliability System with Priority Repair Discipline
    Aihemaitijiang YUMAIER, Ehmet KASIM
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 665-694. https://doi.org/10.1007/s10255-023-1079-y
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    This paper considers a multi-state repairable system that is composed of two classes of components, one of which has a priority for repair. First, we investigate the well-posedenss of the system by applying the operator semigroup theory. Then, using Greiner's idea and the spectral properties of the corresponding operator, we obtain that the time-dependent solution of the system converges strongly to its steady-state solution.
  • A Novel Analysis Approach of Uniform Persistence for an Epidemic Model with Quarantine and Standard Incidence Rate
    Song-bai GUO, Yu-ling XUE, Xi-liang LI, Zuo-huan ZHENG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 695-707. https://doi.org/10.1007/s10255-023-1078-y
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    Inspired by the transmission characteristics of the coronavirus disease 2019 (COVID-19), an epidemic model with quarantine and standard incidence rate is first developed, then a novel analysis approach is proposed for finding the ultimate lower bound of the number of infected individuals, which means that the epidemic is uniformly persistent if the control reproduction number $\mathcal{R}_{c}>1$. This approach can be applied to the related biomathematical models, and some existing works can be improved by using that. In addition, the infection-free equilibrium $V^0$ of the model is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$ and linearly stable if $\mathcal{R}_{c}=1$; while $V^0$ is unstable if $\mathcal{R}_{c}>1$.
  • Long-time Asymptotics for the Reverse Space-time Nonlocal Hirota Equation with Decaying Initial Value Problem: without Solitons
    Wei-qi PENG, Yong CHEN
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 708-727. https://doi.org/10.1007/s10255-024-1121-8
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    In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the $\vartheta(\lambda_{i}) \ (i=0, 1)$ would like to be imaginary, which results in the $\delta_{\lambda_{i}}^{0}$ contains an increasing $t^{\frac{\pm Im\vartheta(\lambda_{i})}{2}}$, and then the asymptotic behavior for nonlocal Hirota equation becomes differently.
  • Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem
    Xiao-yao JIA, Zhen-luo LOU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 728-743. https://doi.org/10.1007/s10255-024-1091-x
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    In this paper, we study the following quasi-linear elliptic equation: \begin{equation}\nonumber \left\{ \begin{aligned} &- {\rm div} (\phi(|\nabla u|)\nabla u)=\lambda \psi(|u|)u + \varphi(|u|)u, ~~~\text{in } \Omega,\\ & u= 0, ~~~\text {on} \partial \Omega, \end{aligned} \right. \end{equation} where $\Omega \subset \mathbb R^N$ is a bounded domain, $\lambda > 0$ is a parameter. The function $\psi(|t|)t$ is the subcritical term, and $\varphi(|t|)t$ is the critical Orlicz-Sobolev growth term with respect to $\phi$. Under appropriate conditions on $\phi$, $\psi$ and $\varphi$, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for $\lambda \in (0,\lambda_0)$, where $\lambda_0>0$ is a fixed constant.
  • Optimal Timing of Business Conversion for Solvency Improvement
    Peng LI, Ming ZHOU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 744-757. https://doi.org/10.1007/s10255-024-1048-0
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    In this paper, we study the optimal timing to convert the risk of business for an insurance company in order to improve its solvency. The cash flow of company evolves according to a jump-diffusion process. Business conversion option offers the company an opportunity to transfer the jump risk business out. In exchange for this option, the company needs to pay both fixed and proportional transaction costs. The proportional cost can also be seen as the profit loading of the jump risk business. We formulated this problem as an optimal stopping problem. By solving this stopping problem, we find that the optimal timing of business conversion mainly depends on the profit loading of the jump risk business. A larger profit loading would make the conversion option valueless. The fixed cost, however, only delays the optimal timing of business conversion. In the end, numerical results are provided to illustrate the impacts of transaction costs and environmental parameters to the optimal strategies.
  • Equilibrium Reinsurance Strategy and Mean Residual Life Function
    Dan-ping LI, Lv CHEN, Lin-yi QIAN, Wei WANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 758-777. https://doi.org/10.1007/s10255-024-1050-6
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    In this paper, we analyze the relationship between the equilibrium reinsurance strategy and the tail of the distribution of the risk. Since Mean Residual Life (MRL) has a close relationship with the tail of the distribution, we consider two classes of risk distributions, Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) distributions, which can be used to classify light-tailed and heavy-tailed distributions, respectively. We assume that the underlying risk process is modelled by the classical Cramér-Lundberg model process. Under the mean-variance criterion, by solving the extended Hamilton-Jacobi-Bellman equation, we derive the equilibrium reinsurance strategy for the insurer and the reinsurer under DMRL and IMRL, respectively. Furthermore, we analyze how to choose the reinsurance premium to make the insurer and the reinsurer agree with the same reinsurance strategy. We find that under the case of DMRL, if the distribution and the risk aversions satisfy certain conditions, the insurer and the reinsurer can adopt a reinsurance premium to agree on a reinsurance strategy, and under the case of IMRL, the insurer and the reinsurer can only agree with each other that the insurer do not purchase the reinsurance.
  • Derivation of Expanded Isospectral-Nonisospectral Integrable Hierarchies via the Column-vector Loop Algebra
    Hai-feng WANG, Yu-feng ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 778-800. https://doi.org/10.1007/s10255-024-1047-1
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    A scheme for generating nonisospectral integrable hierarchies is introduced. Based on the method, we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem. It follows that the corresponding expanded isospectral and nonisospectral integrable hierarchies are deduced based on a 6 dimensional complex linear space $\widetilde{\mathbb{C}}^6$. By reducing these integrable hierarchies, we obtain the expanded isospectral and nonisospectral derivative nonlinear Schrödinger equation. By using the trace identity, the bi-Hamiltonian structure of these two hierarchies are also obtained. Moreover, some symmetries and conserved quantities of the resulting hierarchy are discussed.
  • A Reliable Iteration Algorithm for One-Bit Compressive Sensing on the Unit Sphere
    Yan-cheng LU, Ning BI, An-hua WAN
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 801-822. https://doi.org/10.1007/s10255-024-1046-2
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    The one-bit compressed sensing problem is of fundamental importance in many areas, such as wireless communication, statistics, and so on. However, the optimization of one-bit problem constrained on the unit sphere lacks an algorithm with rigorous mathematical proof of convergence and validity. In this paper, an iteration algorithm is established based on difference-of-convex algorithm for the one-bit compressed sensing problem constrained on the unit sphere, with iterating formula $$ x^{k+1} = \mathop{\arg\min}\limits_{x\in C} \big\{\|x\|_1 + \eta_1 \|x^k\|_1 \max(\|x\|_2^2,1) - 2\eta_2 \|x^k\|_1 \langle x,x^k\rangle\big\}, $$ where $C$ is the convex cone generated by the one-bit measurements and $\eta_1 > \eta_2 > \frac{1}{2}$. The new algorithm is proved to converge as long as the initial point is on the unit sphere and accords with the measurements, and the convergence to the global minimum point of the $\ell_1$ norm is discussed.
  • On the Strong Approximation for a Simple Reentrant Line in Light Traffic Under First-buffer First-served Service Discipline
    Kai-ming YANG, Yong-jiang GUO
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 823-839. https://doi.org/10.1007/s10255-024-1093-8
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    For a 2-station and 3-class reentrant line under first-buffer first-served (FBFS) service discipline in light traffic, we firstly construct the strong approximations for performance measures including the queue length, workload, busy time and idle time processes. Based on the obtained strong approximations, we use a strong approximation method to find all the law of the iterated logarithms (LILs) for the above four performance measures, which are expressed as some functions of system parameters: means and variances of interarrival and service times, and characterize the fluctuations around their fluid approximations.
  • Incidence Coloring of Outer-1-planar Graphs
    Meng-ke QI, Xin ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 840-845. https://doi.org/10.1007/s10255-024-1126-3
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    A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge. It is known that every outer-1-planar graph is a planar partial 3-tree. In this paper, we conjecture that every planar graph $G$ has a proper incidence $(\Delta(G)+2)$-coloring and confirm it for outer-1-planar graphs with maximum degree at least $8$ or with girth at least $4$. Specifically, we prove that every outer-$1$-planar graph $G$ has an incidence $(\Delta(G)+3,2)$-coloring, and every outer-$1$-planar graph $G$ with maximum degree at least $8$ or with girth at least $4$ has an incidence $(\Delta(G)+2,2)$-coloring.
  • Barrier Option Pricing in Regime Switching Models with Rebates
    Yue-xu ZHAO, Jia-yong BAO
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 846-861. https://doi.org/10.1007/s10255-024-1053-3
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    This paper is concerned with the valuation of single and double barrier knock-out call options in a Markovian regime switching model with specific rebates. The integral formulas of the rebates are derived via matrix Wiener-Hopf factorizations and Fourier transform techniques, also, the integral representations of the option prices are constructed. Moreover, the first-passage time density functions in two-state regime model are derived. As applications, several numerical algorithms and numerical examples are presented.
  • Strong Limit Theorems for Weighted Sums under the Sub-linear Expectations
    Feng-xiang FENG, Ding-cheng WANG, Qun-ying WU, Hai-wu HUANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 862-874. https://doi.org/10.1007/s10255-024-1127-2
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    In this article, we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. We establish general strong law and complete convergence theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. Our results of strong limit theorems are more general than some related results previously obtained by Thrum (1987), Li et al. (1995) and Wu (2010) in classical probability space.
  • Testing Linearity in Functional Partially Linear Models
    Fan-rong ZHAO, Bao-xue ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 875-886. https://doi.org/10.1007/s10255-023-1040-0
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    For the functional partially linear models including flexible nonparametric part and functional linear part, the estimators of the nonlinear function and the slope function have been studied in existing literature. How to test the correlation between response and explanatory variables, however, still seems to be missing. Therefore, a test procedure for testing the linearity in the functional partially linear models will be proposed in this paper. A test statistic is constructed based on the existing estimators of the nonlinear and the slope functions. Further, we prove that the approximately asymptotic distribution of the proposed statistic is a chi-squared distribution under some regularity conditions. Finally, some simulation studies and a real data application are presented to demonstrate the performance of the proposed test statistic.
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