中国科学院数学与系统科学研究院期刊网

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  • ARTICLES
    Yong LIU, Zi-yu LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 1-16. https://doi.org/10.1007/s10255-023-1072-5
    We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces. More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous Markov-Feller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior, then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.
  • ARTICLES
    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
    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.
  • ARTICLES
    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
    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.
  • ARTICLES
    Lu-yi LI, Ping LI, Xue-liang LI
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 269-274. https://doi.org/10.1007/s10255-024-1076-9
    Let $\mathbf{G}=\{G_i: i\in[n]\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$, where $\mathbf{G}$ can be seen as an edge-colored (multi)graph and each $G_i$ is the set of edges with color $i$. A graph $F$ on $V$ is called rainbow if any two edges of $F$ come from different $G_i$s'. We say that $\mathbf{G}$ is rainbow pancyclic if there is a rainbow cycle $C_{\ell}$ of length $\ell$ in $\mathbf{G}$ for each integer $\ell\in [3,n]$. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem: If $\delta(G_i)\geq\frac{n}{2}$ for each $i\in[n]$, then there is a rainbow Hamiltonian cycle in $\mathbf{G}$. In this paper, under the same condition, we show that $\mathbf{G}$ is rainbow pancyclic except that $n$ is even and $\mathbf{G}$ consists of $n$ copies of $K_{\frac{n}{2},\frac{n}{2}}$. This result supports the famous meta-conjecture posed by Bondy.
  • ARTICLES
    Meng CHEN, Wang-xue CHEN, Rui YANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 75-90. https://doi.org/10.1007/s10255-024-1104-9
    The traditional simple random sampling (SRS) design method is inefficient in many cases. Statisticians proposed some new designs to increase efficiency. In this paper, as a variation of moving extremes ranked set sampling (MERSS), double MERSS (DMERSS) is proposed and its properties for estimating the population mean are considered. It turns out that, when the underlying distribution is symmetric, DMERSS gives unbiased estimators of the population mean. Also, it is found that DMERSS is more efficient than the SRS and MERSS methods for usual symmetric distributions (normal and uniform). For asymmetric distributions considered in this study, the DMERSS has a small bias and it is more efficient than SRS for usual asymmetric distribution (exponential) for small sample sizes.
  • ARTICLES
    Qing-qing ZHENG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 17-34. https://doi.org/10.1007/s10255-024-1100-0
    In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results.
  • ARTICLES
    Cong-hui ZHANG, Hai-feng ZHANG, Mei-rong ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 275-301. https://doi.org/10.1007/s10255-024-1084-9
    The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper. We first show that there exist both continuous and discontinuous stationary solutions. Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition. In addition, we demonstrate the influences of the diffusion coefficient on stationary solutions. The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem. Finally, some numerical simulations are given to illustrate the theoretical results.
  • ARTICLES
    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
    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 $n\geq5$.
  • ARTICLES
    Peng-fei LI, Jun-hui XIE, Dan MU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 225-240. https://doi.org/10.1007/s10255-024-1111-x
    Let $\Omega$ be a bounded smooth domain in ${\mathbb{R}}^N \ (N\geq3)$. Assuming that 0<s<1, 1p,q)≠(N+2s/N-2s,N+2s/N-2s), and $a,b>0$ are constants, we consider the existence results for positive solutions of a class of fractional elliptic system below, \begin{align*} \left\{\begin{array}{ll} (a+b[u]^2_s)(-\Delta)^su=v^p+h_1(x,u,v,\nabla u,\nabla v), &\quad x\in\Omega,\\ (-\Delta)^sv=u^q+h_2(x,u,v,\nabla u,\nabla v), &\quad x\in\Omega,\\ u,v>0, &\quad x\in\Omega,\\ u=v=0, &\quad x\in \mathbb{R}^N\backslash\Omega. \end{array}\right. \end{align*} Under some assumptions of $h_i(x,u,v,\nabla u,\nabla v)(i=1,2)$, we get a priori bounds of the positive solutions to the problem (1.1) by the blow-up methods and rescaling argument. Based on these estimates and degree theory, we establish the existence of positive solutions to problem (1.1).
  • ARTICLES
    Chang-feng LI, Yi-rang YUAN, Huai-ling SONG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 302-319. https://doi.org/10.1007/s10255-024-1088-5
    In this paper, the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description, numerical simulation and theoretical analysis. Two important factors, heat and magnetic influences are involved. The mathematical model is formulated by four nonlinear partial differential equations (PDEs), determining four major physical variables. The influences of magnetic fields are supposed to be weak, and the strength is parallel to the $z$-axis. The elliptic equation is treated by a block-centered method, and the law of conservation is preserved. The computational accuracy is improved one order. Other equations are convection-dominated, thus are approximated by upwind block-centered differences. Upwind difference can eliminate numerical dispersion and nonphysical oscillation. The diffusion is approximated by the block-centered difference, while the convection term is treated by upwind approximation. Furthermore, the unknowns and adjoint functions are computed at the same time. These characters play important roles in numerical computations of conductor device problems. Using the theories of priori analysis such as energy estimates, the principle of duality and mathematical inductions, an optimal estimates result is obtained. Then a composite numerical method is shown for solving this problem.
  • ARTICLES
    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
    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.
  • ARTICLES
    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
    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.
  • ARTICLES
    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
    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 $\mathcal{R}_{c}>1$.
  • ARTICLES
    Ya-zhou CHEN, Hakho HONG, Xiao-ding SHI
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 45-74. https://doi.org/10.1007/s10255-023-1070-7
    This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/Allen-Cahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time $t\rightarrow+\infty$.
  • ARTICLES
    WANG Wei-fan, WANG Yi-qiao, YANG Wan-shun
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 35-44. https://doi.org/10.1007/s10255-024-1101-z
    An acyclic edge coloring of a graph $G$ is a proper edge coloring such that there are no bichromatic cycles in $G$. The acyclic chromatic index $\chi'_a(G)$ of $G$ is the smallest $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that every simple graph $G$ with maximum degree $\Delta$ has $\chi'_a(G)\le \Delta+2$. A 1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every 1-planar graph $G$ without $4$-cycles has $\chi'_a(G)\le \Delta+22$.
  • ARTICLES
    Nai-dan DENG, Chun-wei WANG, Jia-en XU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 109-128. https://doi.org/10.1007/s10255-024-1102-y
    In this paper, the insurance company considers venture capital and risk-free investment in a constant proportion. The surplus process is perturbed by diffusion. At first, the integro-differential equations satisfied by the expected discounted dividend payments and the Gerber-Shiu function are derived. Then, the approximate solutions of the integro-differential equations are obtained through the sinc method. Finally, the numerical examples are given when the claim sizes follow different distributions. Furthermore, the errors between the explicit solution and the numerical solution are discussed in a special case.
  • ARTICLES
    Aihemaitijiang YUMAIER, Ehmet KASIM
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 665-694. https://doi.org/10.1007/s10255-023-1079-y
    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.
  • ARTICLES
    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
    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.
  • ARTICLES
    Jia-min ZHU, Bo-jun YUAN, Yi WANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 129-136. https://doi.org/10.1007/s10255-024-1103-x
    Let $G$ be a simple graph and $G^{\sigma}$ be the oriented graph with $G$ as its underlying graph and orientation $\sigma$. The rank of the adjacency matrix of $G$ is called the rank of $G$ and is denoted by $r(G)$. The rank of the skew-adjacency matrix of $G^{\sigma}$ is called the skew-rank of $G^{\sigma}$ and is denoted by $sr(G^{\sigma})$. Let $V(G)$ be the vertex set and $E(G)$ be the edge set of $G$. The cyclomatic number of $G$, denoted by $c(G)$, is equal to $|E(G)|-|V(G)|+\omega(G)$, where $\omega(G)$ is the number of the components of $G$. It is proved for any oriented graph $G^{\sigma}$ that $-2c(G)\leqslant sr(G^{\sigma})-r(G)\leqslant2c(G)$. In this paper, we prove that there is no oriented graph $G^{\sigma}$ with $sr(G^{\sigma})-r(G)=2c(G)-1$, and in addition, there are infinitely many oriented graphs $G^{\sigma}$ with connected underlying graphs such that $c(G)=k$ and $sr(G^{\sigma})-r(G)=2c(G)-\ell$ for every integers $k, \ell$ satisfying $0\leqslant\ell\leqslant4k$ and $\ell\neq1$.
  • ARTICLES
    Wen-qing XU, Sha-sha WANG, Da-chuan XU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 91-108. https://doi.org/10.1007/s10255-024-1115-6
    The classical Archimedean approximation of $\pi$ uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in $\mathbb{R}^2 $ and it is well-known that by using linear combinations of these basic estimates, modern extrapolation techniques can greatly speed up the approximation process. % reduce the associated approximation errors. Similarly, when $n$ vertices are randomly selected on the circle, the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to $\pi$ almost surely as $ n \to \infty $, and by further applying extrapolation processes, faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons. In this paper, we further develop nonlinear extrapolation methods for approximating $\pi$ through certain nonlinear functions of the semiperimeter and area of such polygons. We focus on two types of extrapolation estimates of the forms $ \mathcal{X}_n = \mathcal{S}_n^{\alpha} \mathcal{A}_n^{\beta} $ and $ \mathcal{Y}_n (p) = \left( \alpha \mathcal{S}_n^p + \beta \mathcal{A}_n^p \right)^{1/p} $ where $ \alpha + \beta = 1 $, $ p \neq 0 $, and $ \mathcal{S}_n $ and $ \mathcal{A}_n $ respectively represents the semiperimeter and area of a random $n$-gon inscribed in the unit circle in $ \mathbb{R}^2 $, and $ \mathcal{X}_n $ may be viewed as the limit of $ \mathcal{Y}_n (p) $ when $ p \to 0 $. By deriving probabilistic asymptotic expansions with carefully controlled error estimates for $ \mathcal{X}_n $ and $ \mathcal{Y}_n (p) $, we show that the choice $ \alpha = 4/3 $, $ \beta = -1/3 $ minimizes the approximation error in both cases, and their distributions are also asymptotically normal.
  • ARTICLES
    Shao-qiang LIU, Yue-jian PENG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 347-357. https://doi.org/10.1007/s10255-024-1118-3
    For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\cdots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N, N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $1\leq i\leq r$. We show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum\limits^{j+1}_{i=1}\alpha_i$ for $j=1,2,\cdots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\cdots,C_{2\lfloor \alpha_r n\rfloor})=\big(\sum\limits^r_{j=1} \alpha_j+o(1)\big)n.$
  • ARTICLES
    En-wen ZHU, Zi-wei DENG, Han-jun ZHANG, Jun CAO, Xiao-hui LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 320-346. https://doi.org/10.1007/s10255-024-1072-0
    This paper considers the random coefficient autoregressive model with time-functional variance noises, hereafter the RCA-TFV model. We first establish the consistency and asymptotic normality of the conditional least squares estimator for the constant coefficient. The semiparametric least squares estimator for the variance of the random coefficient and the nonparametric estimator for the variance function are constructed, and their asymptotic results are reported. A simulation study is presented along with an analysis of real data to assess the performance of our method in finite samples.
  • ARTICLES
    Ming-hua YANG, Si-ming HUANG, Jin-yi SUN
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 241-268. https://doi.org/10.1007/s10255-024-1119-2
    In this paper, we study a global zero-relaxation limit problem of the electro-diffusion model arising in electro-hydrodynamics which is the coupled Planck-Nernst-Poisson and Navier-Stokes equations. That is, the paper deals with a singular limit problem of \begin{eqnarray*}\label{1.2} \left\{ \begin{array}{ll} u_t^{\epsilon}+u^{\epsilon}\cdot\nabla u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}=\Delta \phi^{\epsilon}\nabla\phi^{\epsilon}, \ \ \ &{\rm in}\ \mathbb{R}^{3}\times(0, \infty), \\[5pt] \nabla\cdot u^{\epsilon}=0, \ \ \ &{\rm in}\ \mathbb{R}^{3}\times(0, \infty), \\[5pt] n_t^{\epsilon}+u^{\epsilon}\cdot\nabla n^{\epsilon}-\Delta n^{\epsilon}=-\nabla\cdot(n^{\epsilon}\nabla \phi^{\epsilon}), &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] c_t^{\epsilon}+u^{\epsilon}\cdot\nabla c^{\epsilon}-\Delta c^{\epsilon}=\nabla\cdot(c^{\epsilon}\nabla\phi^{\epsilon}), &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] \epsilon^{-1} \phi^{\epsilon}_t= \Delta \phi^{\epsilon}- n^{\epsilon}+ c^{\epsilon}, &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] (u^{\epsilon}, n^{\epsilon}, c^{\epsilon},\phi^{\epsilon})|_{t=0}= (u_{0}, n_{0}, c_{0},\phi_{0}), &{\rm in}\ \mathbb{R}^{3} \end{array} \right. \end{eqnarray*} involving with a positive, large parameter $\epsilon$. The present work show a case that $(u^{\epsilon}, n^{\epsilon}, c^{\epsilon})$ stabilizes to $(u^{\infty}, n^{\infty}, c^{\infty}):=(u, n, c)$ uniformly with respect to the time variable as $\epsilon\rightarrow+\infty$ with respect to the strong topology in a certain Fourier-Herz space.
  • ARTICLES
    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
    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.
  • ARTICLES
    Dong-Jie WU, Xin-Jian XU, Chuan-Fu YANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 568-576. https://doi.org/10.1007/s10255-024-1042-6
    The classical Ambarzumyan's theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator $-\frac{d^2}{dx^2}+q$ with an integrable real-valued potential $q$ on $[0,\pi]$ are $\{n^2:n\geq 0\}$, then $q=0$ for almost all $x\in [0,\pi]$. In this work, the classical Ambarzumyan's theorem is extended to the Dirac operator on equilateral tree graphs. We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case, then the potential is identically zero.
  • ARTICLES
    Zhi-min REN, Yong-yi LAN
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 414-429. https://doi.org/10.1007/s10255-024-1120-9
    In this paper, we investigate the following $p$-Kirchhoff equation \begin{eqnarray*} \left\{\begin{array}{ll} \big(a+b\int_{\mathbb{R}^{N}}(|\nabla u|^{p}+|u|^{p})dx\big)\big(-\Delta_{p}u+|u|^{p-2}u\big)=|u|^{s-2}u+\mu u,~x\in \mathbb{R}^{N},\\ \int_{\mathbb{R}^{N}}|u|^{2}dx=\rho, \end{array} \right. \end{eqnarray*} where $a> 0, \,b \geq 0 , \,\rho>0$ are constants, $p^{\ast}=\frac{Np}{N-p}$ is the critical Sobolev exponent, $\mu$ is a Lagrange multiplier, $-\Delta_{p}u=-{\rm div}(|\nabla u|^{p-2}\nabla u), \ 2<p<N<2p, \ \mu\in\mathbb{R}$, and $ s\in(2\frac{N+2}{N}p-2,~p^{\ast})$. We demonstrate that the $p$-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.
  • ARTICLES
    Jian CAO, Yong-jiang GUO, Kai-ming YANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 445-466. https://doi.org/10.1007/s10255-024-1089-4
    The law of the iterated logarithm (LIL) for the performance measures of a two-station queueing network with arrivals modulated by independent queues is developed by a strong approximation method. For convenience, two arrival processes modulated by queues comprise the external system, all others are belong to the internal system. It is well known that the exogenous arrival has a great influence on the asymptotic variability of performance measures in queues. For the considered queueing network in heavy traffic, we get all the LILs for the queue length, workload, busy time, idle time and departure processes, and present them by some simple functions of the primitive data. The LILs tell us some interesting insights, such as, the LILs of busy and idle times are zero and they reflect a small variability around their fluid approximations, the LIL of departure has nothing to do with the arrival process, both of the two phenomena well explain the service station's situation of being busy all the time. The external system shows us a distinguishing effect on the performance measures: an underloaded (overloaded, critically loaded) external system affects the internal system through its arrival (departure, arrival and departure together). In addition, we also get the strong approximation of the network as an auxiliary result.
  • ARTICLES
    Qiang WEN, Guo-qiang REN, Bin LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 164-191. https://doi.org/10.1007/s10255-024-1107-6
    In this paper, we consider a susceptible-infective-susceptible (SIS) reaction-diffusion epidemic model with spontaneous infection and logistic source in a periodically evolving domain.~Using the iterative technique, the uniform boundedness of solution is established.~In addition, the spatial-temporal risk index $\mathcal{R}_0(\rho)$ depending on the domain evolution rate $\rho(t)$ as well as its analytical properties are discussed.~The monotonicity of $\mathcal{R}_0(\rho)$ with respect to the diffusion coefficients of the infected $d_I$, the spontaneous infection rate $\eta(\rho(t)y)$ and interval length $L$ is investigated under appropriate conditions.~Further, the existence and asymptotic behavior of periodic endemic equilibria are explored by upper and lower solution method.~Finally, some numerical simulations are presented to illustrate our analytical results.~Our results provide valuable information for disease control and prevention.
  • ARTICLES
    Pei-yu ZHANG, Li FANG, Zhen-hua GUO
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(4): 954-978. https://doi.org/10.1007/s10255-024-1080-0
    The purpose of this work is to investigate the existence and uniqueness of weak solutions to the initial-boundary value problem for a coupled system of an incompressible non-Newtonian fluid and the Vlasov equation. The coupling arises from the acceleration in the Vlasov equation and the drag force in the incompressible viscous non-Newtonian fluid with the stress tensor of a power-law structure for $p\geqslant\frac{11}{5}$. The main idea of the existence analysis is to reformulate the coupled system by means of a so-called truncation function. The advantage of the new formulation is to control the external force term $G=-\mathbf{\int}_{\mathbb{R}^d}(\mathbf{u}-\mathbf{v})fd\mathbf{v}~(d=2,3)$. The global existence of weak solutions to the reformulated system is shown by using the Faedo-Galerkin method and weak compactness techniques. We further prove the uniqueness of weak solutions to the considered system.
  • ARTICLES
    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
    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.
  • ARTICLES
    Jia-qi YANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 205-210. https://doi.org/10.1007/s10255-024-1114-7
    We consider the relation between the direction of the vorticity and the global regularity of 3D shear thickening fluids. It is showed that a weak solution to the non-Newtonian incompressible fluid in the whole space is strong if the direction of the vorticity is 11-5p/2-Hölder continuous with respect to the space variables when 2

    <11/5.

  • ARTICLES
    Wen WANG, Da-peng XIE, Hui ZHOU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 539-546. https://doi.org/10.1007/s10255-024-1041-7
    In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation $$ u_{t}(x,t)=\Delta u(x,t) +au(x,t)\ln u(x,t)+ bu^{\alpha}(x,t), $$ on $\mathbf{M}\times (-\infty, \infty)$ with $\alpha\in\mathbf{R}$, where $a$ and $b$ are constants. As application, the Harnack inequalities are derived.
  • ARTICLES
    Cong-hua CHENG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 478-507. https://doi.org/10.1007/s10255-024-1044-4
    In this paper, we consider a system which has $k$ statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements with doubly type-II censored scheme. These elements $(X_1, Y_1 ),$ $(X_2, Y_2 ),$ $\cdots$, $(X_k, Y_k)$ follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress $T$ which follows a Kumaraswamy distribution. The system is regarded as operating only if at least $s$ out of $k$ ($1\leq s\leq k$) strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}$=$P$(at least $s$ of the $(Z_1, \cdots, Z_k)$ exceed $T)$ where $Z_i=\min(X_i, Y_i ), \ i=1,\cdots, k.$ The Bayes estimates of $R_{s,k}$ have been developed by using the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and exact Bayes estimates of $R_{s,k}$ are obtained analytically when the common second shape parameter is known. The asymptotic confidence interval and the highest probability density credible interval are constructed for $R_{s,k}$. The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.
  • ARTICLES
    Li-li LIU, Hong-gang WANG, Ya-zhi LI
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 508-525. https://doi.org/10.1007/s10255-024-1049-z
    Considering that HBV belongs to the DNA virus family and is hepatotropic, we model the HBV DNA-containing capsids as a compartment. In this paper, a delayed HBV infection model is established, where the general incidence function and two infection routes including cell-virus infection and cell-cell infection are introduced. According to some preliminaries, including well-posedness, basic reproduction number and existence of two equilibria, we obtain the threshold dynamics for the model. We illustrate numerical simulations to verify the above theoretical results, and furthermore explore the impacts of intracellular delay and cell-cell infection on the global dynamics of the model.
  • ARTICLES
    De-jian TIAN, Shang-ri WU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 430-444. https://doi.org/10.1007/s10255-024-1045-3
    This article analyzes the Pareto optimal allocations, agreeable trades and agreeable bets under the maxmin Choquet expected utility (MCEU) model. We provide several useful characterizations for Pareto optimal allocations for risk averse agents. We derive the formulation descriptions for non-existence agreeable trades or agreeable bets for risk neutral agents. We build some relationships between ex-ante stage and interim stage on agreeable trades or bets when new information arrives.
  • ARTICLES
    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
    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.
  • ARTICLES
    Peng LI, Ming ZHOU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(3): 744-757. https://doi.org/10.1007/s10255-024-1048-0
    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.
  • ARTICLES
    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
    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.
  • ARTICLES
    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
    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.
  • ARTICLES
    Dong-han ZHANG, You LU, Sheng-gui ZHANG, Li ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(1): 211-224. https://doi.org/10.1007/s10255-024-1110-y
    A neighbor sum distinguishing (NSD) total coloring $\phi$ of $G$ is a proper total coloring of $G$ such that $\sum\limits_{z\in E_{G}(u)\cup\{u\}}\phi(z)\neq\sum\limits_{z\in E_{G}(v)\cup\{v\}}\phi(z)$ for each edge $uv\in E(G)$, where $E_{G}(u)$ is the set of edges incident with a vertex $u$. In 2015, Pilśniak and Woźniak conjectured that every graph with maximum degree $\Delta$ has an NSD total $(\Delta+3)$-coloring. Recently, Yang et al. proved that the conjecture holds for planar graphs with $\Delta\ge 10$, and Qu et al. proved that the list version of the conjecture also holds for planar graphs with $\Delta\ge 13$. In this paper, we improve their results and prove that the list version of the conjecture holds for planar graphs with $\Delta\ge 10$.