中国科学院数学与系统科学研究院期刊网

15 April 2024, Volume 40 Issue 2
    

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  • 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
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    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.
  • 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
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    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.
  • 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
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    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.
  • 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
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    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.
  • 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
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    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 $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\cdots,C_{2\lfloor \alpha_r n\rfloor})=\big(\sum\limits^r_{j=1} \alpha_j+o(1)\big)n.$
  • Jin-yan ZHU, Yong CHEN
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 358-378. https://doi.org/10.1007/s10255-024-1109-4
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    The Gerdjikov-Ivanov (GI) hierarchy is derived via recursion operator, in this article, we mainly investigate the third-order flow GI equation. In the framework of the Riemann-Hilbert method, the soliton matrices of the third-order flow GI equation with simple zeros and elementary high-order zeros of Riemann-Hilbert problem are constructed through the standard dressing process. Taking advantage of this result, some properties and asymptotic analysis of single soliton solution and two soliton solution are discussed, and the simple elastic interaction of two soliton are proved. Compared with soliton solution of the classical second-order flow, we find that the higher-order dispersion term affects the propagation velocity, propagation direction and amplitude of the soliton. Finally, by means of a certain limit technique, the high-order soliton solution matrix for the third-order flow GI equation is derived.
  • Tao HAO
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 379-413. https://doi.org/10.1007/s10255-024-1112-9
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    This paper concerns a global optimality principle for fully coupled mean-field control systems. Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear relation is introduced, with which we successfully decouple the fully coupled first-order variational equations. We give a new second-order expansion of $Y^\varepsilon$ that can work well in mean-field framework. Based on this result, the stochastic maximum principle is proved. The comparison with the stochastic maximum principle for controlled mean-field stochastic differential equations is supplied.
  • 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
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    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.
  • 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
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    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.
  • 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
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    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.
  • Ke-Jie LI, Xin ZHANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 467-477. https://doi.org/10.1007/s10255-024-1026-6
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    The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color. An outerplane graph with independent crossings is a graph embedded in the plane in such a way that all vertices are on the outer face and two pairs of crossing edges share no common end vertex. It is proved that every outerplane graph with independent crossings and maximum degree $\Delta$ has strong chromatic index at most $4\Delta-6$ if $\Delta\geq 4$, and at most 8 if $\Delta\leq 3$. Both bounds are sharp.
  • Cong-hua CHENG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 478-507. https://doi.org/10.1007/s10255-024-1044-4
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    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.
  • 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
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    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.
  • Hui-qing LIU, Rui-ting ZHANG, Xiao-lan HU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 526-538. https://doi.org/10.1007/s10255-024-1113-8
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    Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number $b(G)$ of a graph $G$, is defined as the smallest integer $k$ for which there are vertices $x_1,\cdots, x_k$ such that for every vertex $u$ of $G$, there exists $i\in \{1,\cdots, k\}$ with $d_G(u, x_i)\le k-i$, and $d_G(x_i, x_j) \ge j-i$ for any $1\le i<j \le k$. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.
  • 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
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    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.
  • Fang DUAN, Qiong-xiang HUANG
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 547-556. https://doi.org/10.1007/s10255-024-1023-9
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    Let $G$ be a finite group generated by $S$ and $C(G,S)$ the Cayley digraphs of $G$ with connection set $S$. In this paper, we give some sufficient conditions for the existence of hamiltonian circuit in $C(G,S)$, where $G=Z_m\rtimes H$ is a semiproduct of $Z_m$ by a subgroup $H$ of $G$. In particular, if $m$ is a prime, then the Cayley digraph of $G$ has a hamiltonian circuit unless $G=Z_m\times H$. In addition, we introduce a new digraph operation, called $\varphi$-semiproduct of $\Gamma_1$ by $\Gamma_2$ and denoted by $\Gamma_1 \rtimes_\varphi\Gamma_2$, in terms of mapping $\varphi:V(\Gamma_2)\longrightarrow\{1,-1\}$. Furthermore we prove that $C(Z_m, \{a\}) \rtimes_{\varphi}C(H,S)$ is also a Cayley digraph if $\varphi$ is a homomorphism from $H$ to $\{1,-1\}\le Z_m^*$, which produces some classes of Cayley digraphs that have hamiltonian circuits.
  • Jin LIANG, Jia-qi MAO, Zhao-ya LIU
    Acta Mathematicae Applicatae Sinica(English Series). 2024, 40(2): 557-567. https://doi.org/10.1007/s10255-024-1122-7
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    Aiming at the problem that the asset's fluctuation influences the borrower's repayment ability, a loan with a new and flexible repayment method is designed, which depends on the asset value of the borrower. The repayment method can reduce the loan default probability, but it causes the uncertainty of the pay off time. Because the repayment term is related to the regular repayment amount in this method, a boundary for the regular repayment amount is set up in order to avoid too long repayment term. This will balance the benefit of borrowers and lenders and improve the applicability of this method. By establishing a mathematical model of the residual value of the loan, this model can be transformed into an initial-boundary problem of a partial differential equation. The analytic solution and the expected time to pay off the loan are obtained. Finally, numerical analysis are shown.
  • 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
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    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.