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2022, Vol. 42 No. 2   Published date: 28 June 2022
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• Multiplicity of P-cyclic Symmetric Closed Characteristics on Compact Star-shaped P-cyclic Symmetric Hypersurfaces in R²ⁿ

  Li Yangyang , Liu Hui

  2022, 42(2): 1-11. doi: 10.3969/j.issn.1006-­8074.2022.02.001

  Abstract ( 2402 )   PDF (181KB) ( 435 )   　　

  Let $k\geq2$ be a positive integer and

  $P={\rm e}^{\frac{2}{k}\pi J}$, where  J =\left(

  \begin{array}{cc}

  0 & -I_n \\

  I_n & 0 \\

  \end{array}

  \right)

  is the standard symplectic matrix. In this paper we prove that for any $P$-cyclic symmetric compact star-shaped hypersurface $\Sigma$ in $\mathbb{R}^{2n}$ with $n\geq 2$, there exists at least one $P$-cyclic symmetric closed characteristic on $\Sigma$, and moreover, give a sufficient condition on the star-shaped hypersurface for the existence of $n$ geometrically distinct $P$-cyclic symmetric closed characteristics.

  
  

  
  

• Finite­time Flocking Analysis of a Class of Complex Coupling Systems

  Wang Chuxiang, Liu Yicheng, Ru Lining

  2022, 42(2): 12-24. doi: 10.3969/j.issn.1006­-8074.2022.02.002

  Abstract ( 1946 )   PDF (1348KB) ( 612 )   　　

  This paper presents a new particle dynamic system based on the C-S model, studies the sufficient condition of flocking in finite time, and analyzes the factors of convergence time. We show that when the initial values of agents’ position satisfy some specific conditions, finite-time flocking occurs. Furthermore, the convergence time can be estimated by the agents’ number, that is, there is a power-law relationship between them. When the number of agents is large, the convergence time will decrease.

• Subharmonic Bifurcations and Chaos of the Buckled Beam Subjected to Parametrical Excitations

  Zhang Dongmei, Li Feng

  2022, 42(2): 25-34. doi: 10.3969/j.issn.1006-­8074.2022.02.003

  Abstract ( 1893 )   PDF (348KB) ( 513 )   　　

  This paper presents a new particle dynamic system based on the C-S model, studies the sufficient condition of flocking in finite time, and analyzes the factors of convergence time. We show that when the initial values of the agents’ position satisfy some specific conditions, finite-time flocking occurs. Furthermore, the convergence time can be estimated by the agents’ number, that is, there is a power-law relationship between them. When the number of the agents is large, the convergence time will decrease.

• Prediction of Early Summer Rainstorm Days in South China Based on Least Angle Regression on High Correlation Regions

  2022, 42(2): 35-46. doi: 10.3969/j.issn.1006-­8074.2022.02.004

  Abstract ( 1830 )   PDF (467KB) ( 662 )   　　

  In this paper, we study the relationship between the rainstorm days in South China and the early atmospheric circulation factors as well as the external forcing factors, and based on it , build a model to forecast the rainstorm days in the area. Firstly, we select the high correlation regions to construct the prediction characteristics for variable dimension reduction. Then, a least angle regression model is built to forecast the rainstorm days in early summer in South China. Compared to other models in terms of time anomaly correlation coefficient ($TCC$), rate of the same sign ($SS$), coefficient of determination ($CD$), and adjusted prospect anomaly synthetical score ($APs$), it is obtained that the prediction results of the minimum angle regression model based on the high correlation region have a strong time correlation with the observed values and the high $APs$ score, which shows that the method constructed in this paper has a strong practical value.

• Asymptotic Behavior of the Multiscale Stochastic Systems

  Li Nannan , Xie Longjie

  2022, 42(2): 47-60. doi: 10.3969/j.issn.1006-­8074.2022.02.005

  Abstract ( 1918 )   PDF (241KB) ( 997 )   　　

  This paper summarizes the recent progress on the limiting behavior of multiscale stochastic systems with irregular coefficients, focuses on presenting the averaging principle, the normal deviation and the diffusion approximation, in particular, the Smoluchowski-Kramers approximations for the classical stochastic Langevin equation driven by Brownian noise and the Langevin equation driven by L\'evy noise. Unlike the classical equation driven by Brownian noise, there is no noise induced drift in the limit equation of the Langevin euqation driven by L\'evy noise even if the friction is state dependent.

• Free Cocycle Infinitesimal Unitary Hopf Algebra on Decorated Rooted Forests

  Zhang Yi, Cao Jinghan, Wu Xiaoxing

  2022, 42(2): 61-75. doi: 10.3969/j.issn.1006­-8074.2022.02.006

  Abstract ( 1813 )   PDF (181KB) ( 215 )   　　

  In this paper, we equip an infinitesimal bialgebra on rooted forests with an antipode such that it is further an infinitesimal unitary Hopf algebra. Viewing this infinitesimal unitary Hopf algebra in the framework of operated algebras, we then propose the concept of cocycle infinitesimal unitary Hopf algebra, which involves an infinitesimal Hochschild 1-cocycle condition. Finally, we prove that the space of decorated planar rooted forests together with a family of grafting operations is the free cocycle infinitesimal unitary Hopf algebra on a set.

• A compact difference scheme for the time fractional integro-differential equation with Neumann boundary conditions

  Tang Sheng , Mo Yan , Wang Zhibo

  2022, 42(2): 76-89. doi: 10.3969/j.issn.1006-­8074.2022.02.007

  Abstract ( 1835 )   PDF (185KB) ( 639 )   　　

   In this paper, we study the numerical method for a time fractional integro-differential equation with Neumann boundary conditions. A difference scheme combining the $L1$ approximation for the Caputo fractional derivative, the weighted and shifted Gr\"{u}nwald difference formula for the Riemann-Liouville fractional integral and the compact difference approach for the spatial second order derivative is proposed and analyzed. Although the spatial approximation order at Neumann boundary is one order lower than that for interior mesh points, the unconditional stability and global convergence with O$(\tau^{2-\alpha}+h^4)$ were proved rigorously. Finally, numerical experiments are carried out to support the theoretical results.

• The Generalized 3­connectivity of Cayley Graphs Generated by Unicyclic Graphs

  Wang Yanna, Zhou Bo

  2022, 42(2): 90-98. doi: 10.3969/j.issn.1006­-8074.2022.02.008

  Abstract ( 1721 )   PDF (149KB) ( 735 )   　　

  Let $\mbox{Sym}(n)$ be the symmetric group on $\{1,\dots,n\}$ and $\mathcal{T}$ be a set of some transpositions of $\mbox{Sym}(n)$. Let $G(\mathcal{T})$ be the graph with vertex set $\{1,\dots,n\}$ such that there is an edge $ij$ in $G(\mathcal{T})$ if and only if the transposition $[i,j]\in\mathcal{T}$. In this paper we show that, for $n\geq 4$, the generalized $3$-connectivity of the Cayley graph on $\mbox{Sym}(n)$ generated by $\mathcal{T}$ is $n-1$ if $G(\mathcal{T})$ is any unicyclic graph.

• Research on the SIR Model of Two New Coronaviruses with Time Delay in Parallel Propagation 

  Jin Wei, Liao Xinyuan, She Zhifeng

  2022, 42(2): 99-107. doi: 10.3969/j.issn.1006-­8074.2022.02.009

  Abstract ( 2022 )   PDF (278KB) ( 487 )   　　

  In this paper a SIR model for the parallel propagation of two new coronaviruses with time delay is studied. Firstly, the existence and uniqueness of the global positive solution of the system is proved. Secondly, the sufficient conditions for persistence and extinction of the new coronavirus and the variant new coronavirus are obtained by constructing a appropriate Lyapunov function. Finally, the relevant conclusions are verified by numerical simulations.

  
  

• An Improved Elastic Net Estimate for Logistic Regression Models

  2022, 42(2): 108-119. doi: 10.3969/j.issn.1006­-8074.2022.02.010

  Abstract ( 2478 )   PDF (245KB) ( 990 )   　　

  For improving the application performance of Logistic regression models on classification problems, this paper develops a double adaptive elastic net by combining the adaptive Lasso and adaptive Ridge. The double adaptive elastic net has both the oracle property and the adaptive grouping effect, which ensures that it can effectively estimate parameters and accurately select important variables under certain assumed premises and consequently, makes the established Logistic regression model simple and precise. Simulation and case analysis show that the double adaptive elastic net is suitable for medium or high correlation cases with adaptive grouping effect, and its performance of improving Logistic regression is equal to or better than that of the elastic net and other partial improvement methods.

• Extrapolated CSCS Iterations of Large Continuous Sylvester Equations

  Liu Zhongyun, Zhang Fang

  2022, 42(2): 120-126. doi: 10.3969/j.issn.1006-8074.2022.02.011

  Abstract ( 2097 )   PDF (180KB) ( 312 )   　　

  In reference [1], a circulant and skew-circulant splitting iterative method (CSCS iteration) for solving the continuous Sylvester equation $AX+XB=E$ is proposed , where the coefficients $A$ and $B$ are both positive definite Toeplitz matrices. In order to improve the convergence speed of this method, we propose an extrapolated CSCS iteration, discuss its convergence and show its effectiveness by numerical experiments.