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2024, Vol. 44 No. 4   Published date: 28 December 2024
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• (m,n)-coherent Rings and FP_(m,n)-projective Modules

  2024, 44(4): 1-18. doi: 10.3969/j.issn.1006-8074.2024.04.001

  Abstract ( 37 )   PDF (224KB) ( 22 )   　　

  In this paper, we introduce the notions of $(m,n)$-coherent rings and $FP_{(m,n)}$-projective modules for nonnegative integers $m,n$. We prove that ($\mathcal{FP}_{(m,n)}$-Proj, ($\mathcal{FP}_{n}$-id)$_{\leq m}$) is a complete cotorsion pair for any $m,n\geq 0$ and it is hereditary if and only if the ring $R$ is a left $n$-coherent ring for all $m\geq 0$ and $n\geq 1$. Moreover, we study the existence of $\mathcal{FP}_{(m,n)}$-Proj covers and envelopes and obtain that if $\mathcal{FP}_{(m,n)}$-Proj is closed under pure quotients, then $\mathcal{FP}_{(m,n)}$-Proj is covering for any $n\geq2$. As applications, we obtain that every $R$-module has an epic $\mathcal{FP}_{(m,n)}$-Proj-envelope if and only if the left $FP_{(m,n)}$-global dimension of $R$ is at most 1 and $\mathcal{FP}_{(m,n)}$-Proj is closed under direct products.

• Block-transitive  3-designs Associated with Alternating Groups

  Gan Yunsong, Liu Weijun

  2024, 44(4): 19-30. doi: 10.3969/j.issn.1006-8074.2024.04.002

  Abstract ( 39 )   PDF (185KB) ( 20 )   　　

  A~$t$-$(v,k,\lambda)$ design is said to be $G$-point-primitive or $G$-block-transitive, if its automorphism group $G$ acts primitively on the point set or transitively on the block set, respectively. In this paper we begin by extending some results on block-transitive Steiner $2$-designs to block-transitive $3$-designs, and then based on these results, investigate the $G$-point-primitive block-transitive $3$-$(v,k,\lambda)$ designs for alternating or symmetric groups $G$. We prove that when $n\geq\min\{\lambda^2, 30\}$ the point stabilizer in $G$ must be of intransitive type, and specifically, when $n\geq30$ there exists no nontrivial $G$-point-primitive block-transitive $3$-$(v,k,2)$ design.

• Centers and Limit Cycles for a Class of Three-dimensional Cubic Kukles Systems

  2024, 44(4): 31-44. doi: 10.3969/j.issn.1006-8074.2024.04.003

  Abstract ( 35 )   PDF (190KB) ( 21 )   　　

  In this paper the centers and limit cycles for a class of three-dimensional

  cubic Kukles systems are investigated. First, by calculating and analyzing the common zeros of the first

  ten singular point quantities, the necessary conditions for the origin being a center on

  the center manifold are derived, and furthermore, the sufficiency of those conditions is proved using the Darboux

  integrating method. Then, by calculating and analyzing the common zeros of the first three period

  constants, the necessary and sufficient conditions for the origin being an isochronous

  center on the center manifold are given. Finally, by proving the linear independence of

  the first ten singular point quantities, it is demonstrated that the system can bifurcate ten

  small-amplitude limit cycles near the origin under a suitable perturbation, which is a new lower bound for the number of limit cycles around a weak focus in a

  three-dimensional cubic system.

• Bounded and Unbounded Solutions of the SD Oscillator at Resonance

  Bian Jingke, Liu Jie

  2024, 44(4): 45-69. doi: 10.3969/j.issn.1006-8074.2024.04.004

  Abstract ( 38 )   PDF (242KB) ( 17 )   　　

  In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous (SD) oscillator

  \begin{equation*}

  x''+f(x)x'+x-\frac{x}{\sqrt{x^{2}+\alpha^{2}}}=p(t).

  \end{equation*}

  Since $f(x)\neq 0$, the system is non-Hamiltonian, so we have to introduce some reversibility assumptions to apply a suitable twist theorem, for reversible maps with small twist. Moreover, when the nonnegative parameter $\alpha$ decreases to 0, the system becomes discontinuous. In this case, we need to introduce some suitable transformations to overcome the lack of regularity. We will prove that for any nonnegative parameter $\alpha$, when $p(t)$ is an odd periodic function satisfying $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|<4$, all the solutions are bounded; when $p(t)$ satisfies $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|>4$, the SD oscillator has unbounded solutions, and when $p(t)$ satisfies $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|\geqslant 4+|F|_{\infty}$, all the solutions are unbounded.

• Global Stability and Bifurcation Analysis of a Cholera Transmission Model

  Liu Qiumei , Liu Lingling, Xu Fang

  2024, 44(4): 70-87. doi: 10.3969/j.issn.1006-8074.2024.04.005

  Abstract ( 38 )   PDF (1270KB) ( 16 )   　　

  This paper investigates the stability and bifurcation phenomena of a cholera transmission model in which individuals who have recovered from the disease may become susceptible again. The threshold for determining disease prevalence is established, and the parameter conditions for

  the existence of equilibria are discussed. The Routh-Hurwitz criterion is applied to demonstrate the local asymptotic stability of equilibria.

  By utilizing composite matrices and geometric techniques, the global dynamic behavior of the endemic equilibrium is investigated, and the sufficient conditions for its global asymptotic stability are derived. Furthermore, the disease-free equilibrium is a saddle-node when the basic reproductive number is 1, and tthe transcritical bifurcation in this case is discussed.

• Persistence of a Stochastic HPV Epidemic Model with Markov Switching

  Li Jiyuan, Qiu Hong, Ju Xuewei

  2024, 44(4): 88-99. doi: 10.3969/j.issn.1006-8074.2024.04.006

  Abstract ( 34 )   PDF (287KB) ( 15 )   　　

  In order to study the influence of stochastic disturbance and environment switching on the HPV infection and provide a theoretical basis for the development of effective HPV disease prevention measures, in this paper we establish a kind of two-sex stochastic HPV epidemic model with white noise and Markov switching. We show that the model has a unique local positive solution and a unique global positive solution. Then we identify the threshold conditions for the persistence of the HPV epidemic, and verify the persistence of the disease using the Lyapunov method and the It$\hat{\rm o}$ formula. At last, the numerical simulation is carried out to illustrate the rationality of the theoretical results.

• A Sparse Optimal Scoring Model with Adherent Penalty

  Hou Dandan, Liu Yongjin

  2024, 44(4): 100-115. doi: 10.3969/j.issn.1006-8074.2024.04.007

  Abstract ( 43 )   PDF (311KB) ( 22 )   　　

  We consider the task of binary classification in the high-dimensional setting where the number of features of the given data is larger than the number of observations. To accomplish this task, we propose an adherently penalized optimal scoring (APOS) model for simultaneously performing discriminant analysis and feature selection. In this paper, an efficient algorithm based on the block coordinate descent (BCD) method and the SSNAL algorithm is developed to solve the APOS approximately. The convergence results of our method are also established. Numerical experiments conducted on simulated and real datasets demonstrate that the proposed model is more efficient than several sparse discriminant analysis methods.

• Estimations for Determinats of Dashnic-Zusmanovich Type Matrices

  Liu Dan, Wang Shiyun, Zhou Lixin, Lyu Zhenhua

  2024, 44(4): 116-126. doi: 10.3969/j.issn.1006-8074.2024.04.008

  Abstract ( 38 )   PDF (164KB) ( 14 )   　　

  To estimate the determinant of a matrix is a hot issue in numerical algebra. The class of $H$-matrices has a wide range of applications in many fields such as computational mathematics, control theory. In this paper, the estimations for determinants of Dashnic-Zusmanovich type matrices are studied. For a Dashnic-Zusmanovich type matrix, we transform it into a diagonally dominant matrix by using scaling matrices. Then we obtain the upper and lower bounds for the determinant of the Dashnic-Zusmanovich type matrix. Numerical examples are given to illustrate the effectiveness of the proposed results.