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2023, Vol. 43 No. 1   Published date: 28 March 2023
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• Study of Period Functions

  Li Chengzhi

  2023, 43(1): 1-31. doi: 10.3969/j.issn.1006-8074.2023.01.001

  Abstract ( 2042 )   PDF (485KB) ( 575 )   　　

  In this survey article we first briefly introduce some concepts related to the period function of a planar smooth (or analytic) vector field, and its isochronicity, monotonicity, and the number of critical periods. Then, we introduce some important results in this field, especially about the isochronous centers, the period functions associated to the elliptic and hyperelliptic Hamiltonian functions, and the period functions of quadratic integrable systems. Besides these results we list some conjectures and problems in Section 6, which may provide topics for further studies.

• Gradient Estimates and Liouville Theorems for $\Delta u + au^{p+1}=0$ 

  Peng Bo, Wang Youde, Wei Guodong

  2023, 43(1): 32-43. doi: 10.3969/j.issn.1006-8074.2023.01.002

  Abstract ( 1516 )   PDF (177KB) ( 329 )   　　

  In this paper, we employ Li-Yau's method and delicate analysis techniques to provide a unified and simple approach to the gradient estimate of the positive solution to the nonlinear elliptic equation $\Delta u + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p < 4/n $ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, we extend the range of $p$ and improve some results in \cite{J-L, MHL} and supplement the results for the case $\dim(M)= 2$. For the case $a<0$ and $p>0$, we improve or perfect the previous results due to Ma, Huang and Luo \cite{MHL} since one does not need to suppose the positive solutions are bounded. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.

• Noise Induced Chaotic Bursting

  Li Ji, Li Ping

  2023, 43(1): 44-63. doi: 10.3969/j.issn.1006-8074.2023.01.003

  Abstract ( 1328 )   PDF (540KB) ( 353 )   　　

  Periodic or chaotic bursting behavior is commonly observed in neurons and endocrine cells, consisting of recurrent transitions between quiescent states and repetitive spikings, and has been of interest for a long time. Stochastic forcing is known to have non-negligible influence in many cases. In this paper we study the effect of uniformly bounded noise on the spike and burst behavior and explain the mechanism how uniformly bounded noise generates chaotic bursting. Numerical simulation is provided to support the underlying mechanism.

• The Turán  Number of Disjoint Stars in Hypergraphs

  Deng Jinghua, Hou Jianfeng, Zeng Qinghou, Zhang Yixiao

  2023, 43(1): 64-73. doi: 10.3969/j.issn.1006-8074.2023.01.004

  Abstract ( 1296 )   PDF (165KB) ( 237 )   　　

  Given an $r$-uniform hypergraph $\mathcal{F}$, the { Tur\'{a}n number} of $\mathcal{F}$, denoted by $\mathrm{ex}_{r}(n,\mathcal{F})$, is the maximum number of edges in an $\mathcal{F}$-free $r$-uniform hypergraph on $n$ vertices. For $r\ge 3$, determining $\mathrm{ex}_{r}(n,\mathcal{F})$ is known to be notoriously hard especially when $\mathrm{ex}_{r}(n,\mathcal{F})=o(n^r)$. For a graph $F$, the expansion of $F$, denoted by $F^{+}$, is an $r$-uniform hypergraph by adding $r-2$ new elements to each edge of $F$; and the Berge copy of $F$, denoted by Berge-$F$, is an $r$-uniform hypergraph $\mathcal{H}$ with $V(F) \subseteq V(\mathcal{H})$ satisfying that there is a bijection $f$ from $E(F)$ to $E(\mathcal{H})$ such that $e\subseteq f(e)$ for every $e \in E(F)$. In this paper, we determine the Tur\'{a}n numbers of the expansion, and the family of all Berge copy of disjoint union of stars. Both generalize the results given by Khormali and Palmer [14].

• Existence of Perfect Matchings in General Graphs and Bipartite Graphs with Respect to Distance Signless Laplacian Spectral Radius

  Yan Zimo, Liu Chang, Li Jianping

  2023, 43(1): 74-84. doi: 10.3969/j.issn.1006-8074.2023.01.005

  Abstract ( 1308 )   PDF (277KB) ( 286 )   　　

  Let $\mathcal{D}(G)=(D_{i,j})$ be the distance matrix of a connected graph $G$, where $\mathcal{D}_{i,j}$ equals the distance between the vertices $v_i$ and $v_j$ of $G$. Let $\eta_1(G)$ be the distance signless Laplacian spectral radius of $G$, i.e., the largest eigenvalue of the distance signless Laplacian matrix $\mathcal{Q}(G)=Diag(Tr)+\mathcal{D}(G)$, where $Diag(Tr)$ is a diagonal matrix with $Diag(Tr)_{ii}=\sum_{v_iv_j\in E(G)}\mathcal{D}_{i,j}$. In this paper, we investigate the relationships between the perfect matchings and the distance signless Laplacian spectral radius, and give sufficient conditions for the existence of perfect matchings in general graphs and bipartite graphs with respect to the distance signless Laplacian spectral radius, respectively.

• Functional Inequalities of q- Analog of Bi-univalent Function Classes Involving a Particular Integral Operator

  Mai Tingmei, Long Pinhong, Han Huili, He Fuli

  2023, 43(1): 85-99. doi: 10.3969/j.issn.1006-8074.2023.01.006

  Abstract ( 1301 )   PDF (213KB) ( 267 )   　　

  In this paper a

  kind of integral operator $I^{\beta}_{\alpha}f(z)$ related to the

  parameters $\alpha, \beta$ in an open unit disk is investigated. Firstly,

  the bi-univalent function classes $\mathfrak{H}_{\Sigma_{q}}^{\alpha,\beta}(\lambda;\phi)$ and $\mathfrak{L}_{\Sigma_{q}}^{\alpha,\beta}(\mu,\lambda;\phi)$ involving this integral operator and the $q$-derivative operator are defined by applying the subordination principle of analytic functions. Then, the upper bounds of the first two coefficients $a_{2}$ and $ a_{3}$ of the two classes of bi-univalent analytic functions are estimated, and the corresponding Fekete-Szegö  inequalites for these classes are obtained.

• Positive Periodic Solutions for a Sixth-order Variable Coefficient Singular Differential Equation

  Liu Jie, Li Panpan, Cheng Zhibo, Jing Taiyan

  2023, 43(1): 100-114. doi: 10.3969/j.issn.1006-8074.2023.01.007

  Abstract ( 1269 )   PDF (169KB) ( 217 )   　　

  In this paper, the existence of positive periodic solutions for a sixth-order singular differential equation is proved by the properties of Green function of a sixth-order linear differential equation with variable coefficient coupled with the Schauder fixed point theorem. Our results contain both the attractive singularity case and the repulsive singularity case.

• Strong Attractors for Semilinear Reaction-diffusion Equations with Memory

  Tang Zhipiao, Sun Chunyou, Xie Yongqin

  2023, 43(1): 115-125. doi: 10.3969/j.issn.1006-8074.2023.01.008

  Abstract ( 1242 )   PDF (232KB) ( 266 )   　　

  In this paper, we discuss the long time behavior of strong solutions of semilinear reaction diffusion equations with fading memory. First of all, by the regularity of solutions and the control convergence principle, we prove that the semigroup of the solutions is an contractive semigroup on $ H_0^1(\Omega)\times L_\mu^2(\mathbb{R}; D(A))$, which leads to the asymptotic compactness of the semigroup. Then, we show the existence and regularity of global attractor $\mathcal{A}$ on the product space. It is noteworthy that the nonlinearity $f$ satisfies the polynomial growth of arbitrary order and $\mathcal{A}\subset D(A)\times L_\mu^2(\mathbb{R}; D(A))$.