数学季刊

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1. 曲面上的曲率在理论物理中的一些应用

杨亦松

数学季刊 2023, 38 (3): 221-253. DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.001

摘要（538）

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In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

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2. 一类对数p-Laplace抛物型方程中的爆破和衰减估计

李平, 李锋杰

数学季刊 2024, 39 (4): 331-354. DOI: 10.13371/j.cnki.chin.q.j.m.2024.04.001

摘要（248）

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This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian, which could be proposed as a model for the epitaxial growth of thin films. By using the variational method and the
logarithmic type Sobolev inequality, we give some threshold results for blow-up solutions and global solutions, which could be classified by the initial energy. The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained.

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3. 一类具有状态依赖脉冲影响的捕食者-食饵模型动力学分析

李永凤, 朱城志, 刘艳伟

数学季刊 2023, 38 (1): 1-19. DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.001

摘要（242）

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In this paper, a pest-dependent model and integrated pest management
strategy is proposed, that is, when pest populations reach levels that impair economic
development, we will use a combination of strategies, such as biological, cultural and
chemical control strategies reduce pests to a reasonable level. First, we investigated the
system without control measures, and discussed the existence and stability of equilibria,
we also proved the system has no limit cycle. Then, a state feedback impulsive model is
constructed, the existence and uniqueness of the order-one periodic solution are proved
by means of the successor function method to confirm the feasibility of the biological and
chemical control strategy of pest management. Secondly, the stability of system is proved
by the analogue of the Poincar ´ e criterion. Finally, we give an example and numerical
simulations to explain the mathematical conclusions.

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4. 带位势的基尔霍夫方程规范解的存在性

梁艳霞

数学季刊 2023, 38 (2): 196-209. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.007

摘要（239）

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In this paper we discuss the following Kirchhoff equation

\left\{
\begin{array}{lr}
-\left(a+b \int_{\mathbb{R}^3}|\nabla u|^{2} d x\right) \Delta u+V(x)u+\lambda u=\mu|u|^{q-2}u+|u|^{p-2}u \ {\rm in}\ \mathbb{R}^3,&\\
\int_{\mathbb{R}^{3}}u^{2}dx=c^2,
\end{array}
\right.
where a, b, µ and c are positive numbers, λ is unknown and appears as a Lagrange multiplier,

14/3<q<p<6 and V is a continuous non-positive function vanishing at infinity.
Under some mild assumptions on V , we prove the existence of a mountain pass normalized solution. To the author’s knowledge, it is the first time to study the existence of
normalized solution to Kirchhoff equation with potential via the minimax principle.

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5. 有向图的加权解析挠率

任世全, 王冲

数学季刊 2023, 38 (1): 30-49. DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.003

摘要（224）

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In 2020, Alexander Grigor’yan, Yong Lin and Shing-Tung Yau [6] introduced
the Reidemeister torsion and the analytic torsion for digraphs by means of the path
complex and the path homology theory. Based on the analytic torsion for digraphs
introduced in [6], we consider the notion of weighted analytic torsion for vertex-weighted
digraphs. For any non-vanishing real functions f and g on the vertex set, we consider the
vertex-weighted digraphs with the weights ( f,g ). We calculate the ( f,g )-weighted analytic
torsion by examples and prove that the ( f,g )-weighted analytic torsion only depend on
the ratio f/g . In particular, if the weight is of the diagonal form ( f,f ), then the weighted
analytic torsion equals to the usual (un-weighted) torsion.

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6. 具有勢壘的Sturm-Liouville算子的特徵值和特徵函數

SARWAR Qanitah 黄振友 ZAHID Abdul Hannan

数学季刊 2023, 38 (1): 50-61. DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.004

摘要（221）

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We aim to find the eigenvalues and eigenfunctions of the barrier potential
case for Strum-Liouville operator on the finite interval [0 ,π ] when λ> 0. Generally, the
eigenvalue problem for the Sturm-Liouville operator is often solved by using integral
equations, which are sometimes complex to solve, and difficulties may arise in computing
the boundary values. Considering the said complexity, we have successfully developed
a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for
Sturm-Liouville operator with barrier potential. The results are of significant interest in
the field of quantum mechanics and atomic systems to observe discrete energy levels.

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7. 一类非凸优化问题的邻近拟牛顿方法的复杂性

金玲子

数学季刊 2023, 38 (1): 62-84. DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.005

摘要（218）

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This paper studies a class of nonconvex composite optimization, whose
objective is a summation of an average of nonconvex (weakly) smooth functions and a
convex nonsmooth function, where the gradient of the former function has the Hölder
continuity. By exploring the structure of such kind of problems, we first propose a
proximal (quasi-)Newton algorithm wPQN (Proximal quasi-Newton algorithm for weakly
smooth optimization) and investigate its theoretical complexities to find an approximate
solution. Then we propose a stochastic variant algorithm wPSQN (Proximal stochastic
quasi-Newton algorithm for weakly smooth optimization), which allows a random subset
of component functions to be used at each iteration. Moreover, motivated by recent
success of variance reduction techniques, we propose two variance reduced algorithms,
wPSQN-SVRG and wPSQN-SARAH, and investigate their computational complexity
separately.

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8. 一类具有时滞的生态流行病模型的稳定性与 Hopf 分支

白宏芳

数学季刊 2023, 38 (2): 157-183. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.005

摘要（201）

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In this paper, an eco-epidemiological model with time delay is studied. The local stability of the four equilibria, the existence of stability switches about the predation- free equilibrium and the coexistence equilibrium are discussed. It is found that Hopf bifurcations occur when the delay passes through some critical values. Formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Some numerical simulations are carried out to illustrate the theoretical results.

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9. 具有环境容纳量和成熟双时滞的有毒食饵-捕食者收获模型

钟颖, 韦煜明

数学季刊 2023, 38 (4): 331-348. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.001

摘要（197）

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In this paper, a model of predator-prey with dual delay in maturation and carrying capacity is discussed, in which the past activity of the prey should have an impact on the carrying capacity, the mature prey initiates defense mechanisms to release toxins when subjected to predation, and a commercial harvest of the prey is performed. The stability of the equilibrium of the system in the absence of delay is examined and the optimal harvesting strategy of the model is proven. By investigating the roots of the characteristic equation and applying normalized theory, the properties of the coexistence equilibrium of the system and the conditions for the occurrence of the Hopf bifurcation in the neighborhood of the positive equilibrium are described for various combinations of delays. In the end, numerical simulations are used to verify theoretical analysis results.

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10. 群作用在复几何中的一些运用

关庄丹

数学季刊 2023, 38 (3): 254-275. DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.002

摘要（196）

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In this article, we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020, mostly related to my own interests.

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11. 变指标中心Morrey空间上带Dini核的多线性C-Z算子

马腾

数学季刊 2023, 38 (2): 184-195. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.006

摘要（176）

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In this paper, we obtain the boundedness of multilinear Calder´ on-Zygmund operators with kernels of Dini type and commutators with variable exponent λ -central BMO functions in variable exponent central Morrey spaces.

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12. Allen-Cahn方程的虚拟元方法

王培珍, 田旭

数学季刊 2023, 38 (1): 20-29. DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.002

摘要（174）

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In this article, the virtual element method of the Allen-Cahn equation on a
polygon grid is discussed in the fully discrete formulation. With the help of the energy
projection operator, we give the corresponding error estimates in the L² norm and H¹
norm.

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13. 带分数阶Robin边界条件的时间-空间分数阶扩散方程的有限差分方法

唐忠华, 房少梅

数学季刊 2024, 39 (1): 18-30. DOI: 10.13371/j.cnki.chin.q.j.m.2024.01.002

摘要（173）

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In this paper, an efficient numerical method is proposed to solve the Caputo-Riesz fractional diffusion equation with fractional Robin boundary conditions. We approximate the Riesz space fractional derivatives using the fractional central difference scheme with second-order accurate. A priori estimation of the solution of the numerical scheme is given, and the stability and convergence of the numerical scheme are analyzed. Finally, a numerical example is used to verify the accuracy and efficiency of the numerical method.

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14. 关于全局吸引子的一个新的存在性定理以及它对于非经典扩散方程的应用

秦玉明, 陈家乐, 姜慧特

数学季刊 2023, 38 (3): 276-289. DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.003

摘要（169）

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In this paper, we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems, then establish a new existence theorem of global attractors which will be applied to a nonclassical diffusion equation for the norm-to-weak continuous, weakly compact semigroup on H01(Ω) and H2(Ω)∩H01(Ω) respectively. As an application of this new existence theorem of global attractors, we obtain the existence of the global attractors onH01(Ω) and H2(Ω)∩ H01(Ω) respectively for a nonclassical-diffusion equation.

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15. 两类四圈图的奇异性

攸晓杰, 马海成, 张斌, 李雅兰

数学季刊 2023, 38 (4): 349-359. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.002

摘要（166）

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Let G be a finite simple graph and A(G) be its adjacency matrix. Then G is singular if A(G) is singular. The graph obtained by bonding the starting vertices and ending vertices of three paths P_a1, P_a2, P_a3is called θ-graph, represented by θ(a1,a2,a3). The graph obtained by bonding the two end vertices of the path P_s to the vertices of the θ(a₁,a₂,a₃) and θ(b₁,b₂,b₃) of degree three, respectively, is denoted by α(a₁,a₂,a₃,s,b₁,b₂,b₃) and called α-graph. β-graph is denoted when β(a₁,a₂,a₃,b₁,b₂,b₃) =α(a₁,a₂,a₃,1,b₁,b₂,b₃). In this paper, we give the necessary and sufficient conditions for the singularity of α-graph and β-graph, and prove that the probability that a random given α-graph and β-graph is a singular graph is equal to 1423/2048 and 733/1024 , respectively.

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16. 在复二维仿射锥上的完备余齐性一Kähler度量

关庄丹, 梁梦翔

数学季刊 2024, 39 (2): 200-220. DOI: 10.13371/j.cnki.chin.q.j.m.2024.02.008

摘要（162）

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In this paper, we revisit the K¨ahler structures on the affine quadrics M₁={z₁² +z₂² +z₃² = 1} in the paper by Bo Yang and Fang-Yang Zheng. We found that theK¨ahler structures on the complex surface are more complicated than what they havethought. We shall also give some detail calculations and found that our results fit quitewell with earlier papers of the first author, one of them with X. X. Chen.

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17. 凸复Finsler流形上的Hopf-Rinow定理

李鸿军

数学季刊 2024, 39 (1): 31-45. DOI: 10.13371/j.cnki.chin.q.j.m.2024.01.003

摘要（161）

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Suppose (M,F) is a convex complex Finsler manifold. We prove that geodesics of (M,F) are locally minimizing. Hence, F introduces a distance function d such that (M,d) is a metric space from topology. Next, we prove the classical Hopf-Rinow Theorem holds on (M,F).

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18. 富勒烯图的极大共振性

杨瑞, 马燕菲

数学季刊 2024, 39 (1): 1-17. DOI: 10.13371/j.cnki.chin.q.j.m.2024.01.001

摘要（159）

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A {(3,4),4}-fullerene graph S is a 4-regular map on the sphere whose faces are of length 3 or 4. It follows from Euler’s formula that the number of triangular faces is eight. A set H of disjoint quadrangular faces of S is called resonant pattern if S has a perfect matching M such that every quadrangular face in H is M-alternating. Let k be a positive integer, S is k-resonant if any i≤k disjoint quadrangular faces of S form a resonant pattern. Moreover, if graph S is k-resonant for any integer k, then S is called maximally resonant.

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19. 带有对合的素环的微分恒等式

黄述亮

数学季刊 2023, 38 (2): 134-144. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.003

摘要（158）

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Let R be a prime ring of characteristic different from two with the sec- ond involution ∗ and α an automorphism of R . An additive mapping F of R is called a generalized ( α,α )-derivation on R if there exists an ( α,α )-derivation d of R such that F ( xy )= F ( x ) α ( y )+ α ( x ) d ( y ) holds for all x,y∈R. The paper deals with the s- tudy of some commutativity criteria for prime rings with involution. Precisely, we describe the structure of R admitting a generalized ( α,α )-derivation F satisfying any one of the following properties:
( i ) F ( xx^∗) −α ( xx^∗) ∈Z ( R ).
( ii ) F ( xx^∗ )+ α ( xx^∗ ) ∈Z ( R ).
( iii ) F ( x ) F ( xx^∗) −α ( xx^∗) ∈Z ( R ).
( iv ) F ( x ) F (x^∗)+ α ( xx^∗) ∈Z ( R ).
( v ) F ( xx^∗) −F ( x ) F (x^∗ ) ∈Z ( R ).
( vi ) F ( xx^∗) −F (x^∗) F ( x )=0
for all x∈R . Also, some examples are given to demonstrate that the restriction of the various results is not superfluous. In fact, our results unify and extend several well known theorems in literature.

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20. 具有指数临界增长的薛定谔方程的高能正规化解

张小藏, 许丽萍

数学季刊 2025, 40 (1): 1-19. DOI: 10.13371/j.cnki.chin.q.j.m.2025.01.001

摘要（153）

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In this paper, we study high energy normalized solutions for the following Schrödinger equation ……

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