数学季刊

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1. 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（88）

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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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2. 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（85）

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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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3. 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（73）

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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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4. 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（65）

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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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5. 非奇异H-张量的新判定

柳亮, 王亚强

数学季刊 2023, 38 (2): 123-133. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.002

摘要（61）

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H -tensor plays an important role in identifying positive definiteness of even
order real symmetric tensors. In this paper, some definitions and theorems related to
H -tensors are introduced firstly. Secondly, some new criteria for identifying nonsingular
H -tensors are proposed, moreover, a new theorem for identifying positive definiteness
of even order real symmetric tensors is obtained. Finally, some numerical examples are
given to illustrate our results.

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6. 矩阵方程的最小二乘{P,Q,k+1}-自反解

董昌州, 李浩雪

数学季刊 2023, 38 (2): 210-220. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.008

摘要（51）

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Let P ∈C ^m×m and Q∈C ^n×n be Hermitian and {k +1 } -potent matrices,
i.e., P ^k+1 = P = P ^∗, Q^k+1 = Q = Q^∗ , where ( · ) ∗ stands for the conjugate transpose of a
matrix. A matrix X ∈C^m×n is called {P,Q,k +1 } -reflexive (anti-reflexive) if PXQ = X
( PXQ = −X ). In this paper, the least squares solution of the matrix equation AXB = C
subject to {P,Q,k +1 } -reflexive and anti--reflexive constraints are studied by converting
into two simpler cases: k=1 and k=2.

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7. 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（50）

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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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8. 奇异鞍点问题中广义位移分裂迭代方法的半收敛性分析

黄卓红

数学季刊 2023, 38 (2): 145-156. DOI: 10.13371/j.cnki.chin.q.j.m.2023.02.004

摘要（49）

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Recently, some authors (Shen and Shi, 2016) studied the generalized shift-
splitting (GSS) iteration method for singular saddle point problem with nonsymmetric
positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. In this
paper, we further apply the GSS iteration method to solve singular saddle point problem
with nonsymmetric positive semidefinite (1,1)-block and symmetric positive semidefinite
(2,2)-block, prove the semi-convergence of the GSS iteration method and analyze the
spectral properties of the corresponding preconditioned matrix. Numerical experiment is
given to indicate that the GSS iteration method with appropriate iteration parameters is
effective and competitive for practical use.

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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（49）

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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（44）

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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. 具有非线性边界条件的不可压 MHD-Boussinesq方程组初边值问题的整体适定性#br#

王术, 孙瑞

数学季刊 2023, 38 (3): 290-310. DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.004

摘要（42）

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The global well-posedness of another class of initial-boundary value problem on
two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain
with the smooth boundary is studied. The existence of a class of global weak solution to the
initial boundary value problem for two/three-dimensional incompressible MHD-Boussinesq
equation with the given pressure-velocity’s relation boundary condition for the fluid field,
one generalized perfectly conducting boundary condition for the magnetic field and one
density/temperature-velocity’s relation boundary condition for the density/temapture at
the boundary is obtained, and the global existence and uniqueness of the smooth solution
to the corresponding problem in two-dimensional case for the smooth initial data is also
proven.

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12. Stokes 现象与量子群的表示

徐晓濛

数学季刊 2023, 38 (3): 311-330. DOI: 10.13371/j.cnki.chin.q.j.m.2023.03.005

摘要（32）

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This is a survey paper that lists our research works in the study of Stokes
phenomenon of meromorphic ordinary differential equations and its relation with representation theory of quantum groups

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13. 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（32）

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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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14. 具有对数非线性项的伪抛物型 p-Laplacian 型方程的初边值问题

潘嘉慧, 房少梅

数学季刊 2023, 38 (4): 360-369. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.003

摘要（29）

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In this paper, we study the initial boundary value problem of pseudo-parabolic
p-Laplacian type equation, which be use to model some important physical and biological
phenomena. By using the potential well method, we obtain the global existence, asymptotic
behavior and blow up results of weak solution with subcritical initial energy. Then we
also extend these results to the critical initial energy.

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15. 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（28）

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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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16. 一类包含广义Mittag-Leffler函数的积分算子的从属与超从属结果

王小元, 刘亚娟

数学季刊 2023, 38 (4): 379-391. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.005

摘要（26）

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In the paper, by making use of the principle of subordination between analytic
functions with the multiplier transforms defined by generalized Mittag-Leffler function, the
authors investigate subclasses of univalent analytic functions, such as starlike functions,
convex functions, close-to-convex functions and quasiconvex functions. Several inclusion
relationships, inequality properties, subordination and superordination results associated
with the multiplier transforms are proved and the sandwich-type results are also obtained.

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17. 一类 Gerstenhaber 代数的构造

侯波, 扣雯

数学季刊 2023, 38 (4): 370-378. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.004

摘要（25）

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For any K-algebra A, based on Hochschild complex and Hochschild cohomology of A, we construct a new Gerstenhaber algebra, and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.

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18. 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（23）

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In this paper, an efficient numerical method is proposed to solve the CaputoRiesz 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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19. 一个一维二阶拟线性双曲组的精确边界能控性

王珂, 潘盼盼

数学季刊 2023, 38 (4): 424-440. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.010

摘要（22）

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In this paper, we propose a second-order quasilinear hyperbolic system. By
means of the theory on semi-global C¹solution to first-order quasilinear hyperbolic
systems, we establish the existence and uniqueness of semi-global C²solution to this
second-order quasilinear hyperbolic system. After then, the general constructive framework
is utilized to obtain the local exact boundary controllability for this second-order system.
Keywords: First-order quasilinear hyperbolic system; Second-order quasilinear hyperbolic system; Semi-global solution; Exact boundary controllability

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20. 一个2×2块矩阵的Moore-Penrose逆的显式公式

曾敏, 袁永新

数学季刊 2023, 38 (4): 401-409. DOI: 10.13371/j.cnki.chin.q.j.m.2023.04.007

摘要（22）

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The representation for the Moore-Penrose inverse of the matrix
A B
C D
is derived by using the solvability theory of linear equations, where A∈C^m×n, B∈C^m×p, C∈C^q×nand D∈C^q×p, with which some special cases are discussed.

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