数学季刊

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1. 有限元方法的后验误差估计

鄂维南, 穆默, 黄鸿慈

数学季刊 1988, 3 (1): 97-107.

摘要（967）

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有限元的误差分析理论已有丰富的成果,但一般以先验估计为主。这些理论对算法评价和指导计算无疑具有重要意义。但实际计算中常要求给出误差的定量估计,先验估计一般就做不到这一点,因为其中会有如微分方程解的导数等不可计算的量。后验误差估计的研究近年来已开始受到注意。在多重网格技术中基于误差渐近展开的外推可以给出逐点意义下的后验估计。（例如见[1],[2]）另一条途径是通过局部化进行能量模意义下的后验误差估计。其中对一维边值问题的研究已较成熟。[7]就常系数

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2. 关于复几何中的一些新进展—与齐性流形相关的领域

关庄丹

数学季刊 2020, 35 (2): 111-144. DOI: 10.13371/j.cnki.chin.q.j.m.2020.02.001

摘要（701）

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In this article, we give a survey of some progress of the complex geometry, mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years. In particular, we explore some works in the special area in Dierential Geometry, Lie Group and Complex Homogeneous Space. Together with the special area in nonlinear analysis on complex manifolds, they are the two major aspects of my research interests.

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3. 求解低秩列稀疏矩阵恢复问题的嵌套交替方向乘子法

沈楠, 靳正芬, 王秋雨

数学季刊 2021, 36 (1): 90-110. DOI: 10.13371/j.cnki.chin.q.j.m.2021.01.007

摘要（547）

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The task of dividing corrupted-data into their respective subspaces can be well
illustrated, both theoretically and numerically, by recovering low-rank and sparse-column
components of a given matrix. Generally, it can be characterized as a matrix and a
`2,1-norm involved convex minimization problem. However, solving the resulting problem
is full of challenges due to the non-smoothness of the objective function. One of the
earliest solvers is an 3-block alternating direction method of multipliers (ADMM) which
updates each variable in a Gauss-Seidel manner. In this paper, we present three variants
of ADMM for the 3-block separable minimization problem. More preciously, whenever
one variable is derived, the resulting problems can be regarded as a convex minimization
with 2 blocks, and can be solved immediately using the standard ADMM. If the inner
iteration loops only once, the iterative scheme reduces to the ADMM with updates in a
Gauss-Seidel manner. If the solution from the inner iteration is assumed to be exact, the
convergence can be deduced easily in the literature. The performance comparisons with a
couple of recently designed solvers illustrate that the proposed methods are effective and
competitive.

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4. 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（539）

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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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5. 楼与群III

黎景辉

数学季刊 2021, 36 (1): 1-31. DOI: 10.13371/j.cnki.chin.q.j.m.2021.01.001

摘要（461）

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This is the third part of a pedagogical introduction to the theory of buildings of Jacques Tits. We describe the construction and properties of the Bruhat-Tits building of a reductive group over a local field.

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6. 楼与群Ⅱ

周国晖 , 黎景辉

数学季刊 2020, 35 (3): 221-254. DOI: 10.13371/j.cnki.chin.q.j.m.2020.03.001

摘要（385）

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This is the second part of a pedagogical introduction to the theory of buildings of Jacques Tits. We define a(B,N) pair and construct a building out of it. Then we give a description of Chevalley groups, their(B,N) pair and the associated buildings. We illustrates this theory with many examples from classical groups.

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7. 关于数论函数值分布的研究

邵品琮

数学季刊 1987, 2 (2): 27-43.

摘要（328）

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对于数义在自然数集 N 上的数论函数（Arithmetical function）f（n）, n\inN,取值可以是实数, 也可以是复数（一般为实数）, 例如著名的 Euler 函数 \varphi(n)=\Sigma_{(a,n)=1,a<n}1, 除数函数τ（n）=\Sigma_{d/n}1,以及除数和函数σ（n）=\Sigma_{d/n}d

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8. 关于自然数的方幂和

陈景润, 黎鉴愚

数学季刊 1987, 2 (1): 1-18.

摘要（317）

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求自然数的方幂和是一个经典问题，设 S_k(n) 表示 \sigma_m-1ⁿ m⁴, Schwlts 和Scott 都通过复杂的计算给出了当1 \leq k \leq 6 及 7 \leq k \leq 13时 S_k(n) 的计算公式。本文用初等的方法给出了1 \leq k \leq 30时 S_k(n) 的计算公式。

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9. 测度空间上的Lévy-Prohorov距离

曲立敏 , 朱季云

数学季刊 2017, 32 (1): 42-48. DOI: 10.13371/j.cnki.chin.q.j.m.2017.01.005

摘要（315）

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Under the premise of infinitely many pure strategies, by defining the new LP* metric, striking an equivalence of topology and weak* topology, we prove that the existence of the essential component.

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10. r-正则有向图的多数染色

夏伟皓, 石美, 肖明月, 蔡建生, 王纪辉

数学季刊 2022, 37 (2): 142-146. DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.004

摘要（313）

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A majority k-coloring of a digraph D with k colors is an assignment c: V ( D ) →{ 1,2,···,k} , such that for every v∈V (D), we have c(w)=c(v) for at most half of all out-neighbors w∈N⁺( v ). For a natural number k≥2, a
1/k-majority coloring of a digraph is a coloring of the vertices such that each vertex receives the same color as at
most a 1/k proportion of its out-neighbours. Kreutzer, Oum, Seymour, van der Zypen and
Wood proved that every digraph has a majority 4-coloring and conjectured that every
digraph admits a majority 3-coloring. Gir$\widetilde{a}$o, Kittipassorn and Popielarz proved that every
digraph has a 1/k-majority 2k-coloring and conjectured that every digraph admits a
1\k-majority (2k−1)-coloring. We showed that every r -regular digraph D with r>36ln(2n)
has a majority 3-coloring and proved that every digraph D with minimum outdegree
$\delta^+>\frac{2k^2(2k-1)^2}{(k-1)^2}\ln{[(2k-1)n]}$ has a 1/k-majority (2k−1)-coloring. And we also proved that
every r-regular digraph D with $r>\frac{3k^2(2k-1)}{(k-1)^2} \ln(2n)$ has a 1/k-majority (2k−1)-coloring.

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11. 求解箱式约束全局优化问题的新的填充函数

屈德强, 吴丹, 尚有林

数学季刊 2020, 35 (4): 354-362. DOI: 10.13371/j.cnki.chin.q.j.m.2020.04.003

摘要（311）

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本文考虑利用辅助函数法求解箱式约束全局优化问题。首先，提出了一个新的与目标函数具有相同局部极小点的无参数填充函数。该填充函数的局部极小点是目标函数更优的局部极小点，基于这一特征设计的填充函数方法，目标函数只需要被极小化一次。改变了传统填充函数法目标函数和填充函数交替极小化的算法框架，能有效减少算法的迭代次数，加快寻找全局极小点的速度。随后进一步讨论了填充函数的理论性质，并建立了相应的算法。最后，通过数值实验验证了算法的可行性和有效性。

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12. 奇异点附近的复微分方程解的增长性与系数下级的关系

刘元竹, 龙见仁, 曾三桂

数学季刊 2020, 35 (2): 163-170. DOI: 10.13371/j.cnki.chin.q.j.m.2020.02.003

摘要（311）

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We investigate the growth of solutions of the following complex linear differential equation f’’+ A（z）f’+ B（z）f = 0,where A（z） and B（z） are analytic functions in C-{z₀}, z₀∈ C. Some estimations of lower bounded of growth of solutions of the differential equation are obtained by using the concept of lower order.

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13. 广义的Orlicz空间上的Korn不等式与散度方程

吴芮民, 王松柏

数学季刊 2020, 35 (4): 344-353. DOI: 10.13371/j.cnki.chin.q.j.m.2020.04.002

摘要（297）

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Let ϕ be a generalized Orlicz function satisfying (A0), (A1), (A2), (aInc) and (aDec). We prove that the mapping

f →f ^#:=sup_B 1/\int_|B||f(x)-f_B|dx is continuous on L^ϕ(·)(Rⁿ) by extrapolation. Based on this result we generalize Korn’s inequality to the setting of generalized Orlicz spaces, i.e., ||\triangledown f||_{L^{ϕ(·)}(Ω)}\lesssim||DF|||_L^{ϕ}(Ω) . Using the Calder´on–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu=f has a solution u∈(W0^1,ϕ(·)(Ω))ⁿsuch that ||\triangledown u||_L^{_ϕ(·)}(Ω) \lesssim ||f||_L^{ϕ}(Ω).

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14. 一类不确定薛定谔基尔霍夫方程解的存在性和集中性

陈玉松, 常荷洁

数学季刊 2020, 35 (1): 37-45. DOI: 10.13371/j.cnki.chin.q.j.m.2020.01.003

摘要（290）

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This paper is concerned with the nonlinear Schrodinger-Kirchhoff system $-(a+b \int _{R^{3}}|\nabla u|^{2} dx) \triangle u+ \lambda V(x)u=f(x,u)$ in R³, where constants a > 0,b ≥ 0 and λ > 0 is a parameter. We require that V (χ) ∈ C(R³) and has a potential well V ^-1(0). Combining this with other suitable assumptions on K and ƒ, the existence of nontrivisd solutions is obtained via vaxiational methods. Furthermore, the concentration behavior of the nontrivial solution is also explored on the set V ^-1(0) as λ → + ∞ as well. It is worth noting that the (PS )-condition can not be directly got as done in the literature, which makes the problem more complicated. To overcome this difficulty, we adopt different method.

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15. 具非局部非线性阻尼和源项的波方程解的能量衰减估计

张宏伟, 李栋浩, 呼青英

数学季刊 2020, 35 (3): 302-310. DOI: 10.13371/j.cnki.chin.q.j.m.2020.03.005

摘要（288）

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We consider a wave equation with nonlocal nonlinear damping and source terms. We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique. The main difficult is how to handle with the nonlocal nonlinear damping term. Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso（Evolution Equation and Control Theory, 2017（6）:437-470） and Narciso（Evolution Equations and Control Theory, 2020, 9（2）: 487-508）.

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16. 与 Mittag-Leffler 函数有关的多叶亚纯函数的优化和 Fekete-Szegö 问题

龙品红, 甘加达兰•穆鲁古桑德拉莫西, 韩惠丽, 汪文帅

数学季刊 2022, 37 (2): 111-123. DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.001

摘要（286）

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The main object of this present paper is to investigate the problem of majorization of certain class of multivalent meromorphic functions of complex order involving Mittag-Leffler function. Meanwhile, for this subclass the corresponding coefficient estimates and some Fekete-Szeg¨ o type inequalities are obtained. Moreover we point out some new or known consequences of our main results.

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17. 非对称凸体的截面与投影体积不等式

曹子昕, 李爱军

数学季刊 2022, 37 (2): 178-188. DOI: 10.13371/j.cnki.chin.q.j.m.2022.02.007

摘要（264）

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In this paper, we establish volume inequalities for k-dimensional sections and projections of convex bodies (not necessarily symmetric) and their polars in a more general position than John’s position.

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18. 哈林图的导出匹配可扩性

张庆楠, 惠志昊, 杨雨, 王安

数学季刊 2022, 37 (4): 380-385. DOI: 10.13371/j.cnki.chin.q.j.m.2022.04.005

摘要（253）

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Let G be a connected graph having a perfect matching. The graph G is said to be induced matching (IM) extendable if every induced matching M of G is contained in a perfect matching of G. In this paper, we show that Halin graph G =T ∪C is IMextendable if and only if its characteristic tree T is isomorphic to K_1,3, K_1,5, K_1,7or S_2,2.

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19. 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（249）

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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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20.

带有时滞和热扩散效应的 Timoshenko 系统的适定性

秦玉明, 韩宁

数学季刊 2022, 37 (1): 1-9. DOI: 10.13371/j.cnki.chin.q.j.m.2022.01.001

摘要（246）

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In this paper, we study the well-posedness for the thermoelastic Timoshenko system with a constant delay and mass diffusion effects. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam. The C₀ -semigroup theory will be used to prove the well-posedness of the considered problem.

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