数学季刊

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1. 具有非线性应力-应变关系的非线性多孔弹性模型多物理场有限元法的最优误差估计

葛志昊, 李海润, 李婷婷

数学季刊 2025, 40 (3): 271-294. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.004

摘要（331）

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In this paper, we propose a multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relation. Firstly, we reformulate the original problem into a new coupled fluid system-a generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure and a diffusion problem of other pseudo pressure fields. Secondly, a fully discrete multiphysics finite element method is performed to solve the reformulated system numerically. Thirdly, existence and uniqueness of the weak solution of the reformulated model and stability analysis and optimal convergence order for the multiphysics finite element method are proven theoretically. Lastly, numerical tests are given to verify the theoretical results.

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2. Dirichlet L-函数的均值

黄炳荣, 黄俊皓

数学季刊 2025, 40 (4): 360-371. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.003

摘要（204）

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In this paper, we establish asymptotic formulas for a cubic moment of Dirichlet L-functions restricted to a coset, as well as for a mixed moment of Dirichlet L-functions and twists of GL(2) L-functions along a coset. Our main tool is a power-saving estimate of bilinear forms of hyper-Kloosterman sums due to Kowalski–Michel–Sawin.

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3. 具有Riemann-Liouville分数阶积分的强阻尼波动方程的爆破现象

向常永, 段霁松, 龙群飞

数学季刊 2025, 40 (3): 304-312. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.006

摘要（181）

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We investigate the blow-up effect of solutions for a non-homogeneous wave equation
u_tt−∆u−∆u_t=I₀^α+ (|u|^p)+ω(x),
where p >1, 0≤α<1 and ω(x) with \int_{\mathbb{R}^{N}} ω(x)dx >0. By a way of combining the argument by contradiction with the test function techniques, we prove that not only any non-trivial solution blows up in finite time under 0< α <1, N ≥1 and p >1, but also any non-trivial solution blows up in finite time under α= 0, 2≤ N ≤4 and p being the Strauss exponent.

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4. 非线性抛物型积分-微分方程的高阶标量辅助变量方法

闫丽娜, 张根根, 黄琼敖

数学季刊 2025, 40 (3): 262-270. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.003

摘要（153）

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An efficient and accurate scalar auxiliary variable (SAV) scheme for numerically solving nonlinear parabolic integro-differential equation (PIDE) is developed in this paper. The original equation is first transformed into an equivalent system, and the k-order backward differentiation formula (BDFk) and central difference formula are used to discretize the temporal and spatial derivatives, respectively. Different from the traditional discrete method that adopts full implicit or full explicit for the nonlinear integral terms, the proposed scheme is based on the SAV idea and can be treated semi-implicitly, taking into account both accuracy and effectiveness. Numerical results are presented to
demonstrate the high-order convergence (up to fourth-order) of the developed schemes and it is computationally efficient in long-time computations.

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5. 矩阵方程在线性时不变系统的应用

周艳平, 陈艳萍, 张娟

数学季刊 2025, 40 (3): 221-237. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.001

摘要（130）

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With the development of science and technology, the design and optimization of control systems are widely applied. This paper focuses on the application of matrix equations in linear time-invariant systems. Taking the inverted pendulum model as an example, the algebraic Riccati equation is used to solve the optimal control problem, and the system performance and stability are achieved by selecting the closed-loop pole and designing the gain matrix. Then, the numerical methods for solving the stochastic algebraic Riccati equations are applied to practical problems, with Newton’s iteration method as the outer iteration and the solution of the mixed-type Lyapunov equations as the inner iteration. Two methods for solving the Lyapunov equations are introduced, providing references for related research.

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6. 变系数常微分方程边值问题的数值方法

赵婷婷, 蔡伟云

数学季刊 2025, 40 (3): 295-303. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.005

摘要（121）

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In order to solve the problem of the variable coefficient ordinary differential equation on the bounded domain, the Lagrange interpolation method is used to approximate the exact solution of the equation, and the error between the numerical solution and the exact solution is obtained, and then compared with the error formed by the difference method, it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.

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7. 多孔弹性问题多物理场间断有限元方法后验误差估计

葛志昊, 贺文龙, 马梦霞

数学季刊 2025, 40 (3): 238-261. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.002

摘要（114）

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In this paper, we design a new error estimator and give a posteriori error analysis for a poroelasticity model. To better overcome “locking phenomenon” on pressure and displacement, we proposed a new error estimators based on multiphysics discontinuous Galerkin method for the poroelasticity model. And we prove the upper and lower bound of
the proposed error estimators, which are numerically demonstrated to be computationally very efficient. Finally, we present numerical examples to verify and validate the efficiency of the proposed error estimators, which show that the adaptive scheme can overcome “locking phenomenon” and greatly reduce the computation cost.

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8. 带有尖形式Fourier系数的Gauss圆内格点问题

陈锋

数学季刊 2025, 40 (3): 313-323. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.007

摘要（112）

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Let f be a Hecke eigenform of even integral weight k for the full modular group SL₂(Z). Denote by λ_f (n) the nth normalized coefficient of f. The sum of Fourier coefficients of cusp form over the quadratic polynomial m² +n² is considered, i.e.,...

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9. 涉及二项式系数的新级数 (III)

孙智伟

数学季刊 2025, 40 (4): 372-392. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.004

摘要（108）

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We evaluate some series with summands involving a single binomial coefficien……

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10. 华林问题及其部分相关问题的综述

赵立璐

数学季刊 2025, 40 (4): 393-400. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.005

摘要（106）

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The aim of this survey is to introduce Waring’s problem and some related topics including mean value theorems and Diophantine equations in prime variables.

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11. 关于Bottleneck代数的一个公开问题

谭宜家

数学季刊 2025, 40 (3): 324-330. DOI: 10.13371/j.cnki.chin.q.j.m.2025.03.008

摘要（99）

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A bottleneck algebra is a linearly ordered set (B,≤) with two operations a⊕b=max{a,b} and a⊗b=min{a,b}. A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way. In 1996, Cechl´arov´a and Pl´avka posed an open problem: Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2B-independent. In this paper, we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2B-independent and answer this open problem.

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12. 非交换留数和扭化BCV空间的次黎曼极限

李洪峰, 刘克峰, 王勇

数学季刊 2026, 41 (1): 15-37. DOI: 10.13371/j.cnki.chin.q.j.m.2026.01.002

摘要（92）

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In this paper, we derive the sub-Riemannian version of the Kastler-KalauWalze type theorem and the Dabrowski-Sitarz-Zalecki type theorem for the twisted BCV spaces. We also compute the Connes conformal invariants for the twisted product, as well as the sub-Riemannian limits of the Connes conformal invariants for the twisted BCV spaces.

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13. 有限整数和集问题研究综述

汤敏

数学季刊 2025, 40 (4): 352-359. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.002

摘要（90）

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Let A be a finite set of integers. For any integer h≥2, let hA and h ^∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively. In this paper, we survey some significant advances in additive combinatorics concerning the size and structure of sumsets of finite integer sets.

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14. 高阶除数函数在混合幂次上的值

都晨豪, 孙庆峰

数学季刊 2025, 40 (4): 331-351. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.001

摘要（83）

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Let τ_k(n) be the k-th divisor function. In this paper, we derive an asymptotic formula for the sum ……

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15. Hecke-Maass新形式的Hecke特征值的第一个变号问题

唐恒才, 王英男

数学季刊 2025, 40 (4): 401-407. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.006

摘要（81）

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In this short note, we survey the recent development on the first sign change of Hecke eigenvalues of Hecke-Maass new forms.

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16. 常黎曼数量曲率Hermitian结构中的临界Hermitian度量研究

关庄丹, 闫笑峰

数学季刊 2026, 41 (1): 1-14. DOI: 10.13371/j.cnki.chin.q.j.m.2026.01.001

摘要（69）

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It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also
Einstein or isometric to a standard sphere. In the Riemannian case, it’s tangent space satisfies a decomposition. In this paper, we prove that if we only consider the Hermitian metrics, it also have a decomposition. Then we obtain the equation of the critical points among the Hermitian metrics.

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17. 航空器供油问题的优先准则及梯度排序算法

林浩, 何程

数学季刊 2026, 41 (1): 38-49. DOI: 10.13371/j.cnki.chin.q.j.m.2026.01.003

摘要（68）

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The airplane refueling problem can be stated as follows. We are given n airplanes which can refuel one another during the flight. Each airplane has a reservoir volume w_j (liters) and a consumption rate p_j (liters per kilometer). As soon as one airplane runs out of fuel, it is dropping out of the flight. The problem asks for finding a refueling scheme such that the last plane in the air reach a maximal distance. An equivalent version is the n-vehicle exploration problem. The computational complexity of

this non-linear combinatorial optimization problem is open so far. This paper employs the neighborhood exchange method of single-machine scheduling to study the precedence relations of jobs, so as to improve the necessary and sufficiency conditions of optimal solutions, and establish an efficient heuristic algorithm which is a generalization of several existing special algorithms.

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18. 一类高阶基尔霍夫方程的适定性和渐近性质

张辽, 李锋杰

数学季刊 2025, 40 (4): 417-440. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.008

摘要（64）

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This paper focuses on the investigation of a hyperbolic Kirchhoff equation with nonlinear damping and higher-order dissipation terms. Initially, the existence and uniqueness of local weak solutions are rigorously established. Next, within the framework of potential well theory, the classification of solution behaviors, including blow-up and global existence, is systematically analyzed according to the relationships among the exponents of nonlinear source terms. Finally, explicit bounds for the blow-up time and decay estimates for global solutions are presented.

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19. 双曲超曲面与费马型函数方程

陶思俊, 谢利兵, 陈裕先

数学季刊 2025, 40 (4): 408-416. DOI: 10.13371/j.cnki.chin.q.j.m.2025.04.007

摘要（63）

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In this paper, we construct new examples of hyperbolic metasurfaces in CP³ and CP⁴, and discusses the existence of solutions for a class of Fermat type functional equations.

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20. 具位势项的非齐次四阶Schrödinger方程的适定性研究

夏素霞, 李烁

数学季刊 2026, 41 (1): 82-91. DOI: 10.13371/j.cnki.chin.q.j.m.2026.01.007

摘要（59）

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The paper considers the initial value problem of inhomogeneous fourth-order Schrödinger equation with potential in energy space H²(R^d). The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle, Strichartz estimates, Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.

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