中国科学院数学与系统科学研究院期刊网
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15 February 2026, Volume 42 Issue 2
    

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  • Möbius Inversion and Duality for Summations of Stable Graphs
    Zhiyuan Wang, Jian Zhou
    Acta Mathematica Sinica. 2026, 42(2): 269-292. https://doi.org/10.1007/s10114-026-4262-2
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    Using the stratifications of Deligne—Mumford moduli spaces $\overline{\mathcal M}_{g,n}$ indexed by stable graphs, we introduce a partially ordered set of stable graphs by defining a partial ordering on the set of connected stable graphs of genus $g$ with $n$ external edges. By modifying the usual definition of zeta function and Möbius function of a poset, we introduce generalized ($\mathbb Q$-valued) zeta function and generalized ($\mathbb Q$-valued) Möbius function of the poset of stable graphs. We use them to proved a generalized Möbius inversion formula for functions on the poset of stable graphs. Two applications related to duality in earlier work are also presented.
  • Multilinear Maximal Operators on Triebel–Lizorkin Spaces and Besov Spaces
    Feng Liu, Simin Liu, Shifen Wang
    Acta Mathematica Sinica. 2026, 42(2): 293-321. https://doi.org/10.1007/s10114-026-4583-1
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    In the present paper, the authors systematically study the mapping properties of multilinear maximal operators on the Triebel–Lizorkin spaces and Besov spaces. In the global setting, the authors provide a criterion on the boundedness and continuity of a class of multilinear operators on the Triebel–Lizorkin spaces and Besov spaces, which can be used to obtain the boundedness and continuity of the multilinear operators associated to balls, cubes and dyadic cubes, multilinear sharp maximal operator as well as multilinear operators of convolution type on the Triebel–Lizorkin spaces and Besov spaces. The corresponding results for the multilinear maximal operators associated to balls are also proved in the local setting.
  • Virtual Element Method for the Elastic Transmission Eigenvalue Problem with Equal Elastic Tensors
    Jian Meng, Bingbing Xu, Fang Su, Xu Qian, Songhe Song
    Acta Mathematica Sinica. 2026, 42(2): 322-356. https://doi.org/10.1007/s10114-026-4056-6
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    The elastic transmission eigenvalue problem is fundamental to the qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the elastic transmission eigenvalue problem is neither self-adjoint nor elliptic. The aim of this work is to provide a systematic spectral approximation analysis for the VEM of the elastic transmission eigenvalue problem with equal elastic tensors. Considering standard assumptions on polygonal/polyhedral meshes, we prove the stability analysis of the associated VEM bilinear forms, which shall be applied to the well-defined property of the discrete solution operator. Then the correct approximation of spectrum for the proposed VEM scheme is proven. Necessitated by supporting the convergence analysis, a series of numerical examples are reported. In addition, some negative points of the current VEM scheme are considered, including the locking phenomenon and the influence of VEM stabilization parameters. Thanks to the flexibility of construction for the VEM space, the locking-free and stabilization-free VEM approaches are utilized to tackle with these negative aspects.
  • Asymptotic Lower and Upper Bounds for Linear Elasticity Eigenvalues
    Yifan Yue, Hongtao Chen, Shuo Zhang
    Acta Mathematica Sinica. 2026, 42(2): 357-376. https://doi.org/10.1007/s10114-026-4391-7
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    Lower and upper bounds for eigenvalues help estimate the location interval of eigenvalues, which is of practical meanings especially for those problems of which the eigenvalues cannot be exactly obtained. In this paper, we study the lower and upper bounds for linear elasticity eigenvalues by displacement-pressure mixed finite element schemes. By applying expansion identities for the error of eigenvalues, lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. For the schemes studied here, roughly speaking, the accuracy loss of the local approximation of the discrete displacement may lead to lower bound and that of pressure to upper bound. By utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lamé parameters.
  • Twist Monomials of Binary Delta-matroids
    Qi Yan, Xian'an Jin
    Acta Mathematica Sinica. 2026, 42(2): 377-383. https://doi.org/10.1007/s10114-026-4460-y
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    Recently, we introduced the twist polynomials of delta-matroids and gave a characterization of even normal binary delta-matroids whose twist polynomials have only one term and posed a problem: what would happen for odd binary delta-matroids? In this paper, we show that a normal binary delta-matroid whose twist polynomials have only one term if and only if each connected component of the intersection graph of the delta-matroid is either a complete graph of odd order or a single vertex with a loop.
  • New Real-Variable Characterizations of Harmonic Functions with Mixed Morrey Traces and Applications
    Bo Li, Tianjun Shen, Wenchang Sun, Chao Zhang
    Acta Mathematica Sinica. 2026, 42(2): 384-414. https://doi.org/10.1007/s10114-026-4067-3
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    A well-known result of Stein–Weiss in 1971 said that a harmonic function, defined on the upper half-space, is the Poisson integral of a Lebesgue function if and only if it is also a Lebesgue function uniformly in the time variable. We show that a solution to the elliptic equation, defined on the upper half-space, is in the essentially-bounded-Morrey space of mixed type if and only if it can be represented by the Poisson integral of a mixed Morrey function, where a Liouville property is assumed. As applications, some new real-variable characterizations of the solution to the elliptic/parabolic equation related to the Neumann/Dirichlet problem are also considered via the gluing technology.
  • Obstructions for Minimal Distal Actions
    Enhui Shi, Hui Xu, Lizhen Zhou
    Acta Mathematica Sinica. 2026, 42(2): 415-426. https://doi.org/10.1007/s10114-026-4381-9
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    In this paper, we mainly consider the nonexistences of minimal distal actions by some groups on compact manifolds, particularly on surfaces. Suppose that $X$ is a compact manifold and $\Gamma$ is a finitely generated group acting on $X$. We show in the following cases that $\Gamma$ cannot act on $X$ minimally and distally. (1) $X$ is connected and the first Čech cohomology group ${\check H}^{1}(X)$ with integer coefficients is nontrivial and $\Gamma$ is amenable; (2) $X$ is the $2$-sphere $\mathbb S^2$ or the real projective plane $\mathbb {RP}^2$ and $\Gamma$ contains no nonabelian free subgroup; (3) $X$ is a closed surface and $\Gamma $ is a lattice of ${\rm SL}(n,\mathbb{R})\ (n\geq 3)$.
  • Uniform Measure Attractors for Non-autonomous Stochastic Evolution Systems
    Dingshi Li, Ran Li, Tianhao Zeng
    Acta Mathematica Sinica. 2026, 42(2): 427-444. https://doi.org/10.1007/s10114-026-4332-5
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    This paper is concerned with uniform measure attractors for non-autonomous stochastic evolution systems. We first introduce the concept of uniform measure attractor and then provide a sufficient criterion for existence and uniqueness of such attractors. As an application, we prove the existence and uniqueness of uniform measure attractors for the stochastic Navier–Stokes equations with deterministic almost-periodic forcing and nonlinear diffusion terms.
  • Small Positive Values and Limit Theorems for Supercritical Branching Processes with Immigration in Random Environment
    Yinxuan Zhao, Mei Zhang
    Acta Mathematica Sinica. 2026, 42(2): 445-463. https://doi.org/10.1007/s10114-026-4335-2
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    Let $(Z_n)$ be a supercritical branching process with immigration in a random environment. The small positive values and some lower deviation inequalities for $Z_n$ are investigated. Based on these results, the central limit theorem of $\log Z_n$ and the Edgeworth expansion are obtained. The study is taken under the assumption that each individual produces $0$ offspring with a positive probability.
  • On the Boundedness of Degenerate Hypergraphs
    Jianfeng Hou, Caiyun Hu, Heng Li, Xizhi Liu, Caihong Yang, Yixiao Zhang
    Acta Mathematica Sinica. 2026, 42(2): 464-480. https://doi.org/10.1007/s10114-026-4419-z
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    We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex $F$-free $r$-graph with a vertex of degree at least $\alpha\binom{n-1}{r-1}$ has fewer than $(1-\beta) \cdot \operatorname{ex}(n, F)$ edges. The boundedness property is crucial for recent works that aim to extend the classical Hajnal-Szemerédi Theorem (Toward a density Corrádi-Hajnal theorem for degenerate hypergraphs. J. Combin. Theory Ser. B, 172, 221-262 (2025)) and the anti-Ramsey theorems of Erdős-Simonovits-Sós (Tight bounds for rainbow partial $F$-tiling in edge-colored complete hypergraphs. J. Graph Theory, 110(4), 457-467 (2025)). We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for 3-graphs, strengthening a theorem by Kostochka-Mubayi-Verstraëte (Turán problems and shadows III: expansions of graphs. SIAM J. Discrete Math., 29(2), 868-876 (2015)).
  • New Types of Singular Solutions of Conformal Q-curvature Equations
    Zongming Guo, Zhongyuan Liu, Fangshu Wan
    Acta Mathematica Sinica. 2026, 42(2): 481-507. https://doi.org/10.1007/s10114-026-4379-3
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    We construct radial and non-radial singular solutions $u \in C^4(B \backslash\{0\})$ with a non-removable singular point $x=0$ for the conformal Q-curvature equation: $$ \left\{\begin{array}{l} \Delta^2 u=\mathrm{e}^u \quad \text { in } B \backslash\{0\}, \\ \int_{B \backslash\{0\}} \mathrm{e}^{u(x)} d x＜\infty, \end{array}\right. $$ where $B=\left\{x \in \mathbb{R}^4:|x|＜1\right\}$. More precisely, we can construct two different types of singular solutions $u \in C^4(B \backslash\{0\})$ of the equation, satisfying $$ |x|^2 u(x) \rightarrow-D＜0 \quad \text { uniformly as }|x| \rightarrow 0 $$ for some $D_0 \geq 0$ and any $D>D_0 \geq 0$, and satisfying $$ u(x)=o\left(|x|^{-2}\right) \quad \text { uniformly as }|x| \rightarrow 0 . $$ Moreover, detailed asymptotic expansions near $x=0$ of these radial and non-radial singular solutions can be established. As an application, we can also obtain the existence of two types of solutions $u \in C^4\left(\mathbb{R}^4 \backslash \bar{B}\right)$ to the problem $$ \left\{\begin{array}{l} \Delta^2 u=\mathrm{e}^u \quad \text { in } \mathbb{R}^4 \backslash \bar{B}, \\ \int_{\mathbb{R}^4 \backslash \bar{B}} \mathrm{e}^{u(x)} d x＜\infty \end{array}\right. $$ satisfying $|x|^{-2} u(x) \rightarrow-D＜0$ uniformly as $|x| \rightarrow \infty$ for some $D_0 \geq 0$ and any $D>D_0$, and satisfying $u(x)=o\left(|x|^2\right)$ uniformly as $|x| \rightarrow \infty$.
  • Cocycle Hopf Algebra Structures on Free Modified Rota–Baxter Algebras by Vertex-decorated Rooted Trees
    Shanghua Zheng, Yukun Liu
    Acta Mathematica Sinica. 2026, 42(2): 508-540. https://doi.org/10.1007/s10114-026-4240-8
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    A modified Rota–Baxter operator originates from the convolution theorem for the Hilbert transformation by Tricomi. It also satisfies the modified classical Yang–Baxter equation discovered by Semenov-Tian-Shansky, and is applied to Lax equations and affine geometry of Lie groups later. In this paper, we provide an alternative explicit construction of free modified Rota–Baxter associative algebras from the perspective of combinatorial objects. First, we revisit a construction of free operated bialgebra structure on vertex-decorated rooted forests via a variant of the Hochschild 1-cocycle condition. Applying an isomorphism between two kinds of free operated algebras given by different carriers, we construct free modified Rota–Baxter algebras on vertex-decorated rooted forests. We then obtain a combinatorial coproduct on the free modified Rota–Baxter algebra by means of the universal property of free operated algebras, leading to a cocycle bialgebra structure and further a cocycle Hopf algebra structure on it.
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