中国科学院数学与系统科学研究院期刊网
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15 March 2025, Volume 41 Issue 3
    

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  • Strassen’s Law of the Iterated Logarithm under Sub-linear Expectations
    Wangyun Gu, Lixin Zhang
    Acta Mathematica Sinica. 2025, 41(3): 827-846. https://doi.org/10.1007/s10114-025-2759-8
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    We establish the Strassen's law of the iterated logarithm (LIL for short) for independent and identically distributed random variables with $\hat{\mathbb{E}}\left[X_1\right]=\hat{\mathcal{E}}\left[X_1\right]=0$ and $C_{\mathrm{V}}\left[X_1^2\right]<\infty$ under a sub-linear expectation space with a countably sub-additive capacity V. We also show the LIL for upper capacity with $\sigma$=$\bar{\sigma}$ under some certain conditions.
  • Different Hearts on Elliptic Curves
    Yucheng Liu
    Acta Mathematica Sinica. 2025, 41(3): 847-853. https://doi.org/10.1007/s10114-025-3286-3
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    The classical Mumford stability condition of vector bundles on a complex elliptic curve $X$, can be viewed as a Bridgeland stability condition on $D^b({\rm Coh}\,X)$, the bounded derived category of coherent sheaves on $X$. This point of view gives us infinitely many $t$-structures and hearts on $D^b({\rm Coh}\, X)$. In this paper, we answer the question which of these hearts are Noetherian or Artinian.
  • Finite Element Method Resolution and Error Estimators for p-Laplacian Problem with a New Boundary Condition
    Omar El Moutea, Hassan El Amri
    Acta Mathematica Sinica. 2025, 41(3): 854-872. https://doi.org/10.1007/s10114-025-3561-3
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    In this article, we discuss the approach to solving a nonlinear PDE equation, specifically the $p$-Laplacian equation, with a general (nonlinear) boundary condition. We establish the existence and uniqueness of the solution, subject to certain assumptions outlined in this paper. To solve our nonlinear problem using the Finite Element Method (FEM), we derive an appropriate variational formulation. Additionally, we introduce a study of the residual a posteriori-error indicator, establishing both upper and lower bounds to control the error. The upper bound is determined using averaging interpolators in some quasi-norms defined by Barrett and Liu. Furthermore, we prove the equivalence between the residual error and the true error $e=u-u_{h}$. Lastly, we perform a simulation of the $p$-Laplacian problem in the $L$-shape domain using a Matlab program in two-dimensional space.
  • Stochastic Differential Games of Mean-Field Dynamics and Second-Order Bellman–Isaacs Equations on the Wasserstein Space
    Tao Hao, Jie Xiong
    Acta Mathematica Sinica. 2025, 41(3): 873-907. https://doi.org/10.1007/s10114-025-2666-z
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    This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control. In our game, both the (lower and upper) value functions and the (lower and upper) second-order Bellman--Isaacs equations are defined on the Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$ which is an infinite dimensional space. The dynamic programming principle for the value functions is proved. If the (upper and lower) value functions are smooth enough, we show that they are the classical solutions to the second-order Bellman--Isaacs equations. On the other hand, the classical solutions to the (upper and lower) Bellman--Isaacs equations are unique and coincide with the (upper and lower) value functions. As an illustrative application, the linear quadratic case is considered. Under the Isaacs condition, the explicit expressions of optimal closed-loop controls for both players are given. Finally, we introduce the intrinsic notion of viscosity solution of our second-order Bellman--Isaacs equations, and characterize the (upper and lower) value functions as their viscosity solutions.
  • Additively Indecomposable Positive Definite Integral Lattices
    Ruiqing Wang
    Acta Mathematica Sinica. 2025, 41(3): 908-924. https://doi.org/10.1007/s10114-025-2562-6
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    In this paper, we obtain some sufficient and necessary conditions for indecomposable positive definite integral lattices with discriminants 2, 3, 4 and 5 over $\mathbb{Z}$ being additively indecomposable lattices. Using these results, we prove that there exist additively indecomposable positive integral quadratic lattices with discriminants 2, 3, 4 and 5 and rank greater than or equal to 2 but for 35 exceptions. In the exceptions there are no lattices with the desired properties. We also give a lifting theorem of additively indecomposable positive definite integral lattices.
  • Sharp Morrey Regularity for an Even Order Elliptic System
    Changyu Guo, Wenjuan Qi
    Acta Mathematica Sinica. 2025, 41(3): 925-937. https://doi.org/10.1007/s10114-025-3353-9
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    In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivière type: \begin{equation*} \Delta^{m}u=\sum_{l=0}^{m-1}\Delta^{l} \langle V_{l},du \rangle +\sum_{l=0}^{m-2} \Delta^{l}\delta (w_{l}du)+f\quad \text{ in } B^{2m}\label{eq: Longue-Gastel system} \end{equation*} under minimal regularity assumptions on the coefficients functions $V_l, w_l$ and that $f$ belongs to certain Morrey space. This can be regarded as a further extension of the recent $L^p$-regularity theory obtained by Guo--Xiang--Zheng [J. Math. Pures Appl. (9), 165, 286--324 (2022)], and generalizes [Proc. Amer. Math. Soc., 152(10), 4261--4268 (2024)], [Acta Math. Sci. Ser. B (Engl. Ed.), 44(2), 420--430 (2024)] for second and fourth order elliptic systems.
  • Tangent Measures of Homogeneous Cantor Sets Satisfying the Strong Separation Condition
    Xiangyu Liang, Yongtao Wang
    Acta Mathematica Sinica. 2025, 41(3): 938-974. https://doi.org/10.1007/s10114-025-3326-z
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    This paper investigates tangent measures in the sense of Preiss for homogeneous Cantor sets on ${{\mathbb{R}}^d}$ generated by specific iterated function systems that satisfy the strong separation condition. Through the dynamics of “zooming in” on typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of homogeneous Cantor sets on ${{\mathbb{R}}^d}$.
  • Finite Solvable Groups Whose Prime Graphs have Diameter 3
    Guohua Qian
    Acta Mathematica Sinica. 2025, 41(3): 975-984. https://doi.org/10.1007/s10114-025-2021-4
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    Let G be a finite group and $\pi(G)$ be the set of prime divisors of $|G|$. The prime graph G of G is the graph with vertex set $\pi(G)$, and different $p,q\in \pi(G)$ are joined by an edge if and only if G has an element of order $pq$. In this paper, we characterize the finite solvable groups whose prime graphs have diameter $3$.
  • Relative Auslander Bijection in Extriangulated Categories
    Tiwei Zhao
    Acta Mathematica Sinica. 2025, 41(3): 985-1014. https://doi.org/10.1007/s10114-025-2074-4
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    Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Auslander bijection in extriangulated categories. Namely, we introduce the notion of generalized ARS-duality relative to an additive subfunctor, and prove that there is a bijective triangle which involves the generalized ARS-duality and the restricted Auslander bijection relative to the subfunctor. We give the Auslander's defect formula in terms of the generalized ARS-duality, show the interplay of morphisms being determined by objects and a kind of extriangles, and characterize a class of objects which are somehow controlled by this kind of extriangles. We also give a realization for a functor relating the Auslander bijection and the generalized ARS-duality.
  • On Positively Complete Einstein Square Metrics
    Zhongmin Shen, Hongmei Zhu
    Acta Mathematica Sinica. 2025, 41(3): 1015-1022. https://doi.org/10.1007/s10114-025-2578-y
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    The well-known Berwald square metric is a positively complete and projectively flat Finsler metric with vanishing flag curvature. In this paper, we study a positively complete square metric on a manifold. We show a rigidity result that if the Ricci curvature is constant, then it must be isometric to the Berwald square metric. This is not true without assumption on the completeness of the metric.
  • Cotorsion Pairs on m-term Subcategories
    Yu Liu, Panyue Zhou
    Acta Mathematica Sinica. 2025, 41(3): 1023-1036. https://doi.org/10.1007/s10114-025-2286-7
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    Let C be a triangulated category. We define $m$-term subcategories on C induced by $n$-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on $2$-term subcategories $\mathcal{G}$ and support $\tau$-tilting subcategories on an abelian quotient of $\mathcal{G}$. If an $m$-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.
  • List Edge Colorings of Planar Graphs without Non-induced 7-cycles
    Li Zhang, Hajo Broersma, You Lu, Shenggui Zhang
    Acta Mathematica Sinica. 2025, 41(3): 1037-1054. https://doi.org/10.1007/s10114-025-2761-1
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    A graph G is edge-$k$-choosable if, for any assignment of lists $L(e)$ of at least $k$ colors to all edges $e\in E(G)$, there exists a proper edge coloring such that the color of $e$ belongs to $L(e)$ for all $e\in E(G)$. One of Vizing's classic conjectures asserts that every graph is edge-$(\Delta+1)$-choosable. It is known since 1999 that this conjecture is true for general graphs with $\Delta\leq4$. More recently, in 2015, Bonamy confirmed the conjecture for planar graph with $\Delta\geq8$, but the conjecture is still open for planar graphs with $5\leq\Delta\leq7$. We confirm the conjecture for planar graphs with $\Delta\ge 6$ in which every 7-cycle (if any) induces a $C_7$ (so, without chords), thereby extending a result due to Dong, Liu and Li.
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