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15 July 2026, Volume 42 Issue 7
    

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  • Limit Theorems for Product of Spatial Average of Parabolic Anderson Model
    Jingyu Li, Yong Zhang, Zhiqiang Tang
    Acta Mathematica Sinica. 2026, 42(7): 1727-1741. https://doi.org/10.1007/s10114-026-4216-8
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    Let $u(t,x)$ be the solution to the parabolic Anderson model on $\mathbb{R}_+\times \mathbb{R}$ driven by a space-time white Gaussian noise. In this paper, we prove the central limit theorem and almost sure central limit theorem for product of spatial average of the form $\prod_{k=1}^n S_k$ as $n\rightarrow\infty$ for fixed $t>0$, where $S_{k}=\int_0^k u(t, x) dx$.
  • The Relationship Between Meromorphic Functions with Three IM Sharing Values and One CM Sharing Value
    Lan Hu, Min Su, Yuhua Li
    Acta Mathematica Sinica. 2026, 42(7): 1742-1754. https://doi.org/10.1007/s10114-026-4474-5
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    In this paper, we investigate the uniqueness of meromorphic functions sharing four values. We show that if $f(z)$ and $g(z)$ are distinct non-constant meromorphic functions sharing $0,\, 1,\, c$ IM and $\infty$ CM, then either $f(z)$ and $g(z)$ share $0,\, 1,\, c,\, \infty$ CM or $\frac{1}{13}T(r,f)-2\bar{N}(r,f)\leq N^{1)}_{E}(r,0)+N^{1)}_{E}(r,1)+N^{1)}_{E}(r,c)+S(r,f)\leq \frac{4}{3}T(r,f)-\frac{2}{3}\bar{N}(r,f)$ holds.
  • On the Saturated Set of One-dimensional Maps with Some Hyperbolicity
    Cao Zhao
    Acta Mathematica Sinica. 2026, 42(7): 1755-1767. https://doi.org/10.1007/s10114-026-4523-0
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    The paper is mainly concerned with the multifractal analysis for smooth dynamical systems in dimension one without uniform hyperbolicity. We characterize the Hausdorff dimension of the saturated set obtained from the empirical measure by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modeled by an abstract dynamical system.
  • Yamabe-Harmonic Flow on Locally Conformally Flat Manifolds
    Zihao Wang
    Acta Mathematica Sinica. 2026, 42(7): 1768-1784. https://doi.org/10.1007/s10114-026-3129-x
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    In this paper we introduce a new geometric flow consisting of the Yamabe flow coupled with harmonic heat flow of a function on a closed manifold $M$, or shortly Yamabe-harmonic flow. It is define by \[\begin{cases} \displaystyle \frac{\partial}{\partial t}g=-(R-\phi)g,\\ \phi=(n-1)\Delta_g\phi. \end{cases}\] To begin with we establish the short-time extsience via conformal transformation for any smooth initial data. Furthermore we derive interior-in-time derivative estimates for the Riemannian curvature and Lapse function and obtain the global existence of Yamabe-harmonic flow. Moreover, compact gradient Yamabe-harmonic solitons are proved to be manifolds with $R=\phi=\mathrm{Constant}$. Eventually we obtain the classification of singularities by blow-up rate of AC curvature, which is analogous to that of Ricci-harmonic flow and Ricci-connection flow. The relationship between singularity models and solitons shall appear in forthcoming work.
  • Manin’s Question and Hopf Algebras via the Boson-fermionic Quantum Weyl Algebra
    Ge Feng, Naihong Hu, Xiaoting Zhang, Rushu Zhuang
    Acta Mathematica Sinica. 2026, 42(7): 1785-1820. https://doi.org/10.1007/s10114-026-4574-2
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    The aim of this article is to introduce a boson-fermionic quantum Weyl algebra $\mathcal W_q(2(m|n))$ of type $A(m|n)$, which is the quantum differential operators (QDO) algebra (also denoted by $\textrm{Diff}_q(\Omega_q)$) defined on $\Omega_q(m|n)$, the quantum Grassmann super-algebra $\Omega_q(m|n)$ we defined. Here we develop a constructive approach of Radford--Majid bosonization theory to yield some pointed Hopf algebras via QDO. This addresses a basic problem due to Manin, namely, any QDO approach might help to yield new Hopf algebras (see § 2.2. Basic problem, p.1010 of Manin's paper in 1992). Remarkably, $\mathcal W_q(2(m|n))$ itself is not a Hopf algebra, however, it contains some interesting pointed Hopf algebras as its subquotient objects, like the bosonization $\mathfrak A_q(m|n)$ of the quantum Manin $(m|n)$-superspace $A_q^{m|n}$, the bosonization $\mathfrak G_q(m|n)$ of $\Omega_q(m|n)$, the multi-rank Taft algebras of $2(m|n)$-type, the homomorphic image of pointed Hopf algebra $\mathcal U_q(\mathfrak{gl}(m|n))$ over $\Omega_q(m|n)$ or $\Omega_q^!(m|n)$ (the quantum dual Grassmann superalgebra), etc.
  • Normalized Solutions for Nonhomogeneous Chern-Simons-Schrödinger Equations with Critical Exponential Growth
    Chenlu Wei, Sitong Chen, Xin'ao Zhou
    Acta Mathematica Sinica. 2026, 42(7): 1821-1847. https://doi.org/10.1007/s10114-026-4581-3
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    This paper investigates the existence of normalized solutions for the following Chern--Simons--Schrödinger equation: \begin{align*} \left\{\!\! \begin{array}{ll} -\Delta u+\lambda u+\bigg(\dfrac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}+\displaystyle\int_{\vert x\vert}^{\infty}\frac{h(s)}{s}u^{2}(s) ds\bigg)u =({\rm e}^{u^2}-1)u+g(x), \quad x\in \mathbb{R}^2, \\ u\in H_r^1(\mathbb{R}^2),\quad \displaystyle\int_{\mathbb{R}^2}u^2 dx=c, \\ \end{array} \right. \end{align*} where $c>0$, $\lambda\in \mathbb{R}$ acts as a Lagrange multiplier and $g\in \mathcal {C}(\mathbb{R}^2,[0,+\infty))$ satisfies suitable assumptions. In addition to the loss of compactness caused by the nonlinearity with critical exponential growth, the intricate interactions among it, the nonlocal term, and the nonhomogeneous term significantly affect the geometric structure of the constrained functional, thereby making this research particularly challenging. By specifying explicit conditions on $c$, we subtly establish a structure of local minima of the constrained functional. Based on the structure, we employ new analytical techniques to prove the existence of two solutions: one being a local minimizer and one of mountain-pass type. Our results are entirely new, even for the Schrödinger equation that is when nonlocal terms are absent. We believe our methods may be adapted and modified to deal with more constrained problems with nonhomogeneous perturbation.
  • Analytical Smoothing Effect of Solution for the MHD-Boussinesq Equations
    Feng Cheng, Xiuhui Wang
    Acta Mathematica Sinica. 2026, 42(7): 1848-1868. https://doi.org/10.1007/s10114-026-4624-9
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    In this paper we study the analytical smoothing effect of Cauchy problem for the incompressible MHD-Boussinesq equations. Precisely, we used the Fourier Gevrey space method to show that the Sobolev $H^1$-solution to the incompressible MHD-Boussinesq equations in periodic domain is analytical for both spatial variable and time variable.
  • One Note on n-Sensitivity in Mean Forms
    Zhongxuan Yang, Xiaojun Huang
    Acta Mathematica Sinica. 2026, 42(7): 1869-1880. https://doi.org/10.1007/s10114-026-4584-0
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    In 2022, Li, Ye and Yu introduced multivariate sensitivity version of notions of mean $m$-Sensitivity and $m$-sensitivity in the mean for $m\geq 2$. In this manuscript, we mainly focus on the investigation of multivariate sensitivity in mean forms. First, we prove that for a linear system, the equivalence between mean $m$-sensitivity and $m$-sensitivity in the mean holds without any more conditions. Subsequently, we introduce the notion of $m$-equicontinuity in the mean, and obtain an Auslander-Yorke type dichotomy between $m$-equicontinuity in the mean and $m$-sensitivity in the mean for minimal systems. As a consequence, we demonstrate that the equivalence between mean $m$-sensitivity and $m$-sensitivity in the mean is valid under the condition of minimality for a general system, thereby affirmatively resolving the conjecture proposed in [Li, J., Ye, X. D., Yu, T.: Equicontinuity and sensitivity in mean forms. J. Dynam. Differential Equations, 34, 133-154 (2022)].
  • An Lp Estimate for a Class of Flag Trilinear Pseudo-differential Operators
    Jing Qiao, Lu Zhang
    Acta Mathematica Sinica. 2026, 42(7): 1881-1898. https://doi.org/10.1007/s10114-026-4607-x
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    In this work, we prove an $L^{p_1}\times L^{p_2} \times L^{p_3} \to L^p$ estimate for a trilinear pseudo-differential operators with flag-type symbols taking the form of $a(x,\xi_1,\xi_2,\xi_3)b(x,\xi_1,\xi_2+\xi_3)$. The study of such operators is motivated by investigations on the corresponding flag Fourier multipliers studied by Muscalu [Rev. Mat. Iberoam., 23, 705-742 (2007); Contemp. Math., 505, 131-151 (2010)], Germain, Masmoudi, Shatah [J. Math. Pures Appl., 97, 505-543 (2012); Ann. of Math., 175, 691-754 (2012)] and Miyachi, Tomita [Math. Z., 282, 577-613 (2016)].
  • Legendrian Submanifolds of ${\mathbb S^{2n+1}(\tilde{c})}$ with Constant Sectional Curvature and Conformal Maslov Form
    Xiuxiu Cheng, Qian Li, Zejun Hu
    Acta Mathematica Sinica. 2026, 42(7): 1899-1920. https://doi.org/10.1007/s10114-026-5075-z
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    In this paper, we study Legendrian submanifolds of the Sasakian space form $\mathbb S^{2n+1}(\tilde{c})$ with constant sectional curvature and conformal Maslov form. As main results, we first classify the $2$-dimensional Legendrian submanifolds of $\mathbb S^{5}(\tilde{c})$ with constant sectional curvature and conformal Maslov form; then we establish a complete classification of the Legendrian submanifolds of $\mathbb S^{2n+1}(\tilde{c})$ with constant sectional curvature and $C$-parallel mean curvature vector field.
  • On Trace Range of Products of Three Projections
    Linzhe Huang, Wenming Wu, Yifeng Ye
    Acta Mathematica Sinica. 2026, 42(7): 1921-1936. https://doi.org/10.1007/s10114-026-5379-z
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    The trace range of products of three projections in a finite factor $(\mathcal{M},\tau)$, with their trace equal to $\lambda$ is studied in this article. When $\lambda \leq \frac{1}{2}$, the corresponding boundary curve of the trace range is shown to be a degenerate elliptic curve. For $n \geq 3$, when $\mathcal{M}$ is of type $\mathrm{I}_n$ and $\lambda=\frac{n-1}{n}$, it is shown that the corresponding boundary curve of the trace range is an algebraic curve in the complex plane.
  • On the Metric Mean Dimensions of Divergence Points
    Tian Zhan, Yong Ji, Yunping Wang
    Acta Mathematica Sinica. 2026, 42(7): 1937-1956. https://doi.org/10.1007/s10114-026-5407-z
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    For the topological dynamical system with infinite entropy and the specification property, in the context of continuous and affine deformations of empirical measures, the set of divergence points is either empty, or its Bowen and packing metric mean dimensions equal those of the full phase space. We also establish the variational principles of Bowen and packing metric mean dimension for the sup set.
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