15 May 2025, Volume 68 Issue 3

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  Wavefronts of a General Nonlocal Dispersal Equation with Discrete State-dependent Delays

  Ruijun Xie, Rong Yuan

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 397-415. https://doi.org/10.12386/A20230165

  Abstract ( 148 ) Download PDF ( 169 )   Knowledge map   Save

  This paper is concerned with the wavefront solutions of general nonlocal diffusion equations with discrete state-dependent delays. Firstly, the existence of wavefront solutions is proved by constructing the upper and lower solutions and invariant set in conjunction with the Schauder's fixed point theorem; then, the strict monotonicity of the wavefront solutions is given; finally, the asymptotics of the wavefront solutions at the minimum wave speed and the existence of the minimum wave speed are proved by using Ikehara's theorem.
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  List 2-distance Coloring of Planar Graphs without 4- and 5-cycles

  Jiahao Yu, Min Chen, Yiqiao Wang

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 416-432. https://doi.org/10.12386/A20230166

  Abstract ( 66 ) Download PDF ( 76 )   Knowledge map   Save

  Let $G=(V,E)$ be a graph. A $2$-distance $k$-coloring of a graph $G$ is to arrange $k$ colors for all vertices of $G$ such that every pair of vertices with the distance at most $2$ in $G$ is colored differently. A $2$-distance $L$-coloring of a graph $G$ is to give a list assignment $L=\{L(v)\mid v\in V(G)\}$ of $G$ such that $G$ has a $2$-distance coloring $\pi$ with $\pi (v)\in L(v)$ for each $v\in V(G)$. A list $2$-distance $k$-coloring of a graph $G$ is to give any list assignment $L$ of $G$ with $|L(v)|\ge k$ for each $v\in V(G)$ such that $G$ is $2$-distance $L$-colorable. In this paper, we will prove that every planar graph with the maximum degree $\mathrm{\Delta }\ge 24$ containing neither $4$-cycles nor $5$-cycles is $2$-distance $(\mathrm{\Delta }+3)$-choosable.
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  Oscillation of Subfractional Brownian Motion and Its Application

  Nana Luan, Li Wang

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 433-446. https://doi.org/10.12386/A20230167

  Abstract ( 51 ) Download PDF ( 61 )   Knowledge map   Save

  Let $X^H=\{X^H(t),t\in {\mathbb{R}}_{+}\}$ be a subfractional Brownian motion in $\mathbb{R}$ with index $H\in (0,1)$. We study the oscillation of $X^H$ and get the almost sure weak approximation of the occupation measure as an application.
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  Multi-step Inertial Regularized Subgradient Extragradient Method for Solving the Hierarchical Variational Inequality

  Bingnan Jiang, Yuanheng Wang, Jen-Chih Yao

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 447-461. https://doi.org/10.12386/A20230161

  Abstract ( 41 ) Download PDF ( 61 )   Knowledge map   Save

  In this paper, based on Popov-type subgradient extragradient method, We construct a new multi-step inertial regularized algorithm for solving the hierarchical variational inequality problem with the generalized Lipschitz mapping over the common solution set of a variational inequality problem and a null point problem in Hilbert spaces. We prove that this algorithm has a strong convergence theorem under certain conditions. Finally, we give some numerical experiments to illustrate the effectiveness and advantages of our new iterative algorithm. The results obtained here extend and improve many recent ones in the literature.
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  Harish-Chandra Modules over the Two-Parameter Deformed Virasoro Algebra of Hom-Type

  Wen Zhou, Qiuli Fan, Yongsheng Cheng

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 462-476. https://doi.org/10.12386/A20230104

  Abstract ( 33 ) Download PDF ( 39 )   Knowledge map   Save

  In this paper, we mainly study the representation theory of Hom-Lie algebras. More specifically, using the bosonic oscillators, we construct a two-parameter deformed Virasoro algebra, which is a Hom-Lie algebra. With suitable hypotheses in each case, we construct several kinds of Harish-Chandra modules of the two parameters deformed Virasoro algebra, and classify indecomposable Harish-Chandra module of an intermediate series.
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  The Quantum Bernoulli Noise Indexed by Γ

  Yulan Zhou, Cuicui Liu, Qingqing Yang, Wanying Wei, Zhouning Wang

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 477-490. https://doi.org/10.12386/A20230096

  Abstract ( 42 ) Download PDF ( 30 )   Knowledge map   Save

  In this paper, we define a family of bounded linear operators on the square integrable Bernoulli functional space $L^2(M)$, with the finite power set $\mathrm{\Gamma }$ of $\mathbb{N}$ as the index set, which includes QBNs, preserving some properties of QBNs$\{{\partial }_k,{\partial }_k^{\ast };k\ge 0\}$, which is called $\mathrm{\Gamma }$-QBNs; the discussion shows that $\mathrm{\Gamma }$-QBNs has some new properties, such as the quasi-exchangeability, the quasi-nilpotent property, the absorbed anti-commutation relation, the canonical binomial anti-commutation relation and the multi-indicator absorbed anti-commutation relation, which is a multi-index generalization of the canonical anti-commutation relation of QBNs; in particular, $\mathrm{\Gamma }$-QBNs is ``quantum generators" of the full space $L^2(M)$. Also, its isochronous mixture product is an ``orthogonal" projection operator on $L^2(M)$, and $\mathrm{\Gamma }$-QBNs can be represented by the Dirac operator generated by the orthogonal basis of $L^2(M)$.
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  Infinitely Many Solutions for Higher-order Schrödinger Equation with Competing Potentials

  Ting Liu

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 491-510. https://doi.org/10.12386/A20230163

  Abstract ( 66 ) Download PDF ( 68 )   Knowledge map   Save

  In this paper, we consider the following higher-order Schrödinger equation with critical growth: $$(-\mathrm{\Delta }{)}^{m}u+V(y)u=Q(y)u^{m^{\ast }-1},u>0\text{in}{\mathbb{R}}^N,u\in {\mathcal{D}}^{m,2}({\mathbb{R}}^N),$$ where $m^{\ast }=\frac{2N}{N-2m},N\ge 4m+1$, $m\ge 2$ is an integer, $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$ and $V(y)=V(|y^{\prime }|,y^{″})$ and $Q(y)=Q(|y^{\prime }|,y^{″})$ are bounded non-negative functions in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N-2}$. By using finite dimensional reduction argument and local Pohozaev type identities, we show that if $N\ge 4m+1$, $Q(r,y^{″})$ has a critical point $(r_0,y_0^{″})$ satisfying $r_0>0$, $Q(r_0,y_0^{″})>0$, $D^{\alpha }Q(r_0,y_0^{″})=0,$ $|\alpha |\le 2m-1$ and $\mathrm{d}\mathrm{e}\mathrm{g}(\mathrm{\nabla }(Q(r,y^{″})),(r_0,y_0^{″}))\ne 0$, and $\frac{1}{(2m-1)!m^{\ast }}\sum _{|\alpha |=2m}{D}^{\alpha }Q(r_0,y_0^{″}){\int }_{{\mathbb{R}}^N}{y}^{\alpha }{U}_{0,1}^{m^{\ast }}-mV(r_0,y_0^{″}){\int }_{{\mathbb{R}}^N}{U}_{0,1}^2<0$, then the above problem has infinitely many solutions, whose energy can be arbitrarily large. Different from that in [Commun. Contemp. Math., 2022, 24: Paper No. 2050071], in our case, the higher order derivative of $Q(r_0,y_0^{″})$ plays an important role in the construction of bubble solutions. Besides, it implies that the potential $V(r_0,y_0^{″})$ can influence the sign of higher order derivative of $Q(r_0,y_0^{″})$.
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  The Classification of Irreducible Conformal S(p)-Modules of Finite Rank

  Guobo Chen, Fu Liu, Pan Wei

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 511-516. https://doi.org/10.12386/A20240005

  Abstract ( 37 ) Download PDF ( 45 )   Knowledge map   Save

  The classification of irreducible conformal $\mathcal{S}(p)$-modules of finite rank was given by H. Chen. In this paper, we use a different way to give a proof of this classification.
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  Some Counterexamples to Stahl’s Conjecture

  Yangyang Huang, Wenwen Liu, Yichao Chen

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 517-526. https://doi.org/10.12386/A20240017

  Abstract ( 43 ) Download PDF ( 53 )   Knowledge map   Save

  The characters of the symmetry group contain rich information on graph embeddings. In his early paper, Stahl [Region distributions of graph embeddings and string numbers, Discrete Mathematics, 1990, 82: 57-78] proposed a conjecture about the upper bound for a class of characters of the symmetry group in the study of the asymptotic estimation of the number of embeddings of some small diameter graphs. In this paper, we disprove the conjecture by giving some counterexamples.
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  Post Lie–Yamaguti Algebras, Relative Rota-Baxter Operators of Nonzero Weights, and Their Deformations

  Yu Qiao, Senrong Xu, Jia Zhao

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 527-544. https://doi.org/10.12386/B20230209

  Abstract ( 30 ) Download PDF ( 46 )   Knowledge map   Save

  In this paper, we introduce the notions of relative Rota—Baxter operators of weight 1 on Lie—Yamaguti algebras and post-Lie-Yamaguti algebras. A post-Lie-Yamaguti algebras describes an underlying algebraic structure of relative Rota—Baxter operators of weight 1. We clarify the relationship between Lie—Yamaguti algebras and post-Lie-Yamaguti algebras. Besides, we establish the cohomology theory of relative Rota—Baxter operators of weight 1. Consequently, we make use of this cohomology to characterize linear deformations of relative Rota—Baxter operators of weight 1 on Lie—Yamaguti algebras.
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  Conformal Derivations and Representations of a Class of Loop Lie Conformal Superalgebras

  Chunguang Xia, Ying Wu, Huidong Wang

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 545-559. https://doi.org/10.12386/A20230053

  Abstract ( 25 ) Download PDF ( 29 )   Knowledge map   Save

  In this paper, we study conformal derivations and representations of a class of loop Lie conformal superalgebras $S(a,b)$, where $a,b$ are complex parameters. First, we determine conformal derivations of $S(a,b)$, and show that $S(a,b)$ admits outer conformal derivations if and only if $a=1$. Then, we completely classify the conformal modules of rank $(1+1)$ over $S(a,b)$. Finally, we classify the $\mathbb{Z}$-graded free intermediate series modules over $S(a,b)$ for $a\ne 1$.
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  Tauberian Theorem for Asymptotically Almost Periodic Sequences and Its Application

  Weigang Jian, Zheming Zheng

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 560-572. https://doi.org/10.12386/A20240027

  Abstract ( 29 ) Download PDF ( 33 )   Knowledge map   Save

  With the help of Tauberian theorems for asymptotically almost periodic sequences, the asymptotic behavior of solutions to the difference equations: $$x(n+1)=Tx(n)+y(n),n\in \mathbb{J}\in \{\mathbb{Z},{\mathbb{Z}}^{+}\}$$ has been studied in this paper. Here $x(n),y(n)\in X$ and $T$ is a bounded linear operator on a Banach space $X$. We show that if $c_0⊈X$, $y$ is an asymptotically almost periodic sequence, and the intersection of the spectrum set $\sigma (T)$ of $T$ with the unit circle $\mathrm{\Gamma }$ is finite, then the bounded solution $x$ of the equation is remotely almost periodic sequence (weaker than asymptotically almost periodic sequence). It is worth noting that although the conclusions of the established Tauberian theorem for asymptotically almost periodic sequence and the spectral set determination theorem for difference equations are slightly weaker than asymptotically almost periodic, they completely eliminate the assumption of transitivity in the reference [J. Differential Equations, 1995, 122, 282—301].
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  On Extension of Norm-additive Maps between the Positive Unit Spheres of (Σℓp)_c0

  Longfa Sun, Yipeng Zhang, Yinghua Sun

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 573-582. https://doi.org/10.12386/A20230139

  Abstract ( 37 ) Download PDF ( 39 )   Knowledge map   Save

  Assume that $1\le p\le \mathrm{\infty }$ and $S_{(\sum {\ell }_p{)}_{c_0}}^{+}=\{x\in (\sum {\ell }_p{)}_{c_0}:x\ge 0;\parallel x\parallel =1\}$ is the positive unit sphere of $(\sum {\ell }_p{)}_{c_0}$. Let $f:S_{(\sum {\ell }_p{)}_{c_0}}^{+}\to S_{(\sum {\ell }_p{)}_{c_0}}^{+}$ be a norm-additive map (preserving norm of sums), i.e., $$\parallel f(x)+f(y)\parallel =\parallel x+y\parallel ,\mathrm{\forall }x,y\in S_{(\sum {\ell }_p{)}_{c_0}}^{+}.$$ We prove that if $f$ is bijective, then $f$ can be extended to a linear surjective isometry from $(\sum {\ell }_p{)}_{c_0}$ onto itself if and only if 1<p≤∞.
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  Statistical Inference for Two-stage Outcome-dependent Sampling Design

  Jichang Yu, Yongxiu Cao

  Acta Mathematica Sinica, Chinese Series. 2025, 68(3): 583-596. https://doi.org/10.12386/A20210156

  Abstract ( 47 ) Download PDF ( 36 )   Knowledge map   Save

  Outcome-dependent sampling is a well-known cost-effective sampling design in biomedical and epidemiologic studies. In this article, we propose a two-stage outcome-dependent sampling design in the framework of an accelerated failure time model. We develop the smoothed estimated Gehan estimating equation with the kernel function method to estimate the regression parameters for the data collocated by the proposed design. We establish the consistency and asymptotic normality of the proposed estimator. Simulation studies are conducted to evaluate the finite-sample performance of the proposed estimator and the results show the proposed estimator is more efficient than other competitive estimators. A real data set from the National Wilms' Tumor Study Group is analyzed to illustrate the proposed method.