中国科学院数学与系统科学研究院期刊网

15 September 2024, Volume 67 Issue 5
    

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  • Chun Xu XU, Li HE
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 807-829. https://doi.org/10.12386/B20230367
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    We study some properties of Toeplitz operators with positive operator-valued function symbols on the vector-valued exponential weighted Bergman spaces $A^p_{\varphi}(\mathcal{H})\ (1 < p < \infty)$. Firstly, we discuss when the Bergman projection from $L^p_{\varphi}(\mathcal{H})$ onto $A^p_{\varphi}(\mathcal{H})$ is bounded and get the dual of the vector-valued exponential weighted Bergman spaces. Secondly, we obtain several equivalent descriptions of Carleson condition to characterize the boundedness and compactness of Toeplitz operators on $A^p_{\varphi}(\mathcal{H})$. Finally, we consider the Schatten-$p$ class membership of Toeplitz operators acting on $A^2_{\varphi}(\mathcal{H})$.
  • Jia Lei SONG, Bao Gang XU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 830-842. https://doi.org/10.12386/A20230017
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    A hole is an induced cycle of length at least 4, a hole of odd length (resp. even length) is called an odd hole (resp. even hole). An HVN is a graph composed by a vertex adjacent to both ends of an edge in $K_4$. Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al. in [J. Combin. Theory B, 2010, 100: 313—331] proved that every (odd hole, $K_4$)-free graph is 4-colorable and is 3-colorable if it does not contain $H$ as an induced subgraph. In this paper, we use the idea and proving technique of Chudnovsky et al. to generalize this conclusion to $($odd hole, HVN$)$-free graphs. Let $G$ be an $($odd hole, HVN$)$-free graph. We prove that if $G$ contains $H$ as an induced subgraph, then it either has a special cutset or is in two classes of pre-defined graphs. As its corollary, we show that $\chi(G)\le \omega(G)+1$, and the equality holds if and only if $\omega(G)=3$ and $G$ has $H$ as an induced subgraph.
  • Ke Li PU, Shi Chun YANG, Qun Ying LIAO
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 843-858. https://doi.org/10.12386/A20220166
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    Based on the complete classification of the torsion subgroup by Mazur, and results of the related diophantine equation, we determine all elliptic curves defined over $\mathbb{Q}$ with a rational point of the order $n\ (n \geq 6, n\neq 11)$ and the conductor $p^{a}q^{b}r^{c}$, where $p, q, r$ are distinct primes, and $a, b, c$ are positive integers. In particular, an upper bound of the minimal discriminant for these elliptic curves are given.
  • Ting LUO
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 859-877. https://doi.org/10.12386/B20230285
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    This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible planar magnetohydrodynamics system, which governs the motions of a conducting fluid in an electro-magnetic field. Motivated by the relationship between planar magnetohydrodynamics system and Navier—Stokes system, we can prove that the solutions to the compressible planar magnetohydrodynamics system tend time-asymptotically to the traveling wave, provided that the initial disturbance is small and of integral zero.
  • Zheng Dong LI, Wen Rong PAN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 878-888. https://doi.org/10.12386/A20230060
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    The Lorenz-type maps are piecewise expanding maps with discontinuous points, the discontinuity comes from the singularities of Lorenz equations showing butterfly effect, and the observable statistical properties of such maps are given by the absolutely continuous invariant measures. In this paper, we consider the perturbation $f_t=f+tX\circ f$ of an improved Lorenz-type map $f $, and denote by $\mu_t$ the perturbation of the corresponding absolutely continuous measure $\mu$. We prove that if $X $ takes zero on all image sets of the discontinuous point of $f $, then its sensitivity formula $$\Psi(\lambda)=\sum\limits_{n=0}^\infty \lambda^n \int \mu(dx)X(x)\dfrac{\partial(\varphi(f^nx))}{\partial x},\quad\varphi\in C^1, $$ converges at $ \lambda = 1 $, thus the linear response formula $\frac{d}{dt}|_{t=0}\mu_t(\varphi)=\Psi(1)$ is established.
  • Wei Cong YE, Chang Lian LIU, Deng Pin LIU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 889-894. https://doi.org/10.12386/A20230067
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    For any quasitoric-manifold, $\pi :M^{2n}\to P^{n}$, its cohomology ring is expressed as $H^{\ast}(M^{2n},\mathbb{ Z}) =\mathbb{Z}[F_{1},F_{2},\ldots,F_{m}]/(\mathcal{I}_{P^{n}}+\mathcal{J}_ {P^{n}})$, where $\mathcal{F}(P)=\{F_{1}, F_{2}, \ldots,$ $F_{m}\}$ is the set of all co-one-dimensional surfaces in $P^{n}$. Taking any vertex $\upsilon= F_{i1}\cap F_{i2}\cap\cdots\cap F_{in}$ of $ P^{n}$, we prove that $\langle [F_{i1}F_{i2}\cdots F_{in}],[M^{2n}]\rangle=\pm1$, that is, $[F_{i1}F_{i2}\cdots F_{in}]$ is the generator of $H^{2n}(M^{2n},\mathbb{Z})$. Further we use this conclusion to discuss the rigidity of quasitoric-manifolds, and prove the following conclusions: If $f^{*}:H^{\ast}(M_{1}^{2n},\mathbb{Z})\to H^{\ast}(M_{2}^{2n},\mathbb{Z})$ is a ring isomorphism, then there exists a one-to-one mapping $\tilde{f}:{\rm Fix}(M_{1}^{2n})\to {\rm Fix}(M_{2}^{2n})$, where ${\rm Fix}(M^{2n})$ is the fixed point of $T^{n}$-acting on $M^{2n}$.
  • An Lan ZHAO, Jian Jun NIE
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 895-910. https://doi.org/10.12386/A20230050
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    We consider the following prescribed curvature problem of fractional operator: \begin{align} (-\Delta)^s u=K(y)u^{2_s^*-1},\quad u> 0,\quad u\in D^s(\mathbb{R}^N), \nonumber \end{align} where $N\geq 3$, $0 < s < 1$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $K(y)$ is a positive function. When $K(y)$ has a sequence of strictly local maximum points moving to infinity, we use the finite dimensional reduction method to prove the existence of any finitely many multi-bubbling solutions to the above problem. These solutions concentrate at $k$ different local maximum points of $K(y)$.
  • La Mei YUAN, Jia Xin LI
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 911-925. https://doi.org/10.12386/B20210246
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    The $q$-deformed $W(2,2)$ algebra is a Hom-Lie algebra, denoted by $\ W ^ q$. In this paper, we compute its second cohomology with values in the adjoint module by elementary and direct calculations, and obtain that the second cohomology group $H^2(\ W ^q,\ W ^ q)$ is two-dimensional.
  • Hong Jian LI, Yu Qing HE
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 926-938. https://doi.org/10.12386/A20210193
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    In this paper, we investigate the reciprocal sums of the cubes of odd and even terms in the Fibonacci sequence and we obtain two interesting identities for the Fibonacci numbers.
  • Jun HE, Xing Xing HUANG, Guang Yu AN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 939-946. https://doi.org/10.12386/A20230018
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    Let $\mathcal A$ be a commutative unital ${\rm C}^*$-algebra with the unit element $e$ and $\mathcal M$ be a full Hilbert $\mathcal A$-module. Denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$ and by $\mathcal M'$ the set of all bounded $\mathcal A$-linear mappings from $\mathcal M$ into $\mathcal A$. In this paper, we prove that if there exist $x_0$ in $\mathcal M$ and $f_0$ in $\mathcal M'$ such that $f_0(x_0)=e$, then every $\mathcal A$-linear Lie derivation $\delta$ on End$_{\mathcal A}(\mathcal M)$ is standard. That is, $\delta$ can be decomposed into $d+\tau$, where $d$ is a $\mathcal A$-linear derivation, and $\tau$ is a $\mathcal A$-linear mapping of central value such that $\tau(AB)=\tau(BA)$ for any $A,B\in {\rm End}_{\mathcal A}(\mathcal M)$.
  • Jin Lian ZHANG, Xu Hui PENG, Qi Yao CHEN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 947-961. https://doi.org/10.12386/A20230012
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    Sequences of genus polynomials for what became known as linear (or $H$-linear) families of graphs have been studied for more than 30 years. Most of previous papers concerning them aim to find recursions and expressions for genus (and Euler genus) polynomials of specific families, or try to prove the property of log-concavity. Recently, under some conditions, some researches reveal that the embedding distributions of generalized $H$-linear graph families $\{G_n^\circ \}$ will tend to normal distributions when $n$ tends to infinity (see [19]). Based on this previous work, in this article, we prove that the order of the convergence rate is $\frac{1}{\sqrt{n}}$. We also explain that, for the convergence rate obtained in this paper, it can been considered as optimal. In the end, we use some concrete examples to demonstrate our result.
  • George Xian Zhi YUAN
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 962-986. https://doi.org/10.12386/B20220226
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    The goal of this paper is to develop new fixed points, best approximation, and Leray—Schauder alternative for single-valued and (quasi) upper semicontinuous (QUSC) set-valued mapping in $p$-vector spaces and locally $p$-convex spaces, where $p \in (0, 1]$. The fixed point theorem established in this paper is a positive answer to Schauder conjecture in $p$-vector spaces and locally $p$-convex spaces; the corresponding best approximation theorem and the principle of Leray—Schauder alternative are also the fundamental tools in nonlinear functional analysis under the framework of $p$-vector spaces and locally $p$-convex spaces. These new results unify and generalize the theoretical results existing in the current mathematical literature, and they are also the continuation and in-depth development of the recent work did by Yuan [$Fixed$ $Point$ $Theory$ $Algorithms$ $Sci$. $Eng$., 2022, 2022: Paper Nos. 20, 26], and related references.
  • Xiao Dan YUAN, Wen Peng ZHANG
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 987-994. https://doi.org/10.12386/A20220077
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    The main purpose of this paper is using the elementary methods, the number of the solutions of some congruence equations and the properties of the classical Gauss sums to study the calculating problem of the fifth power mean of one kind two-term exponential sums, and give the exact calculating formula for it.
  • Wen Tao HUANG, Qin Long WANG, Chao Xiong DU
    Acta Mathematica Sinica, Chinese Series. 2024, 67(5): 995-1008. https://doi.org/10.12386/B20210641
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    In this paper, we present a method to study isochronous centers in 3-dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and recursive formulas to obtain them are given. The conditions for the isochronicity of a center are determined by the computation of isochronous constants for which there is no need to compute center manifolds of the three dimensional systems. Then the isochronous center conditions of two specific systems are discussed as an application of our method. Our method is a generalization of the formal series method proposed by Yirong Liu for determining the order of a fine focus of planar differential systems. This method with the recursive formulas can be easily implemented on a computer using a computer algebra system.