中国科学院数学与系统科学研究院期刊网

15 January 2022, Volume 65 Issue 1
    

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  • Fang YU, Wen Jing CHEN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 1-14. https://doi.org/10.12386/A2022sxxb0001
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    We consider a class of nonlocal N-Kirchhoff type problems involving a singular exponential critical growth nonlinearity:m(||u||N)(-ΔNu + V (x)|u|N-2u)=f(x,u)/|x|β + ϵh(x), in RN, where N ≥ 2,||u||N=∫RN (|∇u|N + V (x)|u|N)dx, ΔNu=div(|∇u|N-2u) is the N-Laplacian, m:R+ → R+ is a Kirchhoff function, V:RN → R is a continuous potential, f:RN×R → R is a continuous function, and behaves like eα|s|N/N-1 when|s|→ ∞ for some α > 0, 0 ≤ β < N, h(x) ∈ (W01,N(RN))*, h(x) ≥ 0 and h(x) ≢ 0, ϵ is a small positive parameter. Applying variational methods together with singular Trudinger-Moser inequality in the whole RN, when is small enough, we obtain the existence and multiplicity of solutions.
  • Zheng YIN, Zheng YIN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 15-24. https://doi.org/10.12386/A2022sxxb0002
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    We define unstable local entropies for arbitrary Borel probability measures in partially hyperbolic systems. In order to characterize the multifractal spectrum of unstable local entropies, we introduce the concept of unstable (q, μ)-entropy, provide some basic properties of (q, μ)-entropy and establish a relation formula between the Bowen unstable entropy of the multifractal spectrum and the (q, μ)-entropy.

  • Xin SONG, Ce Zhong TONG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 25-32. https://doi.org/10.12386/A2022sxxb0003
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    In this paper, the dominating set of the Bergman space in the unit ball are characterized in terms of the pseudohyperbolic metric ball. Our method is to generalize Luecking's three key lemmas on the unit disc to the unit ball. We then apply those three lemmas to give a complete description of the dominating set of the Bergman space on the unit ball.

  • Xu Jie YANG, Zhao Yin XIANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 33-52. https://doi.org/10.12386/A2022sxxb0004
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    Under the natural assumption of saturation effect, this paper proves the global existence and uniform boundedness of the classical solutions to the 3D initial boundary value problem for a double chemotaxis-Stokes system. Due to the strong nonlinearity in the system, the method developed in this paper can be applied to the related models for the coral spawning, which have attracted much attention recently.

  • Jin WANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 53-66. https://doi.org/10.12386/A2022sxxb0005
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    The present paper is devoted to the study of the so-called bivariate partial theta function which is first introduced by the author and contains the classical partial theta function as a special case. We focus on its possible product formula, recurrence relation, and series expansion and so on. As main results, we establish a product formula of any two bivariate partial theta functions. It is a generalization of Andrews-Warnaar's product formula for the classical partial theta functions. At the same time, we obtain a second order recurrence relation satisfied by this bivariate partial theta function. Finally, we present two series expansions of the bivariate partial theta function θ(q, x; ab) with respect to {θ(q, axqn; b)|n ≥ 0} and {θ(q, xqn; b)|n ≥ 0}, respectively. As further applications of these results, we also find a product formula of two 3φ2 series and a ternary representation of the bivariate partial theta function.

  • Li Li YANG, Xiao Hong CAO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 67-76. https://doi.org/10.12386/A2022sxxb0006
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    Let H be a complex infinite dimensional Hilbert space. B(H) denotes the algebra of all bounded linear operators on H. In this paper, we characterize the operators in B(H) for which f(T) satisfies Weyl's theorem, where f denotes the analytic function on some neighbourhood of the spectrum of T. Also, the relationships between Weyl's theorem for functions of operators and the stability of Weyl's theorem are explored.

  • Dan Ni GUO, Gang CAI
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 77-88. https://doi.org/10.12386/A2022sxxb0007
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    We introduce a new algorithm for solving pseudomonotone variational inequality problems and fixed point problems by using the subgradient extragradient method. A weak convergence theorem of proposed algorithm is obtained under some suitable assumptions imposed on the parameters. The results obtained in this paper extend and improve many recent ones in the literature.

  • Hong Jian LI, Ruo Ting LIU, Ping Zhi YUAN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 89-114. https://doi.org/10.12386/A2022sxxb0008
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    Let Z and N be the set of all integers and positive integers, respectively. Mm (Z) be the set of m×m matrix over Z where m ∈ N. In this paper, by using the result of Fermat's Last Theorem, we show that the following second-order matrix equation has only trivial solutions:Xn + Yn=λnI (λ ∈ Z, λ ≠ 0, X, YM2(Z)), where X has an eigenvalue that is a rational number and n ∈ N, n ≥ 3; By using the result of primitive divisors, we show that the second-order matrix equation Xn +Yn=(±1)nI (n ∈ N, n ≥ 3, X, YM2(Z)) has nontrivial solutions if and only if n=4 or gcd(n,6)=1 and all nontrivial solutions are given; By constructing integer matrix, we show that the following matrix equation has an infinite number of nontrivial solutions:∀n ∈ N, Xn + Yn=λnI (λ ∈ Z, λ ≠ 0, X, YMn(Z)); X3 + Y3=λ3I (λ ∈ Z, λ ≠ 0, m ∈ N, m ≥ 2, X, YMm(Z)).

  • Zhong Yuan LIU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 115-122. https://doi.org/10.12386/A2022sxxb0009
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    We study the following polyharmonic Dirichlet problems in a punctured unit ball

    where B is the unit ball in RN, ν is the unit outward normal vector of ∂B, N > 2k, k ≥ 2. Under certain assumptions on f, we use the moving plane method to show radial symmetry of any singular positive solution provided that 0 is a nonremovable singularity point. As an application, we can obtain nonexistence of positive solutions for a critical Dirichlet biharmonic problem.

  • Xiao Hua LIU, Shi Chao LUO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 123-146. https://doi.org/10.12386/A2022sxxb0010
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    It was proved that a function with exact one discontinuity may have a continuous iterate of second order. It actually shows that its discontinuity may be repaired to be a continuous one by its own pair of functions under iteration. If a function has at least two discontinuities, then each of its discontinuities may be repaired to be a continuous one by either its own pair of functions or the other's pair of functions under iteration. In this paper we investigate those functions having more than one but finitely many discontinuities of the same type and give necessary and sufficient conditions for those functions whose second order iterates are continuous.

  • Sarina, De Yu WU, Alatancang
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 147-152. https://doi.org/10.12386/A2022sxxb0011
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    In this paper, the quadratic numerical radius inequalities of off-diagonal block operator matrixwhose entries are bounded operators on the Hilbert space is studied. According to the classical convexity inequalities of non-negative real numbers, the quadratic numerical radius inequalities of A is generalized.

  • Juan WANG, Lian Ying MIAO, Jian Sheng CAI
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 153-160. https://doi.org/10.12386/A2022sxxb0012
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    For a graph G=(V (G), E(G)), if a mapping ?:E(G) → {1, 2,..., k} such that ?(e1) =?(e2) for any adjacent edges e1, e2, and there are no bicolored cycles in G, then ? is called an acyclic edge coloring of G. For a list assignment L={L(e)|eV (E)}, if there exists an acyclic coloring ? such that ?(e) ∈ L(e) for each eE(G), then ? is called an acyclic L-list coloring of G. If for any L with|L(e)| ≥ k for each eE(G), there exists an acyclic L-list coloring of G, then we say G is acyclically k-edge choosable. The minimum integer k making G is acyclically k-edge choosable is called the acyclic list chromatic index of G, denoted by al'(G). In this paper, it is proved that for a connected graph G with maximum degree Δ ≤ 4 and|E(G)| ≤ 2|V(G)|-1, it follows that al'(G) ≤ 6, which extends the result of Basavaraju and Chandran[J. Graph Theory, 2009, 61(3):192-209].

  • Huan CHANG, Jian Lin LI, Qi WANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 161-170. https://doi.org/10.12386/A2022sxxb0013
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    The iterated function system with two-element digit set is the simplest case and the most important case in the study of spectrality or non-spectrality of self- affine measures. The one-dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable. However, the higher dimensional analogue, especially the two-dimensional case has not been solved completely. Also, there is a conjecture to illustrate that in the plane, the remaining cases correspond to nonspectrality of self-affine measures. Motivated by this problem, we provide in this paper some non-spectral conditions for the planar self-affine measures with two-element digit set. Under one of the conditions, we determine the maximal cardinality of orthogonal exponentials. An application of this result and the validity of the conditions are also presented.

  • Jin Ke HAI, Jian Xia LIU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 171-176. https://doi.org/10.12386/A2022sxxb0014
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    In this note, it is proved that the Coleman outer automorphism group of a generalized quaternion group is either 1 or an elementary abelian 2-group by using the projection limit property of the group.

  • Ji Yang Lin LI, Shou Xia WANG, Jin Hong YOU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(1): 177-204. https://doi.org/10.12386/A2022sxxb0015
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    Periodicity is one of the most common factor in time series analysis. In the time series analysis of discrete-valued response variables, we use maximum likelihood estimation with penalty to establish a consistent estimator of the unknown period. Given the estimator of the period, we take B-spline to approximate the trend term and the additive function, and at the same time obtain the √n-consistent estimator of the periodic term and the initial estimators of the trend term and the additive function. Then based on the idea of back-fitting, we establish the improved estimators of the trend term and additive function, and the asymptotic normality and efficiency of them are also demonstrated. Simulation experiments and empirical analysis confirm that our proposed method performs well for the finite sample.