15 January 2023, Volume 66 Issue 1

    

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  Chaotic Dynamics of Generalized Elliptical Sitnikov (N+1)-body Problem

  Xu Hua CHENG, Yong Quan WANG

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 1-14. https://doi.org/10.12386/A20210070

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  In this paper, the chaotic behavior of the generalized elliptic Sitnikov (N+1)-body problems is analytically studied. First, based on the perturbation method of integrable Hamiltonian systems, the generalized elliptic Sitnikov (N+1)-body problem is regarded as the perturbation of the generalized circular Sitnikov (N+1)-body problem. Then, we prove that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Moreover, since the equilibrium point is a degenerate hyperbolic saddle, the standard Smale- Birkhoff theorem cannot be used directly to prove the existence of Smale horseshoes. We alternatively construct an invertible map f and check that f satisfies the Conley-Moser condition, which shows that the generalized elliptic Sitnikov (N+1)-body problem possess chaotic behaviour of Smale horseshoe type.
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  Embeddings of Circular Ladder-like Graph Families

  Qian Hua SUN, Yi Chao CHEN

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 15-46. https://doi.org/10.12386/A20210108

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  We determine the genus distributions for circular ladder-like sequences of graphs and Möbius ladder-like sequences of graphs. First, we obtain the production matrix for genus distributions of ladder-like sequences of graphs. Then, any circular (Möbius) ladder-like graph is obtained by adding edges to a ladder-like graph by using edge addition rule, and we obtain the genus polynomials for circular ladder-like sequences of graphs and Möbius ladder-like sequences of graphs. In addition, we also verify the asymptotic normality of the classical ladder graphs.
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  Uniform Large Deviations for 2D Incompressible Magneto-hydrodynamics Equations Driven by Multiplicative Noises

  Jin MA

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 47-66. https://doi.org/10.12386/A20210065

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  We establish a Freidlin- Wentzell type large deviation principle for twodimensional incompressible Magneto-hydrodynamics equations driven by multiplicative noises when the noises converge to zero that are uniform with respect to initial conditions in bounded subsets of the infinite dimensional Banach space. The proof is based on the weak convergence approach.
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  The Zeros of Complex Delay-differential Polynomials Related to Hayman Conjecture

  Ying Chun, GAO Kai LIU

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 67-74. https://doi.org/10.12386/A20210079

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  This paper is based on the Hayman Conjecture for the zeros problem of complex differential polynomials. Using an important estimate on the zeros of higher derivative of meromorphic functions given by Yamanoi, we obtain the improved results on the zeros of delay-differential polynomials. For instance, we have that if f is a transcendental meromorphic function with hyper-order less than one and q ≥ p+s+t+1, then [Q(f)P(f(z +c))]^(k) -a has infinitely many zeros, where a is a non-zero constant, P(z) is a polynomial of degree p with t different zeros and Q(z) is a polynomial of degree q with s different zeros. Our results improve the former results which obtained mainly by the second main theorem of Nevanlinna.
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  A Linear Approximate Bregman-type Peaceman-Rachford Splitting Method for Nonconvex Nonseparable Optimization

  Peng Jie LIU, Jin Bao JIAN, Guo Dong MA, Jia Wei XU

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 75-94. https://doi.org/10.12386/A20210081

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  Based on the Peaceman-Rachford splitting method, combined with the linear approximate technique and Bregman distance, in this paper, we present a linear approximation Bregman-type Peaceman-Rachford splitting method for solving the nonconvex nonseparable optimization problem with linear constraints. Under the conventional assumptions, we get the global convergence of the proposed algorithm. On the premise that the merit function satisfies the Kurdyka-Ƚojasiewicz property, the strong convergence of the proposed algorithm is proved. When the associated Kurdyka-Ƚojasiewicz property function has a special structure, the convergence rate results of the proposed algorithm are analyzed and obtained. Finally, some preliminary numerical results show that the proposed algorithm has numerical validity.
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  Further Results on n-Cycle Permutations

  Zhi Lin ZHANG, Ping Zhi YUAN

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 95-104. https://doi.org/10.12386/A20210088

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  In this paper, some new classes of n-cycle permutations of the form x^rh(x^s) over finite fields are presented, which are the further study on a recent work of Chen, Wang and Zhu. In addition, based on some detailed discussions, four interesting problems are proposed.
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  Existence and Multiplicity of Solutions for Fractional Critical Schrödinger Equation

  Yan Ying SHANG, Yu Ting WANG

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 105-124. https://doi.org/10.12386/B20210004

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  We obtain the existence and multiplicity of solutions for the fractional Schrödinger equation with Hardy-Sobolev critical exponent in R^N by Ekeland’s variational principle and Nehari decomposition.
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  The Spherical Density Properties of Self-similar Sets

  Zhi Wei ZHU

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 125-132. https://doi.org/10.12386/A20200054

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  This paper discusses the computations of Hausdorff centered measure and packing measure for self-similar sets. Let E be a self-similar sets with the strong separation condition, s be the Hausdorff dimension of E, μ be the self-similar measure defined on E, In this paper, we obtained the following results: (1) Lettirg x₀ ∈ E, if $\overline{D}^s(\mu,x_0)=\overline{d}$, then $\overline{D}^s(\mu,x) \ge \overline{d}$; for μ-almost all x ∈ E; (2) Lettirg y₀ ∈ E, if $\underline{D}^s(\mu,y_0)=\underline{d}$, then $\underline{D}^s(\mu,y) \le \underline{d}$ for μ-almost all y ∈ E. Some computational problems of measures of the self-similar sets are discussed by the use of those results.
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  On Simultaneous Pell Equations x²-(m²-1)y²=z²-(n²-1)y²=1

  Xun Gui GUAN

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 133-142. https://doi.org/10.12386/A20210003

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  Let m, n, L be positive integer. The following conclusion are proved: If m<n≤m+Lm^ε, ε∈(0,1), and m>(123789L√L)1/1-ε, or j>10.25×10¹²log⁴(2(L+1)(123789L√L)1/1-ε, then positive integer solutions of simultaneous Pell equations $x^{2}-(m^{2}-1)y^{2}=z^{2}-(n^{2}-1)y^{2}=1$ satisfy $1≤k\leq\delta L^{2}$, where $\delta\in[\frac{1}{2}(123787L\sqrt{L})^{\frac{1}{\varepsilon-1}},1]$,$ and $$y=\frac{(m+\sqrt{m^{2}{-}1})^{j}{-}(m{-}\sqrt{m^{2}{-}1})^{j}}{2\sqrt{m^{2}{-}1}}=\frac{(n{+}\sqrt{n^{2}{-}1})^{k}{-}(n{-}\sqrt{n^{2}{-}1})^{k}}{2\sqrt{n^{2}{-}1}},$ and j$j=k=1$ or $k+2\leq j<\frac{1}{3}(5-2\varepsilon)k$,$2\,|\,(j+k)$, $k>\frac{3}{1-\varepsilon}$. It improves the previous work of [Proc. Amer. Math. Soc., 2015, 143(11): 4685-4693].
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  The Equivalent Properties of Markov Chains Indexed by a Tree Taking Value on R

  Cheng Jun DING, Ying JING, Wei Guo YANG

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 143-148. https://doi.org/10.12386/A20210010

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  In the paper, we give the definition of the Markov chains indexed by a tree taking value on R. Then, we prove the equivalent properties of it.
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  g-Starlike Mappings of Complex Order λ on the Unit Ball $\mathscr {B}^{n}$

  Chun Ying HU, Tai Shun LIU, Jian Fei WANG

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 149-160. https://doi.org/10.12386/A20210029

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  In this paper, a subclass of g-starlike mappings of complex order λ on the unit ball $\mathscr {B}^{n}\subset \mathbb{C}^{n}$ is introduced, which unifies almost starlike mappings of complex order λ and g-starlike mappings. The growth theorem for g-starlike mappings of complex order λ is established by using the parameter method of Loewner chains. By giving scalar conditions of homogenous polynormials of degree k, we prove that the modified Roper-Suffridge extension operator on the unit ball $\mathscr {B}^{n}$ given by Φ_Pk(f)(z) = (f(z₁) + P_k(z₀)f' (z₁), [f' (z₁)] 1/k z₀)^T preserves the property of g-starlike mappings of complex order λ. Our results not only generalize some well-known growth theorems of different classes of starlike mappings on $\mathscr {B}^{n}$, but also give more concise geometric characteristics of polynomial P_k.
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  The Estimate of the Conformal Dimension of the Sierpinski Carpet S_p

  Yun Gui DANG, Yu Xia DAI, Sheng You WEN

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 161-172. https://doi.org/10.12386/A20210099

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  In this note, we prove that the conformal dimension of the Sierpinski carpet S_p satisfies $1{+}\frac{\log (p{-}1)}{logp}\!\leq\dim_{C}S_{p}\leq \frac{\log((p^{2}-1)^{4}-8)}{4\log p}$ , where $p\geq3$ is odd. This result implies that S_p is not a quasisymmetrically minimal set.
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  The Existence for A Class of Composite Involution Polynomials Over Finite Fields

  Zhao Hui ZHANG, Qun Ying LIAO

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 173-180. https://doi.org/10.12386/A20210115

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  In 2021, Zhang and Liao gave a necessary and sufficient condition for that the composite polynomial of two involution polynomials over finite fields is also an involution polynomial. Since the involution polynomial is a special class of permutation polynomials, based on the relationship between the sets of fixed points and non-fixed points of polynomials, we obtain a necessary and sufficient condition for the composition of an involution polynomial and a permutation polynomial to be involuted.
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  The Eigenvalues and Eigenvectors of the Toeplitz Operators

  Xuan Hao DING, Lin HOU, Yong Ning LI

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 181-186. https://doi.org/10.12386/A20210121

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  Let L² be the set consisting of all the Lebesgue integrable functions on the unit circle T. Define the Hardy space to be the closed subspace spanned by the analytic polynomials in L². For any point z in the unit disk D of the complex plane, K_z(w) = 1/1-zw ˉ is the reproducing kernel function in H². It is well known that T_fK_z = f(z)K_z, that is, K_z is the eigenvector of T_f corresponding to the eigenvalue f(z). Conversely, if there exists some z ∈ D (or, for any z ∈ D), K_z is an eigenvector of T_f, whether there must be $f\in \overline{H^{\infty}}$? For the above questions, in this paper, we give a complete characterization of the Toeplitz operators as well as the bounded linear operators which take the reproducing kernels K_z as their eigenvectors. Moreover, we partially describe the Toeplitz operators with f(z) (z ∈ D) as their eigenvalues.
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  Upper Triangular Matrix Groups Over Two Kinds of Domains (I)

  He Guo LIU, Jing ZHAO

  Acta Mathematica Sinica, Chinese Series. 2023, 66(1): 187-198. https://doi.org/10.12386/A20220038

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  We construct two 3-generated metabelian groups from the upper triangular matrices of order 2 over two kinds of domains. Their structures are clear and their residual finiteness is studied. One of the groups with infinite rank is a residually finite p-group, where p is a prime. And the other group with finite rank does not have this neat residual finiteness property, although its structure is simpler.