中国科学院数学与系统科学研究院期刊网

Acta Mathematica Sinica, Chinese Series 2021 Vol.64

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Global Regularity of Classical Solutions to the Planar MHD Equations with Temperature-dependent Viscosity
Zhao Yang SHANG, Kai Fang REN, Fu Quan TANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 1-40.   DOI: 10.12386/A2021sxxb0001
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We study the global regularity of classical solutions to the Cauchy problem of planar magnetohydrodynamics equations when the viscosity coefficients λ, μ, magnetic diffusion coefficient η and the heat conductivity coefficient κ depend on the specific volume v and the temperature θ which are proportional to h(v)θα for certain non-degenerate smooth function h. Under the condition of regularity criterion ∫0 +∞||b||L2ds < +∞, when α is small, we prove the existence of global classical solution with large initial data.
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The Prescribing Problem for Symmetric Function of Ricci Tensor
Yan HE, Wei Wei ZHANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 41-46.   DOI: 10.12386/A2021sxxb0002
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We consider the prescribing problem for symmetric function of Ricci tensor. Suppose a closed Einstein manifold (M, g) is not σ2(Ric) singular. Let fC(M) and it changes sign. We prove that there exists a metric g* such that σ2(Ricg*)=f. Then, as a corollary, we have an existence result for the prescribing problem for Einstein manifold with negative scalar curvature.

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Para-isotropic Spacelike Hypersurfaces in R1m+1
Xiu JI, Tong Zhu LI
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 47-58.   DOI: 10.12386/A2021sxxb0003
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Let f:Mm → R1m+1 be an umbilic-free spacelike hypersurface. Four basic conformal invariants of Mm are the conformal metric g, the conformal 1-form C, the conformal second fundamental form B, and the conformal Blaschke tensor A. Mm is called the para-isotropic spacelike hypersurface, if A + μB=λg for some constant μ and smooth function λ. We not only constructed examples which are para-isotropic spacelike hypersurfaces but also classified completely all para-isotropic spacelike hypersurfaces under the conformal transformal group of R1m+1 in this paper.

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Lusternik–Schnirelmann Category of Small Cover
Xue Wei MU, Deng Pin LIU
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 59-64.   DOI: 10.12386/A2021sxxb0004
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We proved that the Lusternik-Schnirelmann category of any n-dimensional small cover is n and that the Lusternik-Schnirelmann category of any 2n-dimensional toric manifold is also n. Our result depends on the two well-known facts, one is the inequality cup(M) ≤ cat(M) ≤ dim(M)/r which leads us to compute cup(M). The other is explicit expression on the cohomology ring of small cover which makes the calculation of cup(M) easier. Furthermore, we generalized the relevant conclusions.
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Erdös–Kac Type Theorem for Ideal Counting Function over Gaussian Field in Short Intervals
Xiao Li LIU, Zhi Shan YANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 65-76.   DOI: 10.12386/A2021sxxb0005
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Let aK(n) be the number of non-zero integral ideals in Z[i] with norm n, l ∈ Z+. In this paper, we establish an Erdös-Kac type theorem with weight aK(n)l in short intervals, and we get an asymptotic formula for the average behavior of aK(n)l in short intervals.

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The Exponents and Order of Convergence of Small Functions of Meromorphic Functions Concerning Derivatives and Differences
Pin Ling WANG, Shi Wei YANG, Ming Liang FANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 77-86.   DOI: 10.12386/A2021sxxb0006
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Let f(z) be a meromorphic function in the complex plane, let c be a nonzero finite complex number, and let a(z) be a small function with respect to f(z). It is studied that the relationship between the exponent of convergence of zeros of f(z)-a(z), f(z + c)-a(z), and Δcn f(z)-a(z) (n ∈ N+) and the order of f(z). This improves some results in value distribution of meromorphic functions concerning derivatives and differences.

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Kodaira Dimension of Nearly Kähler 6-manifolds
Hao Jie CHEN, Guan Ming WANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 87-98.   DOI: 10.12386/A2021sxxb0007
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We investigate the canonical bundle and Kodaira dimension of nearly Kähler 6-manifolds. We prove that the canonical bundle of a strictly nearly Kähler 6-manifold is pseudoholomorphically trivial. Therefore, the Kodaira dimension is zero. As a corollary, we show the existence of non-integrable almost complex structure on CP3 whose Kodaira dimension is not -∞. We also construct explicit generating sections of the canonical bundle of homogeneous strictly nearly Kähler 6-manifolds and prove that the Hodge numbers h1,0, h2,0, h2,3, h1,3 of the homogeneous strictly nearly Kähler F3 and CP3 are all zeros.
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The Numbers of Nontrivial Weak Solutions to Fourth-order Impulsive Elastic Beam Equations
Jian LIU, Zeng Qin ZHAO, Wen Guang YU
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 99-106.   DOI: 10.12386/A2021sxxb0008
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In this paper, the numbers of nontrivial solutions to superlinear fourthorder impulsive elastic beam equations are obtained. We get two theorems via variational methods and corresponding two-critical-points theorems. Combining with the Newton-iterative method, two examples are presented to illustrate the value of the obtained theorems.

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Global Well-posedness for the 2D MHD System with Partial Dissipation and Magnetic Diffusion in a Bounded Domain
Ming Yu ZHANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 107-122.   DOI: 10.12386/A2021sxxb0009
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We consider the initial boundary value problem of the two-dimensional incompressible magnetohydrodynamic (MHD) equations with partial dissipation and magnetic diffusion. The global and unique strong solution of the model in a bounded domain is justified when the dissipation and magnetic diffusion coefficient in all directions are nonnegative. In addition, the global well-posedness of the system can be extended into the periodic boundary.

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Commutators of Bilinear Hardy Operators on Weighted Herz–Morrey Spaces with Variable Exponents
Sheng Rong WANG, Jing Shi XU
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 123-138.   DOI: 10.12386/A2021sxxb0010
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We give a novel characterization of BMO functions via weighted norm. As an application, we obtain the boundedness of commutators generated by bilinear Hardy operator and BMO functions on products of weighted Herz-Morrey spaces with variable exponents.

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Linear Combinations of Composition Operators on Weighted Bloch Type Space
Li ZHANG, Xiu Jiao CHU
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 139-144.   DOI: 10.12386/A2021sxxb0011
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For i=1, 2, …, N, let λi be nonzero number, and D the unit open disk of complex plane C, φi is analytic self-maps of D. In this paper, the compactness of linear combinations of composition operators ∑i=1N λiCφi on the weighted Bloch space is discussed.

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Large Family of Pseudorandom Binary Lattices by Using the Quadratic Character in Finite Fields
Hua Ning LIU, Ke Yao LI
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 145-150.   DOI: 10.12386/A2021sxxb0012
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In this paper, we construct a large family of pseudorandom binary lattices by using the quadratic character in finite fields, and study the cryptography properties:pseudorandom measure, collision and avalanche effect.

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A Note on Solutions of an Equation Relating to Smarandache Function
Li LIU, Yu LI
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 151-154.   DOI: 10.12386/A2021sxxb0013
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For any positive integer n, let φ(n) be the Euler function and S(n) be the Smarandache function. Bai and Liao[On the solutions for several classes of equations related to the smarandache function, Acta Math. Sin., Chin. Series, 2019, 62(2):247-254] proved that the equation φ(n)= ∑d|n S(d) has only two solutions:n=25 and n=3×25. In this paper, we point out that both numbers are not solutions of the equation, and point out that this mistake was caused by that the authors misunderstood the Möbius inversion formula.

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The Isometry on bp(2) Space
Rui Dong WANG, Pu WANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 155-166.   DOI: 10.12386/A2021sxxb0014
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Metric and linear properties are significant properties of normed spaces, so the study of the relationship between linear operators and isometric operators has become an important research topic in the field of functional analysis. In this paper, we will study a special F-space, bp(2) space, and give the representation theorem for the onto isometric mapping on the unit sphere of the bp(2) space, then give a result about the isometric linear extension from unit sphere in the bp(2).

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Toral Ranks of Restricted Lie Superalgebras
Li Ping SUN, Wen De LIU
Acta Mathematica Sinica, Chinese Series    2021, 64 (1): 167-176.   DOI: 10.12386/A2021sxxb0015
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In this paper, the definitions and properties of torus and toral rank are generalized from modular Lie algebras to super-case. Taking advantage of the restricted envelopes, we obtain certain important results concerning toral rank of modular Lie superalgebras. As an application, the absolute toral ranks of the classical linear Lie superalgebra slm|n, the finite-dimensional restricted Lie superalgebras W(m,n,1) and S(m,n,1) of Cartan type and the toral rank of S(m,n,1) in W(m,n,1) are computed.

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Best Proximity Point Theorems for Generalized Weak Contractive Mappings in Partially Ordered Menger PM-spaces
Zhao Qi WU, Chuan Xi ZHU, Cheng Gui YUAN
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 177-188.   DOI: 10.12386/A2021sxxb0016
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In this paper, some best proximity point theorems for generalized weakly contractive mappings which satisfy certain conditions by using three control functions in partially ordered Menger PM-spaces are obtained, and sufficient conditions to guarantee the uniqueness of the best proximity points are also given. Moreover, some corollaries are derived as consequences of the main results.

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Random Uniform Exponential Attractor for Non-autonomous Stochastic FitzHugh-Nagumo System
Zong Fei HAN, Sheng Fan ZHOU
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 189-218.   DOI: 10.12386/A2021sxxb0017
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We first introduce the concept and the existence criterion of a random uniform exponential attractor for a non-autonomous random dynamical system. Then we prove the existence of a random uniform exponential attractor for FitzHugh-Nagumo system with additive noise and quasi-periodic external forces defined in Rn.

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Generalized Central α-Armendariz Rings
Da Jun LIU, Jia Qun WEI
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 219-224.   DOI: 10.12386/A2021sxxb0018
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In this paper, the notion of a generalized central α-Armendariz ring is introduced, and this paper also obtains some basic properties of such rings. At the same time, the relations between generalized central α-Armendariz rings and other rings are studied.

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Chaotic Criteria in Complete Metric Spaces
Xiao Ying WU, Yuan Long, CHEN Fen WANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 225-230.   DOI: 10.12386/A2021sxxb0019
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In this paper, the chaotic criteria of discrete dynamical systems is studied in complete metric spaces. It is showed that if f is continuous map from a complete metric space X into itself and has regular nondegenrate snap-back repellers or heteroclinic repellers, then there exists an invariant subset Λ of f such that (Λ, f) is topologically conjugate to the one-side symbolic system (Σ+2, σ). Therefore, f is Devaney's chaos, distributional chaos, ω-chaos and has positive entropy. These results improve the related results.
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A Sharp Subcritical Adams Inequality in Lorentz Sobolev Space
Mao Chun ZHU, Yu Hang LIU
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 231-242.   DOI: 10.12386/A2021sxxb0020
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We obtain a sharp second order subcritical Adams inequality in Lorentz Sobolev space W 2L2,q(R4). Moreover, the lower and upper bounds asymptotically for the subcritical Adams functional is obtained. Our approach is based on the rearrangement free argument developed by Lam and Lu[A new approach to sharp MoserTrudinger and Adams type inequalities:a rearrangement-free argument, J. Diff. Equ., 2013, 255(3):298-325].

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K-g-frames and Their Duality
Chun Nian DAI, Jin Song LENG, Miao HE
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 243-254.   DOI: 10.12386/A2021sxxb0021
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We mainly discuss K-g-frames and its duality. First, we explore the relationship between K-g-frames and g-frames. Then, we give some sufficient conditions under which the sum of K-g-frames and g-Bessel sequences with bounded linear operator or nonzero complex bounded sequence is still K-g-frames. In addition, we also give two special forms about the sum of K-g-frames. Finally, we research the duality of K-g-frames in closed subspace R(K), and the ways of constructing the K-g-frames by using the approximate duality.
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A Scaling Framework for the Non-isentropic Gas Dynamics System with Large Initial Data
Shu Jun LIU
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 255-260.   DOI: 10.12386/A2021sxxb0022
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There are two open problems on the global existence results of non-isentropic gas dynamics. One is whether the weak solutions exist globally with small initial data containing vacuum, the other is whether the global existence results hold with arbitrary large initial data. By introducing a scaling framework, we give the equivalence of the two problems above. For vanishing viscosity solutions, the positive answer to the first question naturally implies the positive answer to the second one. And this scaling framework can be applied to most systems of conservation laws with physical background.

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Convergence of Bi-shift Localized Baskakov Operators
Lin Sen XIE, Ting Fan XIE, Hong DU
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 261-268.   DOI: 10.12386/A2021sxxb0023
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We consider a new form of the localized Baskakov operators, and obtain some convergence properties of the new operators. We also obtain a new estimate for the kernel of the Baskakov operators by making use of one of the central limit theorems in probability theory.

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Existence of Positive Solutions for Fractional Schrödinger-Poisson System with Critical or Supercritical Growth
Wen Bo WANG, Jian Wen ZHOU, Yong Kun LI, Quan Qing LI
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 269-280.   DOI: 10.12386/A2021sxxb0024
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We study the following fractional Schrödinger-Poisson system

where s ∈ (4/3, 1), t ∈ (0, 1), the continuous function f is superlinear at zero and subcritical at infinity and the exponent q ≥ 2s*=(3-2s)/6. We obtain a positive solution of the above problem for small λ > 0 via the variational method. Our main contribution is that we can deal with the supercritical case.

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Some Geometric Constants and Fixed Points for Multivalued Nonexpansive Mappings
Zhan Fei ZUO
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 281-288.   DOI: 10.12386/A2021sxxb0025
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Some geometric conditions in terms of the characteristic of convexity, the normal structure coefficient, the James type constant and the García-Falset coefficient were considered in the paper, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some well known results and give affirmative answers to some open questions.

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The Multidimensional Hausdorff Operators on Hp(Rn)
Shao Yong HE, Xiang Rong ZHU
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 289-300.   DOI: 10.12386/A2021sxxb0026
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We consider the following Hausdorff operator Hφf(x)=∫Rnφ(u1,..., un) · f(u1x1,..., unxn)du1 · · · dun, where φ can be considered as a distribution on Rn. When n ≥ 2 and φ is a Schwartz function, we show that Hφ is bounded on Hp(Rn) for some p ∈ (0, 1) if and only if φ ≡ 0. Furthermore, when n ≥ 2, if φ is just a continuous function and Hφ can be defined suitable, then we can also prove that Hφ is bounded on Hp(Rn) for some p ∈ ((n+1)/n, 1) if and only if φ equals to a constant. These facts mean that Hφ is very complicated on Hp(Rn) (n ≥ 2). Moreover, we establish a result of the boundedness of Hφ on Lp(Rn), p > 1. The key idea used here is to reformulate Hφ as a convolution operator.

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Some Equivalent Descriptions of p-fusion Frames on Banach Spaces
Li Qiong LIN, Yun Nan ZHANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 301-310.   DOI: 10.12386/A2021sxxb0027
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We describe a close relation between the p-fusion frames and the p-frames on Banach spaces. Using the analysis operators and the synthesis operators, we provide the equivalent descriptions of the p-fusion Bessel sequences, p-fusion frames and q-fusion Riesz bases.

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A Homological Characterization of Strong Prüfer Rings
Fang Gui WANG, Lei QIAO, De Chuan ZHOU
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 311-316.   DOI: 10.12386/A2021sxxb0028
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Let R be a commutative ring. Then the small finitistic projective dimension of R is defined as fPD(R)=sup{pdRM|M ∈ FPR}. In this paper, it is shown that if R is a connected strong Prüfer ring, then fPD(R) ≤ 1. It is also shown that if R is a strong Prüfer ring, and if M is a Q-torsion module with M ∈ FPR, then pdRM ≤ 1.

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Semi-classical Solutions of Fractional Kirchhoff-type Equations with Critical Growth
Shun Neng ZHAO, Fu Kun ZHAO
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 317-342.   DOI: 10.12386/A2021sxxb0029
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In this paper, we study the following fractional Kirchhoff-type equation with critical growth ε2sM(ε2s-3 ∫∫R3×R3(|u(x)?u(y)|2)/(|x-y|3+2s)dxdy)(-Δ)su + V (x)u=λW(xf(u) + K(x)|u|2s*-2u, x ∈ R3, where M is a continuous and positive Kirchhoff function, λ > 0 is a parameter, (-Δ)s is the fractional Laplace operator with 3/4 < s < 1, V (x), W(x) and K(x) are all positive potentials. Under some assumptions on potentials, we obtain the existence of a positive ground state solution for ε > 0 small and λ large. Moreover, we show that these ground state solutions concentrate at a special set characterized by potentials. Finally, we study the decay estimate of ground state solutions.

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Quantitative Weighted Bounds for a Class of Singular Integral Operators
Wen Hua GAO
Acta Mathematica Sinica, Chinese Series    2021, 64 (2): 343-352.   DOI: 10.12386/A2021sxxb0030
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Let T be the singular integral operator whose kernel satisfies a variant of Lipschitz regularity introduced by Grubb and Moore, and T* be the associated maximal singular integral operator. In this paper, by establishing the weak type endpoint estimates for the grand maximal operators corresponding to T and T*, the author establishes some quantitative weighted bounds and weighted weak type endpoint bounds in terms of the Ap constants for the operators T and T*.
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Positive Toeplitz Operators on Bergman Space of Annular Induced by Regular-weight
Zhong Hua HE, Jin XIA, Xiao Feng WANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 353-374.   DOI: 10.12386/A2021sxxb0031
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This paper is devoted to studing Bergman spaces induced by regular-weight Aω1,2p(M) (1 < p < ∞) on annular and positive Toeplitz operators on these spaces. The dual spaces of Bergman spaces induced by regular-weight are characterized. We also obtain equivalent conditions for boundedness and compactness of positive Toeplitz operators between these regular-weighted Bergman spaces.

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Numerical Range Estimation of Block Operator Matrices
Hui Ting WU, De Yu WU, Alatancang
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 375-384.   DOI: 10.12386/A2021sxxb0032
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In this paper, the numerical radius of bounded block operator matrices on Hilbert space is studied. First, the generalized form of numerical radius inequalities of off-diagonal block operator matrix is studied, and taking advantage of the unitary similarity invariance of numerical radius and the generalized mixed Schwarz inequality, the inequalities of the numerical radius of sum of two bounded linear operators are considered. Then, numerical radius inequalities for 2×2 bounded block operator matrices are given. Finally, the conclusion is applied in the bounded infinite dimensional Hamiltonian operator and the inequalities of its numerical radius are obtained.

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Well-posedness and Convergence of the Vector Field Regularization Model in Image Registration
Xiao Jun ZHENG, Zhong Dan HUAN, Jun LIU
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 385-404.   DOI: 10.12386/A2021sxxb0033
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Image registration is fundamental to image processing. The vector field regularization model performs relatively well among a large number of registration methods. However, it still can't correspond to all interested regions across images correctly. Therefore, we hope to study the theory of the vector field regularization model to see whether there are some problems with the design of the model. Moreover, as there are two unknowns which are related by an initial value problem in the regularization model, it is novel in mathematics. The vector field regularization model takes the form minv {α||v||H2 + ρ(T (yv(τ)), S)}, where T is a template image, S is a reference image, yv(τ):x ? yv(τ;0, x) is a transformation determined by the solution yv(s;0, x) of the initial value problem dy/ds=v(s, y), y(0)=x, ρ is a similarity functional, α> 0 is a regularization parameter and H is a Hilbert space. In this paper, we firstly show the vector field regularization model has stable solutions and then demonstrate its convergence. The above results can be obtained by the standard arguments of regularized problems together with the convergence relation of yv(τ) and v. However, the requirements for ρ, S and T are relatively strong under the existing regularization theory. We give relatively weak conditions for ρ, S and T by taking full advantage of the good properties of yv(τ). In addition, we verify that three commonly used similarity functionals in image registration satisfy the given conditions.

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Order Continuity of Positive Tensor Products of Banach Lattices
Shao Yong ZHANG, Qi LIU, Yong Jin LI
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 405-412.   DOI: 10.12386/A2021sxxb0034
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For a Banach sequence lattice λ and a Banach lattice X, let λ|π| X (resp. λ|ε| X) denote the positive projective (resp. injective) tensor product of λ and X. In the paper we prove that if λ is a σ-Levi space then λ|π| X (resp. λ|ε| X) is order or σ-order continuous if and only if both λ and X are order or σ-order continuous. We also prove that if λ is σ-order continuous then λ|π| X is a Levi or σ-Levi space if and only if both λ and X are Levi or σ-Levi spaces.
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On a Schwarz-Pick Type Inequality for Quasiconformal Mappings Inhomogeneous Polyharmonic Equation
De Guang ZHONG, Fan Ning MENG, Wen Jun YUAN
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 413-426.   DOI: 10.12386/A2021sxxb0035
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Let ?nC (D), ?jC (T) and K ≥ 1, where n ≥ 2 is an integer, j ∈ {1,..., n -1}. In this paper, we establish a Schwarz-Pick type inequality for the K-quasiconformal self-mapping f of the unit disk D satisfying the inhomogeneous polyharmonic equation Δnf=?n with the associated Dirichlet boundary value condition:Δn-1f|T=?n-1,..., Δ1f|T=?1 and f(0)=0. Furthermore, we prove that this result is asymptotically sharp in the sense that||?j|| → 0 (j=1,..., n) and K → 1+, where||?n||:=supz∈D|?n(z)|and||?j||:=supz∈T|?j(z)|(j=1, 2,..., n -1).

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Higher-Order Convergence of Solutions of Initial Value Problem for Set Differential Equations in Fréchet Space
Pei Guang WANG, Zhen Yu XING, Xi Ran WU
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 427-442.   DOI: 10.12386/A2021sxxb0036
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This paper investigates the initial value problem for a class of set differential equations in Fréchet space F. Based on that the set Kc(F) of all compact convex subsets of a Fréchet space F is considered as a projective limit of semilinear metric spaces Kc(Ei), and the properties of projective limit, we introduce the notions of the Fréchet partial derivative, hyperconcave and hyperconvex of set-valued functions. By using the method of quasilinearization and comparison principle, we construct two monotone iterative sequences in Kc(F), and obtain the sequences of approximate solutions which converge uniformly and rapidly to the unique solution of the problem. The obtained results enrich and develop the theory of set-valued differential equations in Fréchet space F.

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Asymptotic Solution to a Problem about Graphic Sequences with a Realization Containing Cycles C3,..., Cl
Guang Ming LI, Jian Hua YIN
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 443-454.   DOI: 10.12386/A2021sxxb0037
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A non-increasing sequence π=(d1,...,dn of nonnegative integers is said to be graphic if it is realizable by a simple graph G on n vertices. A graphic sequence π=(d1,...,dn is said to be potentially 3Cl-graphic if there is a realization of π containing cycles of every length r, 3 ≤ rl. It is well-known that if the nonincreasing degree sequence (d1,..., dl) of a graph G on l vertices satisfies the Pósa condition that dl +1-ii + 1 for every i with 1 ≤ i < l/2, then G is either pancyclic or bipartite. In this paper, we obtain a Pósa-type condition of potentially 3Cl-graphic sequences, that is, we prove that if l ≥ 5 is an integer, nl and π=(d1,...,dn is a graphic sequence with dl +1-ii + 1 for every i with 1 ≤ i < l/2, then π is potentially 3Cl-graphic. We show that this result is an asymptotic solution to a problem due to Li et al.[Adv. Math. (China), 2004, 33(3):273-283]. As an application, we also show that this result completely implies the value σ(Cl, n) for l ≥ 5 and nl due to Lai[J. Combin. Math. Combin. Comput., 2004, 49:57-64].

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A Note on Surface Singularities of Multiplicity Four
Jie HONG, Jun LU
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 455-462.   DOI: 10.12386/A2021sxxb0038
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Let P be an isolated singularity of multiplicity 4 of a complex surface Y. It is well-known that there is a locally irreducible finite covering π:(Y, P) → (X, p) with π-1(p)=P, and a Jung's resolution f:?Y. Let Wp be the exceptional divisor of (π?f)-1(p). We will prove that Wp has a unique decomposition into fundamental cycles Wp=2Z1 or Wpα=1l Zα satisfying some conditions. We will define a local index wp for π at p and compute it by the above decomposition of Wp. In particular, we will show that (Y, P) is singular iff wp ≥ 1. As another application of the decomposition of Wp, we also compute the number of blown-downs needed to get the minimal resolution from ?.

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The Crossing Number of 2-planar Graphs and Its Application
Xin ZHANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 463-470.   DOI: 10.12386/A2021sxxb0039
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An optimal planar drawing of a graph is an embedding in the plane so that the number of crossings is as small as possible. The number of crossings in an optimal planar drawing of a graph G is the crossing number cr(G) of G. A graph is k-planar if it can be embedded in the planar so that each edge is crossed at most k times. Zhang et al. (2012) proved that the crossing number of any 1-planar graph on n vertices is at most n -2, and this upper bound is best possible. Czap, Harant and Hudák (2014) proved that the crossing number of any 2-planar graph on n vertices is at most 5(n-2). In this paper, we give a better upper bound for the crossing number of 2-planar graphs and show from the point of view of combinatorics that Kn is 2-planar if and only if n ≤ 7 (surprisedly, this was an open problem until 2019, in when Angelini solved it with computer assistance).

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Entire Solutions to a Certain Type of Differential-difference Equations
Li Hao WU, Ran Ran ZHANG, Zhi Bo HUANG
Acta Mathematica Sinica, Chinese Series    2021, 64 (3): 471-478.   DOI: 10.12386/A2021sxxb0040
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We investigate the nonlinear differential-difference equations of form f(z)n+ L(z, f)=q(z)ep(z), where n ≥ 2, L(z, f)(≢ 0) is a linear differential-difference polynomial in f(z), with small functions as its coefficients, p(z) and q(z) are non-vanishing polynomials. We first obtain that n=2 and f(z) satisfies λ(f)=σ(f)=deg p(z) if the equation possesses a transcendental entire solution of hyper order σ2(f)< 1. Furthermore, the exact form of the entire solutions of the equation is also obtained.
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