15 December 2017, Volume 33 Issue 12

    

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  Rees-Shishikura's Theorem for Geometrically Finite Rational Maps

  Wen Qiang SHEN

  Acta Mathematica Sinica. 2017, 33(12): 1587-1596. https://doi.org/10.1007/s10114-017-6420-z

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  In this paper, we generalize Rees-Shishikura's theorem to the class of geometrically finite rational maps.
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  On Meromorphic Solutions of a Certain Type of Nonlinear Differential Equations

  Xiao Qing LU, Liang Wen LIAO, Jun WANG

  Acta Mathematica Sinica. 2017, 33(12): 1597-1608. https://doi.org/10.1007/s10114-017-6484-9

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  We consider transcendental meromorphic solutions with N(r, f)=S(r, f) of the following type of nonlinear differential equations:#br#fⁿ+P_n-2(f)=p₁(z)e^α₁(z)+p₂(z)e^α₂(z),#br#where n ≥ 2 is an integer, P_n-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p₁(z), p₂(z) are nonzero small functions of f, and α₁(z), α₂(z) are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.
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  Complementary Inequalities to Improved AM-GM Inequality

  Hamid Reza MORADI, Mohsen Erfanian OMIDVAR

  Acta Mathematica Sinica. 2017, 33(12): 1609-1616. https://doi.org/10.1007/s10114-017-7118-y

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  Following an idea of Lin, we prove that if A and B are two positive operators such that 0 < mI ≤ A ≤ m'I ≤ M'I ≤ B ≤ MI, then
  
  where K(h)=((h+1)²)/(4h) and h =M/m and Φ is a positive unital linear map.

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  Sub-additive Topological and Measure-theoretic Tail Pressures

  Jing Lan PENG, Yun Hua ZHOU

  Acta Mathematica Sinica. 2017, 33(12): 1617-1631. https://doi.org/10.1007/s10114-017-6594-4

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  In this paper, we give some definitions of the topological tail pressures for sub-additive potentials and prove that they are equivalent if the potentials are continuous. Under some assumptions, we get a variational principle which exhibits the relationship between topological tail pressure and measure-theoretic tail entropy. Finally, we define a new measure-theoretic tail pressure for sub-additive potentials and some interesting properties of it are obtained.
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  On Partial Regularity of Suitable Weak Solutions to the Stationary Fractional Navier-Stokes Equations in Dimension Four and Five

  Xiao Li GUO, Yue Yang MEN

  Acta Mathematica Sinica. 2017, 33(12): 1632-1646. https://doi.org/10.1007/s10114-017-7125-z

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  In this paper, we investigate the partial regularity of suitable weak solutions to the multidimensional stationary Navier-Stokes equations with fractional power of the Laplacian (-Δ)^α (n/6 ≤ α < 1 and α ≠1/2). It is shown that the n + 2 -6α (3 ≤ n ≤ 5) dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu[19] to four and five dimension. Moreover, the pressure in ε-regularity criteria is an improvement of corresponding results in[1, 13, 18, 20].
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  Mapping Properties of Certain Oscillatory Integrals on Modulation Spaces

  Mei Fang CHENG, Hui Hui LIU, Ai Wen SUN

  Acta Mathematica Sinica. 2017, 33(12): 1647-1658. https://doi.org/10.1007/s10114-017-6505-8

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  Suppose β₁ > α₁ ≥ 0, β₂ > α₂ ≥ 0 and (k, j) ∈ R². In this paper, we mainly investigate the mapping properties of the operator#br#T_α,βf(x, y, z)=∫_Q² f(x -t, y -s, z -t^ks^j)e^{-2πit-β_1s-β₂}t^{-1-α_1s-1-α₂}dtds#br#on modulation spaces, where Q²=[0, 1] ×[0, 1] is the unit square in two dimensions.
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  Strong 3-Commutativity Preserving Maps on Standard Operator Algebras

  Mei Yun LIU, Jin Chuan HOU

  Acta Mathematica Sinica. 2017, 33(12): 1659-1670. https://doi.org/10.1007/s10114-017-6145-z

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  Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ:A → A is said to be strong 3-commutativity preserving if[Φ(A), Φ(B)]₃=[A, B]₃ for all A, B ∈ A, where[A, B]₃ is the 3-commutator of A, B defined by[A, B]₃=[[[A, B], B], B] with[A, B]=AB -BA. The main result in this paper is shown that, if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h:A → F and a scalar λ ∈ F with λ⁴=1 such that Φ(A)=λA + h(A)I for all A ∈ A.
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  Similarity and Parameterizations of Dilations of Pairs of Dual Group Frames in Hilbert Spaces

  Xun Xiang GUO

  Acta Mathematica Sinica. 2017, 33(12): 1671-1683. https://doi.org/10.1007/s10114-017-7078-2

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  In this paper, firstly, in order to establish our main techniques we give a direct proof for the existence of the dilations for pairs of dual group frames. Then we focus on proving the uniqueness of such dilations in certain sense of similarity and giving an operator parameterization of the dilations of all pairs of dual group frames for a given group frame. We show that the operators which transform different dilations are of special structured lower triangular.
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  Wandering Subspaces and Quasi-wandering Subspaces in the Hardy-Sobolev Spaces

  Jie Sheng XIAO, Guang Fu CAO

  Acta Mathematica Sinica. 2017, 33(12): 1684-1692. https://doi.org/10.1007/s10114-017-7041-2

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  In this paper, we prove that for -(1/2) ≤ β ≤ 0, suppose M is an invariant subspaces of the Hardy-Sobolev spaces H_β²(D) for T_z^β, then M⊖zM is a generating wandering subspace of M, that is, M=[M⊖zM]_Tz^β. Moreover, any non-trivial invariant subspace M of H_β2(D) is also generated by the quasi-wandering subspace P_MT_z^βM^⊥, that is, M=[P_MT_z^βM^⊥]_Tz^β.
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  Cluster Structures in 2-Calabi-Yau Triangulated Categories of Dynkin Type with Maximal Rigid Objects

  Hui Min CHANG

  Acta Mathematica Sinica. 2017, 33(12): 1693-1704. https://doi.org/10.1007/s10114-017-6504-9

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  In this paper, we consider two kinds of 2-Calabi-Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called A_n,t=D^b(KA_{(2t+1)(n+1)-3)}/τ^t(n+1)-1[1], where n, t ≥ 1, and D_n,t=D^b(KD_2t(n+1))/τ⁽ⁿ⁺¹⁾φⁿ, where n, t ≥ 1, and φ is induced by an automorphism of D_2t(n+1) of order 2. Except the categories A_n,1, they all contain non-zero maximal rigid objects which are not cluster tilting. A_n,1 contain cluster tilting objects. We define the cluster complex of A_n,t (resp. D_n,t) by using the geometric description of cluster categories of type A (resp. type D). We show that there is an isomorphism from the cluster complex of A_n,t (resp. D_n,t) to the cluster complex of root system of type B_n. In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan-Palu-Reiten:Let R_An,t, resp. R_Dn,t, be the full subcategory of A_n,t, resp. D_n,t, generated by the rigid objects. Then R_An,t ≃ R_An,1 and R_Dn,t ≃ R_An,1 as additive categories, for all t ≥ 1.