15 September 2015, Volume 58 Issue 5

    

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  Rademacher Series with Vector-Valued Coefficients and Its Level Sets

  Chun Tai LIU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 705-716. https://doi.org/10.12386/A2015sxxb0072

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  He and Liu initiated a study of the intersection of level sets of Rademacher series through the Rademacher series with vector-value coefficients. Their result based on an estimation about the sum of five vectors which are less than one in norm. In this paper, we continue such investigation. Since the estimation is invalid if the dimension of vectors is larger than 2, we use the mask of a vector instead of employing the estimation to discuss when the Rademacher range of a sequence is dense or equal to the whole space. We then apply this result to consider the Hausdorff dimension of level sets.
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  K-theory and Approximation of a Class of Operators

  Shi Lin WEN, Hua HE

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 717-730. https://doi.org/10.12386/A2015sxxb0073

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  We construct a class of strongly irreducible Cowen-Douglas operators and characterize the K₀-groups of their commutant algebras. Then we show that for each bounded linear operator T on a complex separable Hilbert space and ε > 0, there exists an operator S which can be written as a direct sum of finitely many strongly irreducible Cowen-Douglas operators with nice properties such that ‖T-S‖< ε.
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  On the Hermitian Matrix with Distinct Eigenvalues and Its Applications

  Xue Liang LI, Jian Feng WANG, Qiong Xiang HUANG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 731-738. https://doi.org/10.12386/A2015sxxb0074

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  We first investigate the Hermitian matrices with k distinct eigenvalues, and then give an algebraic characterization to a graph with k distinct eigenvalues with respect to the adjacency and (normalized) Laplacian matrix.
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  Small Prime Solutions of an Nonlinear Equation

  Wei Ping LI, Feng ZHAO, Tian Ze WANG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 739-764. https://doi.org/10.12386/A2015sxxb0075

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  The present paper proved that the prime variables of nonlinear equation a₁p₁+a₂p₂²+a₃p₃²+a₄p₄² = b is soluble if integers a₁,...,a₄, b satisfy certain conditions, and it gave the upper bound for small prime solutions.
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  The Logarithmic Derivative Lemma on Zero Order Meromorphic Functions Composed with Polynomials and Its Applications

  Zhi Bo HUANG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 765-772. https://doi.org/10.12386/A2015sxxb0076

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  We investigate the lemma on the logarithmic derivative of zero order meromorphic functions composed with polynomials. As its applications, we also study the Nevanlinna characteristic and the second main theorem for zero order meromorphic functions composed with polynomials.
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  The Value Distribution of the Random Dirichlet Series in the Whole Plane

  Hong Shen ZHANG, Dao Chun SUN

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 773-780. https://doi.org/10.12386/A2015sxxb0077

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  For random Dirichlet series of ρ (0 < ρ < +∞) order in the whole plane, given some conditions, almost surely there is a Borel line of ρ order with no exceptional small functions in any closed horizontal strip of width π/ρ.
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  New Proofs of Local Torelli Theorems of Riemann Surfaces

  Quan Ting ZHAO, Sheng RAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 781-796. https://doi.org/10.12386/A2015sxxb0078

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  By use of the Kuranishi coordinates on the Teichmüller space of closed surfaces of genus g with g ≥ 2 and the explicit deformation formulae of holomorphic one-forms on close Riemann surfaces, we give explicit expressions of the period map from the Teichmüller space to Siegel upper half space and obtain new proofs of two local Torelli theorems of closed Riemann surfaces.
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  Bernstein Theorem in Q_p Spaces

  Ying Wei CHEN, Guang Bin REN

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 797-814. https://doi.org/10.12386/A2015sxxb0079

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  In Q_p spaces in the unit disc of the complex plane, the Jackson theorem has been established recently. In this article we further consider its inverse theorem, i.e., the Bernstein theorem. This will require the Q_p version of the Beinstein inequality and a derivative-free characterization for Q_p norm. The derivative-free characterization is realized by invoking the Riesz interpolation formula which interprets derivative as translation operators. As applications, the Lipschitz and Zygmund subspaces in Q_p spaces can be characterized in terms of rates of approximation.
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  Differences of Generalized Composition Operators Form F(p,q,s) to B_μ Space

  Li ZHANG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 815-824. https://doi.org/10.12386/A2015sxxb0080

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  Let B be the unit ball of the complex vector space C^N, φ is a holomorphic self-mapping of B, and g is a holomorphic function on B with g(0) = 0, we define the generalized composition operator as follows C_φ^g(f)(z) = ∫₀¹Rf(φ(tz))g(tz)(dt/t). In this paper, we characterize the boundedness and compactness of difference of generalized composition operators, acting from F(p,q,s) space to weighted Bloch space B_μ on the unit ball B.
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  Uniqueness Theorems on Entire Functions and Their Difference Operators

  Hui Fang LIU, Zhi Qiang MAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 825-832. https://doi.org/10.12386/A2015sxxb0081

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  Let f be a non-constant entire function of finite order, and α be a small function with respect to f. We prove that if f and its difference operator Δ_ηⁿf share α CM and 0 is a deficient value of f, then (Δ_ηⁿf-α)/(f-α) ≡ c for some non-zero constant c. We also consider the uniqueness of entire function f sharing α CM with its difference operators Δ_ηf and Δ_ηⁿf.
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  The Dual Space of Pluriharmonic Function Space

  Li HE, Guang Fu CAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 833-840. https://doi.org/10.12386/A2015sxxb0082

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  We consider the pluriharmonic function space and the composition operators on it, mainly about characterizing the dual space of pluriharmonic function space, describing the boundedness, compactness and Fredholmness of composition operators.
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  Some Infinite Sums Equal Zero

  Zhong Feng ZHANG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 841-846. https://doi.org/10.12386/A2015sxxb0083

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  We investigate the infinite convergent sum T = ∑_n=0^∞(P(n)/Q(n)) = 0, where P(x) ∈ Q[x], Q(x) ∈ Q[x], gcd(P(x), Q(x))=1 and Q(x) has only simple rational zeros which are all in the interval [-1, 0), and prove that for each integer m ≥ 5, there are infinite Q(x) with degree m such that T is not a transcendental number.
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  2-Local Derivations of Finite-Dimensional Simple Lie Algebras

  Xuan LAI, Zheng Xin CHEN

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 847-852. https://doi.org/10.12386/A2015sxxb0084

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  Let F be an algebraically closed field of characteristic 0, g a finite-dimensional simple Lie algebra over F. Amap φ on g is called a 2-local derivation, if for any x, y ∈ g, there is a derivation D_{x, y}: g→g, such that φ(x)=D_{x, y}(x), φ(y)=D_{x, y}(y). We prove that any 2-local derivation of g is an inner derivation.
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  On the Continuity of Double Derivations

  Bing YANG, Cheng Jun HOU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 853-860. https://doi.org/10.12386/A2015sxxb0085

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  The double derivations are the generalized forms of ordinary derivations. For two mappings δ and ε from a complex linear algebra A into itself, a linear mapping d from A into itself is called a (δ, ε)-double derivation, if d(ab) = d(a)b+ad(b)+δ(a)ε(b)+ ε(a)δ(b) for all a, b ∈ A. We studies the problem on the automatic continuity of double derivations of Banach algebras. We prove that, if δ and ε are two continuous at zero mappings from a unital C^*-algebra A into itself, then every (δ, ε)-double derivation of A is automatically continuous.
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  Higher All-Derivable Points in Nest Algebras

  Lei LIU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 861-870. https://doi.org/10.12386/A2015sxxb0086

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  Let N be a nontrivial complete nest on a Hilbert space H. We say that φ = {φ⁽ⁿ⁾}_n∈N is a higher derivable linear mapping at G if φ⁽ⁿ⁾(ST) = ∑_#em/em#+j=nφ⁽ⁱ⁾(S)φ^j(T) for all n ∈ N and S, T ∈ algN with ST = G. An element G ∈ algN is called a higher all-derivable point of algN if every higher derivable linear mapping φ = {φ⁽ⁿ⁾}_n∈N at G is a higher derivation. In this paper, we show G ∈ algN is a higher all-derivable point if and only if G ≠ 0.
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  Differences of Generalized Integration Operators from F(p,q,s) Spaces to Bloch-Type Spaces

  Zhong Hua HE

  Acta Mathematica Sinica, Chinese Series. 2015, 58(5): 871-880. https://doi.org/10.12386/A2015sxxb0087

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  A generalized integration operator is defined by I_g,φ⁽ⁿ⁾f(z)=∫₀^zf⁽ⁿ⁾(φ(ζ))g(ζ)dζ induced by holomorphic maps g and φ of the unit disk D, where φ(D) ⊂ D and n is a positive integer. In this paper, we investigate the boundedness and the compactness of the differences of two generalized integration operators from F(p, q, s) spaces to Bloch-type spaces.