15 May 2015, Volume 58 Issue 3

    

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  Highest Weight Representations of the Completed Witt Algebra

  Jun Ping LIANG, Guang Zhe FAN

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 353-358. https://doi.org/10.12386/A2015sxxb0035

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  We investigate the irreducibility of Verma modules over the completed Witt algebra. By embedding the Virasoro algebra into the completed Witt algebra, we give realizations of two classes of irreducible non-weight Virasoro-modules. We also obtain some interesting combinatorial identities.
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  Quasirecognition of the Simple Group ²G₂(3²ⁿ⁺¹)

  Li Li LI

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 359-364. https://doi.org/10.12386/A2015sxxb0036

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  Let G be finite group such that M(G) = M(²G₂(3²ⁿ⁺¹)). Then G has a normal subgroup isomorphic to ²G₂(3²ⁿ⁺¹). Especially, if |G| = |²G₂(3²ⁿ⁺¹)|, then G ≌²G₂(3²ⁿ⁺¹).
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  Multiplicative Hom-Structures on Finite Dimensional Modular Lie Superalgebras of Cartan Type

  Xiu Ying HUA, Li Ming TANG, Wen De LIU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 365-372. https://doi.org/10.12386/A2015sxxb0037

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  Let F be an algebraically closed field of characteristic p > 3. Using the invariance of the natural filtration and calculating the actions on the -1 components for the Hom-structures of the modular Lie superalgebras under consideration, we prove that there exists only the trivial multiplicative Hom-structures on the finite dimensional Z-graded modular Lie superalgebras of Cartan type.
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  A Note on the Bound of Castelnuovo's Inequality

  Zheng Jun ZHAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 373-378. https://doi.org/10.12386/A2015sxxb0038

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  We present some explicit formulae for genus of the composite field of Kummer and Artin-Schreier function fields in some specific cases. Meanwhile, we point out in this note that the genus of function field we addressed reaches the upper bound of Castelnuovo's inequality.
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  On the Infinite-Dimensional Modular Lie Superalgebras of L-Type

  Jing ZHAO, Jin WANG, Xiao Ning XU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 379-400. https://doi.org/10.12386/A2015sxxb0039

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  The infinite-dimensional Lie superalgebras of L-type over a field of positive characteristic are studied. The simplicity is proved and the natural filtration of Lie superalgebras of L-type is proved to be invariant under automorphism group by using ad-nilpotent elements. We are thereby able to obtain a property of Lie superalgebras of L-type.
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  Classification and General Forms of Operator Algebras on Pontrjagin Space

  Hai Tao YANG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 401-418. https://doi.org/10.12386/A2015sxxb0040

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  We classify the general symmetric operator algebras on Pontrjagin space into six classes by using of symmetric ideals, and give the general forms for all the SM operator algebras in each class.
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  Finite Groups All of Whose Non-trivial Automorphisms Are Fixed-Point-Free

  Zheng Xing LI, Jin Ke HAI

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 419-422. https://doi.org/10.12386/A2015sxxb0041

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  Let G be a finite group. It is proved that every non-trivial automorphism of G is fixed-point-free if and only if G is an abelian simple group. As an application, it is proved that the holomorph of G is a Frobenius group with G as its Frobenius kernel if and only if G is of odd prime order.
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  Perturbations of the Point Spectrum, Continuous Spectrum and Residual Spectrum for Third-Order Upper Triangular Operator Matrices

  Xiu Feng WU, Jun Jie HUANG, Alatancang

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 423-430. https://doi.org/10.12386/A2015sxxb0042

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  Let H₁, H₂, H₃ be infinite-dimensional complex separable Hilbert spaces, and write M_D,E,F = ∈ B(H₁ ⊕ H₂ ⊕ H₃). Given the operators A ∈ B(H₁), B ∈ B(H₂), C ∈ B(H₃), some complete descriptions of perturbations of the point spectrum, continuous spectrum and residual spectrum for the partial operator matrix M_D,E,F are given, based on the analysis method and block operator technique.
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  Notes on Levin Type Inequalities of Harmonic Functions

  Guo Chang WU, Yu Dong REN, Bai Yun SU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 431-434. https://doi.org/10.12386/A2015sxxb0043

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  For a class of harmonic functions in a half space, we give a generalized Levin type inequality, which generalizes the result obtained by Zhang-Deng-Gao.
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  The Age Structure of Nodes in Markov Random Branching Tree

  Han Xing WANG, Rui Jie DU, Yun Bin FU, Yun Zhi YAN

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 435-446. https://doi.org/10.12386/A2015sxxb0044

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  This paper studies the random branching tree, on which child nodes arrival process is time-homogeneous Poisson process, mainly gets analytical analysis of some characteristics: the number of real nodes and the number of virtual nodes in branching tree, the number of each generation in branching tree, the number of real nodes in every connected component; the number of real nodes and the number of virtual nodes in different age interval in whole branching tree, the number of real nodes in different age interval on a single real component, the number of real nodes in different age interval on each generation of branching tree; the number of nodes or real nodes born at suit-age or at overage.
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  Parabolic Monge-Ampère Equations on Exterior Domains

  Li Mei DAI

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 447-456. https://doi.org/10.12386/A2015sxxb0045

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  We study the existence of solutions to the parabolic Monge-Ampère equations-u_tdet(D²u)=f on the exterior domains. Using the Perron method, we get the existence and uniqueness of solutions with the asymptotic behavior to the exterior problems of parabolic Monge-Ampère equations.
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  Existence of Multiple Solutions for a Quasilinear Elliptic Equation with Sign-Changing Nonlinearities

  Jia Fa XU, Zhong Li WEI, Wei DONG

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 457-462. https://doi.org/10.12386/A2015sxxb0046

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  We utilize the mountain pass theorem to study the quasilinear elliptic equation -div=λf(x,u) with Dirichlet boundary condition and obtain the existence result of multiple solutions; furthermore, the nonlinear term f is allowed to be sign-changing.
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  The Representation Dimensions of Skew Algebra and Its Trivial Extension

  Qian Hong WAN, Li Jing ZHENG, Jin Yun GUO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 463-468. https://doi.org/10.12386/A2015sxxb0047

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  Let Λ be a finite-dimensional selfinjective algebra over k, G be a finite group and the order of G be invertible in Λ, Λ * G be a skew algebra, T(Λ) be its trivial extension, ∧V be the exterior algebra. In this paper, we show the dimensions of stable module categories of selfinjective algebra and skew algebra over it are equal. In particular we obtain the representation dimensions of ∧V * G and T(∧V * G).
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  Explicit Factorization of x^2apbrc-1 over a Finite Field

  Fen LI Xi Wang CAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 469-478. https://doi.org/10.12386/A2015sxxb0048

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  Let F_q be a finite field of odd order q. In this paper, the irreducible factorization of x^2apbrc-1 over F_q is given in a very explicit form, where a, b, c are positive integers and p, r are odd prime divisors of q - 1. It is shown that all the irreducible factors of x^2apbrc-1 over F_q are either binomials or trinomials. In general, denote by v_p(m) the degree of prime p in the standard decomposition of the positive integer m. Suppose that every prime factor of m divides q -1, one has: (1) if v_p(m) ≤ v_p(q - 1) holds true for every prime number p | q - 1, then every irreducible factor of x^m -1 in F_q is a binomial; (2) if q ≡ 3 (mod 4), then every irreducible factor of x^m -1 is either a binomial or a trinomial.
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  Non-Commutative Poisson Algebra Structures on Some Extended Affine Lie Algebras

  Hong XIANG, Bo ZENG, You An CAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 479-490. https://doi.org/10.12386/A2015sxxb0049

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  We determine the non-commutative Poisson algebra structures on the extended affine Lie algebras of type A with nullity v whose coordinate algebra is the quantum torus. It is shown that the associative products on the derived Lie subalgebras are trivial, and the associative products of the scaling elements are described as well.
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  A Diophantine Inequality with Prime Variables and Mixed Power

  Quan Wu MU, Yun Yun QU

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 491-500. https://doi.org/10.12386/A2015sxxb0050

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  Let k be an integer with k ≥ 2 and η be any real number. Suppose that λ₁, λ₂, λ₃ are nonzero real numbers, not all of the same sign and λ₁/λ₂ is irrational. It is proved that the inequality |λ₁p₁+λ₂p₂+λ₃p₃^k +η| < (max p_j)^-σ has infinitely many solutions in prime variables p₁, p₂, p₃, where 0 < σ < 1/2(2^k+1+1) for 2 ≤ k ≤ 3, 0 < σ < 5/(6k2^k) for 4 ≤ k ≤ 5, and 0 < σ < 20/(21k^k) for k ≥ 6.
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  p-adic Estimates for Exponential Sums of Polynomials over Finite Fields

  Bo Hui DING, Wei CAO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 501-506. https://doi.org/10.12386/A2015sxxb0051

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  Let f be a multivariate polynomial over a finite field. Its degree matrix D_f is defined to be the matrix whose columns are the degree vectors appearing in f. In this paper, we obtain a p-adic estimate for the exponential sum of f in terms of D_f, which improves and generalizes some previously known results.
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  A Skewness-Adaptive Projection Depth

  Xiao Hui LIU, Shi Hua LUO

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 507-520. https://doi.org/10.12386/A2015sxxb0052

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  This paper extends the usual projection depth to a version adjusted for possible skewness in the data cloud. The proposal has convex depth regions, and satisfies all the properties of a general statistical depth function. Its computing issue is also investigated. It is shown that the exact computation of the proposal, as well as the related depth regions, is still possible in any dimension, and technically much simpler to be implemented than the usual projection depth. Finally, data examples are provided to illustrate the finite sample performance of this skewness-adaptive projection depth.
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  Quasi Uniform Disconnectedness of Fractals

  Fan LÜ, Ying XIONG, Li Feng XI

  Acta Mathematica Sinica, Chinese Series. 2015, 58(3): 521-528. https://doi.org/10.12386/A2015sxxb0053

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  The uniform disconnectedness plays an important role in the study of fractal geometry. In this paper, we discuss the quasi uniform disconnectedness. For a class of sets defined by sequence, we obtain the necessary and sufficient condition for them to be (quasi) uniformly disconnected. We also introduce Moran sets with slow change and give a sufficient condition to insure their quasi uniform disconnectedness.