15 January 2013, Volume 56 Issue 1

    

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  Asymptotic Behavior of Stochastic Boussinesq Equations with Additive White Noises

  Wen Qiang ZHAO, Yang Rong LI

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 1-14. https://doi.org/10.12386/A2013sxxb0001

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  This paper is devoted to the investigation of asymptotic behavior of stochastic Boussinesq equations influenced by random exterior forced white noises both in the velocity and in the temperature fields. It is showed that the stochastic Boussinesq equations can be solved path-wise and the unique solution generates a random dynamical system. Then the existence of a random attractor for the generated random dynamical system is established.
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  Numerical Study of Stabilization of the Lower Order Finite Volume Methods for the Incompressible Flows

  Jian LI, Xin ZHAO, Jian Hua WU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 15-26. https://doi.org/10.12386/A2013sxxb0002

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  In this paper, the stabilized finite volume method is considered for the R^d,d=2, 3 Stokes equations approximated by the lower order finite element pairs (P₁-P₀ and P₁-P₁), which do not satisfy the so-called inf-sup condition. This method applies the local pressure projection to stabilize the lower order finite element. The convergence analysis also shows an important superclose result O(h²) between the conforming mixed finite element solution and the finite volume solution using the same finite element pair for the incompressible flow. Based on the relationship between the finite element method and finite volume method, optimal estimate and superconvergenc result are obtained. Moreover, the stabilized finite volume method has a convergence rate of the same order as that of the usual stabilized finite element method of the stationary Stokes equations with additional regular assumption on the body force f ∈H¹.
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  The Generalized Acyclic Chromatic Number of Graphs with Large Girth

  Jian Sheng CAI, Guang Hui WANG, Gui Ying YAN

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 27-30. https://doi.org/10.12386/A2013sxxb0003

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  A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle C receives at least min{|C|, r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree Δ and with girth at least 2(r ? 1)Δ is at most 6(r ? 1)Δ.
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  Finite Solvable Groups with Irreducible Character Degrees of Arithmetic Number

  Sheng An CHEN

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 31-40. https://doi.org/10.12386/A2013sxxb0004

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  Let G be a finite group, cd(G) be the set of all complex irreducible character degrees of G. In this paper, we investigate the finite solvable groups with irreducible character degrees of arithmetic number, and obtain two results: If cd(G) = {1,1 + d,1 + 2d, ...,1 + kd}, then k ≤ 2 or cd(G) = {1, 2, 3, 4}; if cd(G) = {1, a, a + d, a + 2d, ..., a + kd}, |cd(G)| ≥ 4, (a, d) = 1, then cd(G) = {1, 2, 2^e + 1, 2^e+1}, and present the group structure of case d > 1.
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  Oblique Dual g-Frames in Hilbert Spaces

  Xiang Chun XIAO, Yu Can ZHU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 41-52. https://doi.org/10.12386/A2013sxxb0005

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  We generalize the oblique principles to the more general g-frames in Hilbert spaces. We give the equivalent condition for a g-frame {Λ_j: j ∈ J} to be an oblique dual g-frame of {Γ_j: j ∈J}. We also give the sufficient condition for a pair of oblique dual g-frames to be symmetric. At the end of this paper we construct several pairs of oblique dual g-frames under different conditions.
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  The Laplace-Stieltjes Transform of Infinite Order in the Whole Complex Plane

  Yin Ying KONG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 53-60. https://doi.org/10.12386/A2013sxxb0006

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  By using a Type-function of Sun, the growth of the Laplace-Stieltjes transform convergent in the whole complex plane are obtained, which cover and extend some results of Dirichlet series.
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  One Certain Type of Differential and Difference Equations Related to Brück Conjecture

  Jie ZHANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 61-66. https://doi.org/10.12386/A2013sxxb0007

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  We investigate the question that an entire function and its k-th derivative share some functions by utilizing Nevanlinna theory and Wiman-Valiron theory. We obtain some uniqueness theorems which are closely related to the Brück's conjecture in this direction.
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  Global Existence and Blow-up Properties for a Parabolic System with Localized Source and Absorption

  Jun ZHOU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 67-86. https://doi.org/10.12386/A2013sxxb0008

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  The aim of this paper is to investigate the following parabolic system
  
  under homogeneous Dirichlet boundary condition, where x₀(t):R⁺→(0,a) is Hölder continuous, and the constants 0≤α, β <1, p₁,p₂,q₁,q₂,k₁,k₂>0. Under appropriate hypotheses, we first prove the local existence of classical solution by a regularization method. Then we discuss the global existence and blow-up of positive solutions by using a comparison principle. Finally, we give the precise blow-up estimates and the uniform blow-up profiles.
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  Classification of Embeddings of Closed-end Multi-Ladders on Surfaces

  Ting GUO, Yuan Qiu HUANG, Xin Qiu LIU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 87-96. https://doi.org/10.12386/A2013sxxb0009

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  In this paper, by using the embedding model of joint tree introduced by Yanpei Liu, we obtain a recurrence relation about the genus distribution of closedend double-ladders on orientable surfaces, and compute the number of embeddings of closed-end multi-ladders on the projective plane.
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  A Hilbert-Type Integral Inequality with the Kernel of Hyperbolic Cosecant Function

  Qiong LIU, Shun Chao LONG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 97-104. https://doi.org/10.12386/A2013sxxb0010

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  By introducing Γ-function and ζ-function, and using the way of weight function and the method of real analysis, a Hilbert-type integral inequality with the kernel of hyperbolic cosecant function is given. The equivalent form is considered and their constant factor are proved being the best possible. By taking the special parameter values, some meaningful results are obtained.
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  Tight K-Frames in Hilbert Spaces

  Ming Ling DING, Xiang Chun XIAO, Xiao Ming ZENG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 105-112. https://doi.org/10.12386/A2013sxxb0011

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  K-frames are the generalizations of frames. In this paper, we introduce the definition of a tight K-frame in a Hilbert space, then we give a characterization for tight K-frames by the synthesis operator and the operator K. By means of the operator K, we also obtain a necessary and sufficient condition for tight K-frames to be tight frames. In the end we make a discussion on the construction for tight K-frame and the correlations between two tight K-frames and the involved operator K.
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  Some Sharp Hardy Inequalities on Half Space in Heisenberg Group

  Ying Xiong XIAO

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 113-120. https://doi.org/10.12386/A2013sxxb0012

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  We prove some sharp Hardy inequalities on half spaces for the sublaplacian in Heisenberg group. As a plication, we obtain some sharp Rellich type inequalities for the same operator.
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  Solutions of a Resonant Semilinear Elliptic Equation with Hardy Singular Terms

  Yong Yi LAN, Chun Lei TANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 121-134. https://doi.org/10.12386/A2013sxxb0013

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  We consider the following semilinear elliptic equation ?Δu-μ(u/|x|² = f(x, u) with the Dirichlet boundary value, and under suitable assumptions on the nonlinear term f. Some existence results of solutions are given via the variational method.
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  Voronovskaya-Type Formulas for ω, q-Bernstein Polynomials

  Hai Xin JIANG, Yun WU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(1): 135-144. https://doi.org/10.12386/A2013sxxb0014

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  We discuss Voronovskaya-type formulas and saturation of convergence for ω, q-Bernstein polynomials. We give explicit formulas of Voronovskaya-type for ω, q- Bernstein polynomials for 0 < q < 1, 0 ≤ ω ≤ 1. If 0 < q < 1, we show that the rate of convergence for ω, q-Bernstein polynomials is o(qⁿ) if and only if {f(1-q^k-1)-f(1-q^k)}/{(1-q^k-1)-(1-q^k)}=f'(1-q^k), k = 1, 2, .... We also prove that if f is convex on [0, 1] or analytic on (?ε,1+ε) for some ε > 0, then the rate of convergence for ω, q-Bernstein polynomials is o(qn) if and only if f is linear.