中国科学院数学与系统科学研究院期刊网

15 July 2014, Volume 57 Issue 4
    

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  • He Guo LIU, Ji Ping ZHANG, Tao XU
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 625-656. https://doi.org/10.12386/A2014sxxb0059
    Abstract ( ) Download PDF ( )   Knowledge map   Save
    Let G be a nilpotent group of finite rank and G = KP, where P is a nilpotent π-group, K is a π'-free normal subgroup, and π does not belong to the spectrum of K. Suppose that1=ζ0G1G<…<ζCG=G is the upper central series of G, that α and β are two automorphisms of G, that αi and βi are the automorphisms of ζiGi-1G induced by α and β, respectively. Put Ii = Im(αi-βi βiαi), then
    (i) in case Ii is a finite cyclic group, and I:= (αβ(g))(βα(g))-1|gG is a finite subgroup of G, then α, β is a soluble abelian-by-finite group.
    (ii) in case Ii is a finite cyclic group, or a divisible group of rank 1, or CD, where C is a cyclic group and D is a divisible group of rank 1, or a torsion-free locally cyclic group, or Ii has a normal series 1 < Ji < Ii, one of whose factors is a finite cyclic group and another is torsion-free locally cyclic group, or Ii = D ⊕ Ji, where D is a divisible group of rank 1 and Ji is a torsion-free locally cyclic group, or Ii has a normal series 1 < Ki < Ji < Ii,whose 3 factors are a finite cyclic group, a divisible group of rank 1 and a torsion-free locally cyclic group, respectively, then α, β is a soluble nilpotent-by-abelian-by-finite group.
    In particular, if α and β are two π'-automorphisms of G. Then
    (iii) in case Ii is a finite cyclic group, and I:= (αβ(g))(βα(g))-1|gG is a finite subgroup of G, then α, β is an extension of a finite nilpotent π-group by a finite abelian π -group. (iv) in case Ii is a cyclic group, or a divisible group of rank 1, or CD, where C is a cyclic group and D is a divisible group of rank 1, then α, β is both a soluble residually finite π ∪ π'-group and is an extension of a finitely-generated torsion-free nilpotent group by a finite π ∪ π'-group.
    (v) in case Ii is a finite cyclic group, or a divisible group of rank 1, or CD, where C is a cyclic group and D is a divisible group of rank 1, or a torsion-free locally cyclic group, or Ii has a normal series 1 < Ji < Ii, one of whose factors is a finite cyclic group and another is a torsion-free locally cyclic group, or Ii = DJi, where D is a divisible group of rank 1 and Ji is a torsion-free locally cyclic group, or Ii has a normal series 1 < Ki < Ji < Ii, whose 3 factors are a finite cyclic group, a divisible group of rank 1 and a torsion-free locally cyclic group, respectively, then α, β is a soluble residually finite π ∪ π'-group and its nilpotent length is at most 4.
    In (v), if K is a FC-group, then α, β is an extension of a finitely-generated torsion-free nilpotent group by a finite π ∪ π'-group.
    In addition, if G = KP and K is a FC-group, we investigate the dual problem for the lower central series of G and obtain the dual results.
  • Yao Ping HOU, Ai Xiang FANG
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 657-664. https://doi.org/10.12386/A2014sxxb0060
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    Let G be a simple undirected graph. The graph G is said to have forcing (resp., admissible) reciprocal skew eigenvalue property if each eigenvalue λ of skewadjacency matrix S() and its reciprocal 1/λ have the same multiplicity as eigenvalues of S() for all (resp., some) orientations of G. In this paper, all unicyclic graphs with forcing (resp., admissible) reciprocal skew eigenvalue property are determined.
  • He Yan JIN
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 665-674. https://doi.org/10.12386/A2014sxxb0061
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    We consider in this paper the estimation of the optimal constant āp in π(|f-π(|f|p-2f)|p)≤āpπ(a|f'|p),fD(Dp) under Neumann boundary condition when π is a finite measure. We can get the estimations of upper and lower bounds by using cutting method to transform this problem to Dirichlet boundary condition. When π is infinite measure, we consider the upper and lower bound of the constant in π(|f|p)≤?pπ(a|f'|p),fD(Dp) under Neumann boundary condition. We give the variational formula and the explicit estimation.
  • Jun YAN
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 675-680. https://doi.org/10.12386/A2014sxxb0062
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    We study the moderate deviation principle for the partly shifted risk process by constructing a martingale. A numerical simulation is shown to support our result.
  • Ling Di WANG, Yu Hui ZHANG
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 681-692. https://doi.org/10.12386/A2014sxxb0063
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    In the paper, the explicit criteria for zero-exit (resp., zero-entrance) of (possible non-conservative) single birth (resp., single death) Q-matrices are presented with an analytic proof and a unified treatment is given for the conservative and non-conservative cases. Moreover, the criteria for zero-exit and zero-entrance of nonconservative birth-death Q-matrices are obtained again. Meanwhile, a similar treatment is given for the corresponding problem of single birth Q-matrices with immigration.
  • Yong Min LIU, Jian GUO
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 693-708. https://doi.org/10.12386/A2014sxxb0064
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    In this paper, using the estimate of the higher order derivative of the function in Hardy spaces, the properties of the analytic function and operator theory, the authors characterize the boundedness and compactness of the Riemann-Stieltjes operator from Hardy spaces Hp to Zygmund-type spaces by means of constructing some test functions, and some necessary and sufficient conditions of the Riemann-Stieltjes operators to be bounded and compact are obtained.
  • Wei Chao GUO, Xiao Mei WU
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 709-718. https://doi.org/10.12386/A2014sxxb0065
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    We study the Cauchy problem for the high order Schrödinger equation, and obtain the global well-posedness with small data.
  • Miao Kun WANG, Ying Qing SONG, Yu Ming CHU
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 719-726. https://doi.org/10.12386/A2014sxxb0066
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    We present several inequalities for zero-balanced Gaussian hypergeometric function F(aba+bx) with a, b > 0, which generalize an identity involving the inverse hyperbolic tangent function.
  • Jia Liang QIU, Zong Xuan CHEN
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 727-736. https://doi.org/10.12386/A2014sxxb0067
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    We investigate the problems of zeros of solutions for some higher order nonhomogeneous linear differential equations. When A0 is a meromorphic function that satisfies δ(∞,A0) = δ(> 0) and limr→∞ logT(rA0)/log r=∞, if f1 and f2 are two linearly independent meromorphic solutions of equation fk + Ak-1fk-1+…+A0f=F,we obtain max{λ(f1),λ(f2)}=∞.And we also investigate the cases that σ(F) = ∞, or some Ad (1≤ dk-1) satisfies limr→∞ logm(rAd)/log r=∞.
  • Deng Feng LI, Shu Yan CHEN
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 737-744. https://doi.org/10.12386/A2014sxxb0068
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    This paper investigates the sparsity of dual frames by using the concept sparkj(Φ) of matrix Φ. First, a new expression of sparsity of optimal sparse dual frames for spectral tetris frame is established. Second, a sufficient and necessary condition of a finite frame, whese dual frame's sparsity is minimal, is presented.
  • Ti Ren HUANG
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 745-750. https://doi.org/10.12386/A2014sxxb0069
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    Let G be a Carnot group and D={e1e1} be a bracket generating left invariant distribution on G. In this paper, we can construct a class of n-dimensional (n > 5) Carnot groups such that there exist strictly abnormal extremals in it. So we know that the first Carnot group with singular geodesics which constructed by Golé, Karidi is only a particular example in present paper.
  • Lin Fang LIU, Fu Qi YIN
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 751-766. https://doi.org/10.12386/A2014sxxb0070
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    We focus on the asymptotic behavior of solution for reaction-diffusion equation with multiplicative noise on unbounded domains. The uniform estimates is used to obtain the Lp-random absorbing set. A new priori estimate method, called asymptotic priori estimate, is applied to the unbounded part of solution to establish the Lp-asymptotic compactness of the corresponding random dynamical system, then the existence of Lp-random attractor is proved. This random attractor is not only a compact set but an invariant tempered set which attracts every tempered random subset of L2 in the topology of Lp. The nonlinearity term f is allowed to possess some growth of arbitrary order p - 1, where p ≥ 2.
  • Chong Shan YUE
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 767-774. https://doi.org/10.12386/A2014sxxb0071
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    Peter and Duan studied the bitangent cicle problem of plane curves, Duan researched into the bitangent sphere problem of (n-1)-dimensional closed submanifolds in Rn. In this paper, we study the bitangent circle problem of space curves, and by geometrical analysis method, show that if it is a good curve, a space curve must have biangent circle at every point.
  • Fang Juan ZHANG
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 775-784. https://doi.org/10.12386/A2014sxxb0072
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    Let M be a unital prime *-ring containing a nontrivial projection P. It is shown that a nonlinear bijective map Ø:MM satisfies Ø(AB - ξBA*) = Ø(A)Ø(B) - ξØ(B)Ø(A)* for all A,BM. If ξ = 1, then Ø is a linear or conjugate linear *-isomorphism; if ξ≠ 1, then Ø is a *-ring isomorphism and ØA) = ξØ(A) for all AM.
  • Lei QIAO
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 785-794. https://doi.org/10.12386/A2014sxxb0073
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    In this paper, using related properties of Whitney cubes, we not only give growth properties for Green potentials at infinity in a cone, but also prove covering theorems for exceptional sets of them.
  • Wen Qiang ZHAO
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 795-810. https://doi.org/10.12386/A2014sxxb0074
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    This paper is concerned with the random dynamics of stochastic three dimensional viscous Camassa-Holm model with additive noises. By proving the stochastic Flattening Condition on the solution orbits the w-limits compactness for the generated random dynamical system is followed. Then by using the recent results about quasi-continuous random dynamical system developed by Li and Guo recently, the random dynamics for this is captured by an H2-attractor.
  • Lin CHEN, Qiong Xiang HUANG
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 811-824. https://doi.org/10.12386/A2014sxxb0075
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    The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We prove that there exist exactly eleven tetracyclic graphs with maximum Laplacian spread among all tetracyclic graphs on n(n≥12) vertices.
  • Bei LIU, Rui LIU
    Acta Mathematica Sinica, Chinese Series. 2014, 57(4): 825-832. https://doi.org/10.12386/A2014sxxb0076
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    The customary definition of redundancy for finite frames in a Hilbert space was improved recently. In this paper, we generalize the quantitative notion of redundancy for finite frames in Hilbert space to finite p-frames in Banach space. We consider the maximum and minimum of a redundancy function which was defined on the sphere of the Banach space and present the properties of upper and lower redundancy.