15 September 2014, Volume 30 Issue 9

    

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  Least Number of Periodic Points of Self-maps of Lie Groups

  Jerzy JEZIERSKI

  Acta Mathematica Sinica. 2014, 30(9): 1477-1494. https://doi.org/10.1007/s10114-014-3193-5

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  There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M→M of a compact smooth manifold of dimension at least 3:NF_n(f)=min{#Fix(gⁿ); g~f; g is continuous} and NJD_n(f)=min{#Fix(gⁿ); g~f; g is smooth}. In general, NJD_n(f) may be much greater than NF_n(f). If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds ⇔ all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.
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  Graded Hochschild Cohomology of a Path Algebra with Oriented Cycles

  Fang LI, De Zhan TAN

  Acta Mathematica Sinica. 2014, 30(9): 1495-1512. https://doi.org/10.1007/s10114-014-2661-2

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  The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.
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  Decomposition Theorems and Conjugate Pair in D_K Spaces

  Ji Zhen ZHOU, Yu Tian WU

  Acta Mathematica Sinica. 2014, 30(9): 1513-1525. https://doi.org/10.1007/s10114-014-3245-x

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  Let K be a right-continuous and nondecreasing function. A function f analytic in the unit disk D belongs to the space D_K if ∫D|f'(z)²K(1-|z|²)dA(z) < ∞. Decomposition theorems for D_K spaces are established in this paper. As an application, we obtain a characterization of interpolation by functions in D_K spaces. Furthermore, we characterize functions in D_K spaces by conjugate pairs.
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  On a New Geometric Constant Related to the Modulus of Smoothness of a Banach Space

  Yasuji TAKAHASHI, Mikio KATO

  Acta Mathematica Sinica. 2014, 30(9): 1526-1538. https://doi.org/10.1007/s10114-014-2782-7

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  We shall introduce a new geometric constant A(X) of a Banach space X, which is closely related to the modulus of smoothness ρX(τ), and investigate it in relation with the constant A₂(X) by Baronti et al., the von Neumann-Jordan constant C_NJ(X) and the James constant J(X). A sequence of recent results on these constants as well as some other geometric constants will be strengthened and improved.
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  Complete and Complete Moment Convergence for Weighted Sums of Widely Orthant Dependent Random Variables

  De Hua QIU, Ping Yan CHEN

  Acta Mathematica Sinica. 2014, 30(9): 1539-1548. https://doi.org/10.1007/s10114-014-3483-y

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  In this paper, we establish a complete convergence result and a complete moment convergence result for weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results for weighted sums of extended negatively orthant dependent random variables are also obtained, which generalize and improve the related known works in the literature.
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  Coleman Automorphisms of Holomorphs of Finite Abelian Groups

  Zheng Xing LI, Jin Ke HAI

  Acta Mathematica Sinica. 2014, 30(9): 1549-1554. https://doi.org/10.1007/s10114-014-2143-6

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  Let K be a finite abelian group and let H be the holomorph of K. It is shown that every Coleman automorphism of H is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalizer property holds for H.
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  A LIL and Limit Distributions for Trimmed Sums of Random Vectors Attracted to Operator Semi-stable Laws

  Wen Sheng WANG

  Acta Mathematica Sinica. 2014, 30(9): 1555-1565. https://doi.org/10.1007/s10114-014-3487-7

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  Let θ∈ R^d be a unit vector and let X,X₁,X₂,... be a sequence of i.i.d. R^d-valued random vectors attracted to operator semi-stable laws. For each integer n≥ 1, let X_1,n≤ … ≤,X_n,n denote the order statistics of X₁,X₂,...,X_n according to priority of index, namely |<X_1,n, θ>| ≥ … ≥ |<X_n,n,θ>|, where <·, ·> is an inner product on R^d. For all integers r≥ 0, define by ^(r)S_n=∑_i=1^n-r X_i,n the trimmed sum. In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums ^(r)S_n. Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded. A stochastically compactness of ^(r)S_n is obtained.
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  Convergence in L^p-norm of Fourier Series on the Complete Product of Quaternion Groups with Bounded Orders

  György GÁT, Rodolfo TOLEDO

  Acta Mathematica Sinica. 2014, 30(9): 1566-1578. https://doi.org/10.1007/s10114-014-3360-8

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  A well-known result for Vilenkin systems is the fact that for all 1 < p < ∞ the n-th partial sums of Fourier series of all functions in the space L^p converge to the function in L^p-norm. This statement can not be generalized to any representative product system on the complete product of finite non-abelian groups, but even then it is true for the complete product of quaternion groups with bounded orders and monomial representative product system ordered in a specific way.
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  The Distance Coloring of Graphs

  Lian Ying MIAO, Yi Zheng FAN

  Acta Mathematica Sinica. 2014, 30(9): 1579-1587. https://doi.org/10.1007/s10114-014-3238-9

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  Let G be a connected graph with maximum degree Δ ≥ 3. We investigate the upper bound for the chromatic number χ_γ(G) of the power graph Gγ. It was proved that χ_γ(G)≤ Δ((Δ-1)^γ-1)/(Δ-2)+1=:M+1, where the equality holds if and only if χ_γ(G) is a Moore graph. If χ_γ(G) is not a Moore graph, and G satisfies one of the following conditions:(1) G is non-regular, (2) the girth g(G) ≤ 2γ-1, (3) g(G) ≥ 2γ+2, and the connectivity κ(G) ≥ 3 if γ ≥ 3, κ(G)≥ 4 but κ(G) > 6 if γ=2, (4) Δ is sufficiently larger than a given number only depending on γ, then χ_γ(G)≤M-1. By means of the spectral radius λ₁(G) of the adjacency matrix of G, it was shown that χ₂(G) ≤ λ₁(G)₂+1, where the equality holds if and only if G is a star or a Moore graph with diameter 2 and girth 5, and χ_γ(G) < λ₁(G)γ+1 if γ≥ 3.
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  On Coprime G-conjugacy Class Sizes in a Normal Subgroup

  Xian He ZHAO, Hai Peng Qu, Gui Yun CHEN

  Acta Mathematica Sinica. 2014, 30(9): 1588-1594. https://doi.org/10.1007/s10114-014-2803-6

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  Let N be a normal subgroup of a group G. Suppose that the positive integers m > n are two longest non-central G-conjugacy class sizes of N with (m,n)=1. The purpose of this paper is to determine the structure of N and give the N-conjugacy class sizes of the elements in N under that assumption that m is square free.
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  A Remark on Strong Law of Large Numbers for Weighted U-Statistics

  Hyung-Tae HA, Mei Ling HUANG, De Li LI

  Acta Mathematica Sinica. 2014, 30(9): 1595-1605. https://doi.org/10.1007/s10114-014-1601-5

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  Let {X,X_n; n≥ 1} be a sequence of i.i.d. random variables with values in a measurable space (S, S) such that E|h(X₁,X₂,...,X_m)| < ∞, where h is a measurable symmetric function from S^m into R=(-∞,∞). Let {w_{n,i1,i2,...,im}; 1 ≤ i₁ < i₂ < … < i_m ≤ n, n ≥m} be a matrix array of real numbers. Motivated by a result of Choi and Sung (1987), in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m. We show that 
  
  whenever sup n≥m max 1≤i₁2<…m≤n |w_{n,i1,i2,...,im}|< ∞, where θ=Eh(X₁,X₂,..., X_m). The proof of this result is based on a new general result on complete convergence, which is a fundamental tool, for array of real-valued random variables under some mild conditions.
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  Efficient Statistical Inference for Partially Nonlinear Errors-in-Variables Models

  San Ying FENG, Gao Rong LI, Jun Hua ZHANG

  Acta Mathematica Sinica. 2014, 30(9): 1606-1620. https://doi.org/10.1007/s10114-014-1358-x

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  In this paper, we consider the partially nonlinear errors-in-variables models when the nonparametric component is measured with additive error. The profile nonlinear least squares estimator of unknown parameter and the estimator of nonparametric component are constructed, and their asymptotic properties are derived under general assumptions. Finite sample performances of the proposed statistical inference procedures are illustrated by Monte Carlo simulation studies.
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  Deterministic Homogenization of Nonlinear Degenerate Elliptic Operators with Nonstandard Growth

  Hubert NNANG

  Acta Mathematica Sinica. 2014, 30(9): 1621-1654. https://doi.org/10.1007/s10114-014-2131-x

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  We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth. One fundamental in this topic is to extend the classical compactness results of the ∑-convergence method to the Orlicz spaces. We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.