15 July 2011, Volume 54 Issue 4

    

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  Maximal Commutators for Multilinear Singular Integrals with Non-Smooth Kernels

  Xiao Li CHEN, Jie Cheng CHEN

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 529-540. https://doi.org/10.12386/A2011sxxb0054

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  This paper is concerned with the Cotlar’s inequality for the maximal commutators of the m-linear Calderón-Zygmund operator with non-smooth kernels and the weighted norm inequalities for the commutators and the maximal commutators of multilinear singular integrals are established.  
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  Completeness of Root Vector Systems of a Class of Infinite-Dimensional Hamiltonian Operators

  Hua WANG, Jun Jie HUANG, Alatancang

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 541-552. https://doi.org/10.12386/A2011sxxb0055

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  This paper deals with the eigenvalue problem of the Infinite-Dimensional Hamiltonian operators with the diagonal elements being constant. Based on certain properties of their eigenvalues and eigenvectors of the product of the off-diagonal elements , the location of their eigenvalues, symplectic orthogonal relationship between eigen or root vectors, and the completeness of the eigen or root vectors system are characterized.  
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  Existence Analysis of the Positive Steady-State Solutions for a Glycolysis Model

  Mei Hua WEI, Jian Hua WU

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 553-560. https://doi.org/10.12386/A2011sxxb0056

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  This paper deals with a representative glycolysis model in biochemical reaction. We study the non-existence of non-constant positive steady-state solutions by using a priori estimates. A necessary condition for the existence of non-constant positive steady-state solutions is obtained. On the basis of Turing instability of constant steady-state solutions, the degree theory is combined with a priori estimates to give a sufficient condition for the existence of non-constant positive steady-state solutions.  
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  Non-commutative Poisson Algebra Structures on Extended Affine Lie Algebras

  Jie TONG, Quan Qin JIN

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 561-570. https://doi.org/10.12386/A2011sxxb0057

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  Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In the paper, the non-commutative poisson algebra structures on extended affine Lie algebras are determined.  
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  A BMO Characterization of the Boundedness for the Commutator of Parabolic Fractional Integral

  Yan Ping CHEN

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 571-580. https://doi.org/10.12386/A2011sxxb0058

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  We proved that if the commutator [b,T_Ω,β] of parabolic fractional integral is bounded from L^p to L^q for some 1 < p, q < ∞ satisfying (1/q)=(1/p)-(β/α), then b ∈ BMO(Rⁿ,ρ). The results in this paper improve and extend some known conclusions.  
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  Fredholm Perturbation of Spectra of 2 × 2-Upper Triangular Matrices

  Shi Fang ZHANG, Huai Jie ZHONG, Jun De WU

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 581-590. https://doi.org/10.12386/A2011sxxb0059

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  Let H and K be complex infinite dimensional separable Hilbert spaces, A ∈ B(H), B ∈ B(K), C ∈ B(K,H) and M_C = (₀^A_B^C). In this paper, we characterize completely the Fredholm perturbation for the Weyl spectrum, essential spectrum, spectrum, left spectrum, right spectrum, lower semi-Fredholm spectrum, lower semi-Weyl spectrum and upper semi-Weyl spectrum of the upper triangular operator matrices M_C.  
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  Regular Semigroups with Regular ^*-Transversals

  Shou Feng WANG, Di ZHANG

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 591-600. https://doi.org/10.12386/A2011sxxb0060

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  In this paper, some remarkable properties of regular semigroups with a regular ^*-transversal are given. A construction method of a regular semigroup with a quasi-ideal regular ^*-transversal is obtained. Furthermore, congruences on this class of semigroups are also considered by applying our structure theorem.  
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  The Minimal Signless Lapacian Spectral Radius of Graphs with Diameter n - 4

  Xiao Li WU, Jing Jing JIANG, Ji Ming GUO, Shang Wang TAN

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 601-608. https://doi.org/10.12386/A2011sxxb0061

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  In the paper [The minimal Lapacian spectral radius of trees with a given diameter, Theoretical Computer Science 410: 78-83], Liu, Lu and Shu (2009) determine the trees with minimal Lapacian spectral radii for the diameter belongs to {1, 2, 3, 4, n - 3, n - 2, n - 1}, respectively. This paper gives the unique graph with minimal signless Lapacian spectral radius among graphs with diameter n - 4. Furthermore as a corollary, we determine the tree with minimal Lapacian spectral radius among trees with diameter n - 4.  
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  Divisibility Properties of Smith Matrices on the Gcd-Closed set S with |S| = 6

  Jian Rong ZHAO, Wei ZHAO, Mao LI

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 609-618. https://doi.org/10.12386/A2011sxxb0062

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  In this paper, we give the necessary and sufficient conditions on the gcd-closed set S with |S| = 6 such that the power GCD matrix (S^e) divides the power LCM matrix [S^e] in the ring M_n(Z) of n × n matrices over the integers. This solves partially an open question raised by Hong in 2002.  
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  Finite Groups with Exactly Two Conjugacy Class Sizes of Subgroups

  Feng TANG

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 619-622. https://doi.org/10.12386/A2011sxxb0063

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  Let G be a finite group and Ns(G) denote the set of conjugacy class sizes of all subgroups of G. The aim of this paper is to investigate the finite groups G with Ns(G) exactly having two elements. We classfy the groups for the nonnilpotent case, and also obtain some properties for the nilpotent case.  
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  Uniqueness Theorems of Meromorphic Functions Sharing Sets IM on Annuli

  Ting Bin CAO, Hong Xun YI

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 623-632. https://doi.org/10.12386/A2011sxxb0064

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  The main purpose of this paper is to deal with the uniqueness problem for meromorphic functions on annuli. We obtain two general uniqueness theorems of meromorphic functions sharing sets from which an analog of Nevanlinna’s famous five-value theorem is proposed.  
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  The Representation of the Total Variation and the Metric Derivative for Fuzzy Bounded Variation Functions

  Zeng Tai GONG, Yu Juan BAI

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 633-642. https://doi.org/10.12386/A2011sxxb0065

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  The metric derivative of the fuzzy-number-valued functions and the representation of the total variation for the fuzzy-number-valued function which is of bounded variation are defined and discussed. It is proved that the fuzzy absolutely continuous functions are metrically differentiable almost everywhere, and the integration of its metric derivative equals to the total variation of the primitive. Finally, the representation of the total variation for the fuzzy-number-valued functions which is of bounded variation is given.  
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  On Completely J-Simple Semigroups

  Si Yao MA, Xue Ming REN, Ying YUAN

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 643-650. https://doi.org/10.12386/A2011sxxb0066

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  Completely J-simple semigroups are natural generalizations of completely simple semigroups and completely J^*-simple semigroups. In this paper, it is proved that a semigroup S is completely J-simple if and only if S is isomorphic to a normal Rees matrix semigroup M(T; I, Λ; P) over amonoid T. This result not only generalizes the well-known Rees theorem of completely simple semigroups, but also extends the structure theorem given by Ren and Shum in 2004 of completely J^*-simple semigroups.  
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  The Automorphism Group of a Generalized Extraspecial p-Group (II)

  Yu Lei WANG, He Guo LIU

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 651-658. https://doi.org/10.12386/A2011sxxb0067

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  In this paper, the automorphism group of a generalized extraspecial p-group G is determined again, where p is a prime number. Assume that |G| = p^2n+m and |ζG| = p^m, where n ≥ 1 and m ≥ 2. Let Aut_cG be the normal subgroup of AutG consisting of all elements of AutG which act trivially on ζG. Then
  (i) If p is odd, then AutG = 〈θ〉  Aut_cG, where θ is of order (p - 1)p^m-1; If p = 2, then AutG = 〈θ₁, θ₂〉 Aut_cG, where 〈θ₁, θ₂〉 = 〈θ₁〉 × 〈θ₂〉 Z_2^m-2 × Z₂.
  (ii) If the exponent of G is equal to p^m, then Aut_cG/InnG Sp(2n, p).
  (iii) If the exponent of G is equal to p^m+1, then Aut_cG/InnG K Sp(2n - 2, p), where K is an extraspecial p-group of order p^2n-1 (If p is odd) or an elementary abelian 2-group of order 2^2n-1. In particular, Aut_cG/InnG Zp when n = 1. 
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  Some Problems on the Convergence of Distributional Denjoy Integral

  Qiao Ling LIU, Guo Ju YE

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 659-664. https://doi.org/10.12386/A2011sxxb0068

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  This paper discusses some problems on the convergence of distributional Denjoy integral. Firstly, we give the relations between strong convergence, weak convergence, weak^* convergence and distributional Denjoy integral convergence in the space of Denjoy integrable distributions. We prove the quasi-uniform convergence is a sufficient and necessary condition of the convergence of distributional Denjoy integral. Finally, we show that weak^* convergence and strong convergence are equivalent if and only if the sequence of the primitive of Denjoy integrable distribution are equicontinuous.  
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  Some Basic Properties of J_β Space and Relations with l^β

  Jian Ze LI

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 665-668. https://doi.org/10.12386/A2011sxxb0069

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  We find the conjugate space of J_β (0 < β < 1) space. Furthermore, we introduce some basic properties of this space and relations with l^β.  
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  Several Results on a Problem of Setting Sites

  Yu Yang ZHU

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 669-676. https://doi.org/10.12386/A2011sxxb0070

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  The following problem of setting sites is considered in the paper. Let S ⊂ E^mbe a set consisting of n points A₁,A₂, . . . ,A_n, d(A_i,A_j)stands for the distance between A_i and A_j, and σ(S) = Σ_{1 ≤ i < j ≤ n} d(A_i,A_j), d(S) = min_{1 ≤ i ≠ j ≤ n}{d(A_i,A_j)}, μ(m, n) = (σ(S))/(d(S)) (S ⊂ E^m, |S| = n), inf μ(m, n) = min{(σ(S))/(d(S)) |S ⊂ E^m, |S| = n}. We proved that inf μ(n, n + 2) = C_n² - 1 + 2, inf μ (m,m + 1) = C_m², and obtained upper bound and lower bound of inf μ(m, n). In additional we also showed that inf μ(1, n) = ((n+1)/3) C_n², and gave some guesses. + 1+2  
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  A Note on the Diophantine System a²+b²=c^r and x²+b^y=c^z

  Yong Zhong HU, Mao Hua LE

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 677-686. https://doi.org/10.12386/A2011sxxb0071

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  Let r be an odd integer with r > 1, and let a, b, c be coprime positive integers satisfying a²+b²=c^r. It is proved that if r ≡ 5 (mod8), c > 10¹²r⁴ and b is an odd prime power, then the equation x²+b^y=c^z has only the positive integer solution (x, y, z) = (a, 2, r).  
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  Extension of Isometries Between the Unit Spheres of l^p(Γ) (1 < p < ∞) and a Banach Space E

  Yan JIANG, Shao Xiong CHEN

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 687-696. https://doi.org/10.12386/A2011sxxb0072

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  We mainly study linearly extension of isometric mapping V₀ between the Unit spheres of l^p(Γ) (1 < p < ∞) and a Banach space E. We show that if V₀ satisfies some hypotheses, then V₀ can be extended to a linear isometry defined on the whole space.  
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  Weak Doi-Hopf Group-Modules for Weak Hopf Group T-Coalgebras

  Quan Guo CHEN, Jian Gang TANG

  Acta Mathematica Sinica, Chinese Series. 2011, 54(4): 697-704. https://doi.org/10.12386/A2011sxxb0073

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  Weak quantum Yetter-Drinfeld modules for weak Hopf group T-coalgebras are introduced. Then, it is proved that weak quantum Yetter-Drinfeld modules are special case of weak Doi-Hopf group-modules. Also, a pair of adjoint functors between the category of weak quantum Yetter-Drinfeld modules and the category of the right B-module are constructed, where B is a subalgebra of coinvariance for a weak Hopf group-bicomodule algebra. Finally, the quantum integrals relative to weak quantum Yetter-Drinfeld modules are discussed.