10 April 2010, Volume 26 Issue 4

    

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  Multi-parameter Triebel–Lizorkin and Besov Spaces Associatedwith Flag Singular Integrals

  Yong DING, Guo Zhen LU, Bo Lin MA

  Acta Mathematica Sinica. 2010, 26(4): 603-620. https://doi.org/10.1007/s10114-010-8352-8

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  Though the theory of one-parameter Triebel–Lizorkin and Besov spaces has been verywell developed in the past decades,the multi–parameter counterpart of such a theory is still absent.The main purpose of this paper is to develop a theory of multi-parameter Triebel–Lizorkin and Besovspaces using the discrete Littlewood–Paley–Stein analysis in the setting of implicit multi–parameterstructure.It is motivated by the recent work of Han and Lu in which they established a satisfactorytheory of multi–parameter Littlewood–Paley–Stein analysis and Hardy spaces associated with the flagsingular integral operators studied by Muller–Ricci–Stein and Nagel–Ricci–Stein.We also prove theboundedness of flag singular integral operators on Triebel–Lizorkin space and Besov space.Our methodshere can be applied to develop easily the theory of multi-parameter Triebel–Lizorkin and Besov spacesin the pure product setting.
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  Square-Free Integers as Ideal Norms

  Shigeru KANEMITSU, Antanas LAURINčIKAS, Jing MA

  Acta Mathematica Sinica. 2010, 26(4): 621-628. https://doi.org/10.1007/s10114-010-8450-7

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  Elaborating on Krätzel's result,Zhai obtained a sharp short interval result for the numberof ways of expressing a square-free integer as sums of two squares.In this note we view Zhai's decom-position as one by degree 1 prime ideals in the Gaussian field and extend Zhai’s approach to a widerrange of quadratic fields etc.
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  Pseudo-C-rpp Semigroups

  Xiao Jiang GUO, Young Bae JUN, Ming ZHAO

  Acta Mathematica Sinica. 2010, 26(4): 629-646. https://doi.org/10.1007/s10114-010-8030-x

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  The aim of this paper is to study a class of right pp semigroups,so-called pseudo-C-rpp semigroups.After obtaining some characterizations of pseudo-C-rpp semigroups,we establish aconstruction of such semigroups in terms of left normal bands and right C-rpp semigroups.In particular,a special case is considered.
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  Panconnectivity for Interconnection Networks with Faulty Elements

  Mei LU, Hui Qing LIU

  Acta Mathematica Sinica. 2010, 26(4): 647-652. https://doi.org/10.1007/s10114-010-8383-1

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  Let G₀ and G₁ be two graphs with the same vertices.The new graph G(G₀,G₁;M)is agraph with the vertex set V(G0)∪V(G₁)and the edge set E(G₀)∪E(G₁)∪M,where M is an arbitraryperfect matching between the vertices of G₀ and G₁,i.e.,a set of cross edges with one endvertex in G₀ and the other endvertex in G₁.In this paper,we will show that if G₀ and G₁ are f-fault q-panconnected,then for any f≥2,G(G₀,G₁;M)is(f+1)-fault(q+2)-panconnected.
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  Sharp L2 Boundedness of the Oscillatory Hyper-HilbertTransform along Curves

  Jie Cheng CHEN, Da Shan FAN, Xiang Rong ZHU

  Acta Mathematica Sinica. 2010, 26(4): 653-658. https://doi.org/10.1007/s10114-010-7396-0

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  Consider the oscillatory hyper-Hilbert transformH_n,α,βf(x)=∫¹₀f(x-Γ(t))e^it-βt^-1-αdt (1)along the curveΓ(t)=(t^p1,t^p2,...,t^pn),whereβ>α≥0 and 0<p1<p2<···<pn.We prove that H_n,α,βis bounded on L² if and only ifβ≥(n+1)α.Our work extends and improves some known results.
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  The BMO_L Space and Riesz TransformsAssociated with Schrödinger Operators

  Jian Feng DONG, He Ping LIU

  Acta Mathematica Sinica. 2010, 26(4): 659-668. https://doi.org/10.1007/s10114-010-8115-6

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  Let L=-Δ+V be a Schrödinger operator in Rd,^d≥3,where the nonnegative potentialV belongs to the reverse Hölder class B_d.We establish the BMO_L-boundedness of Riesz transforms ∂/∂x_#em/em#L^-1/2,and give the Fefferman–Stein type decomposition of BMO_Lfunctions.
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  Lower Bound for Quantum Integration Erroron Anisotropic Sobolev Classes

  Pei Xin YE

  Acta Mathematica Sinica. 2010, 26(4): 669-678. https://doi.org/10.1007/s10114-010-7546-4

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  We study the approximation of the integration of multivariate functions in the quantummodel of computation.Using a new reduction approach we obtain a lower bound of the n-th minimalquery error on anisotropic Sobolev class B(W^r_p([0,1]^d))(r∈R^d₊).Then combining this result withour previous one we determine the optimal bound of n-th minimal query error for anisotropic Hölder–Nikolskii class B(H^r_∞([0,1]^d))and Sobolev class B(W^r_∞([0,1]^d)).The results show that for thesetwo types of classes the quantum algorithms give significant speed up over classical deterministic andrandomized algorithms.
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  Complete Moment Convergence for Sequence of IdenticallyDistributed φ-mixing Random Variables

  Ping Yan CHEN, Ding Cheng WANG

  Acta Mathematica Sinica. 2010, 26(4): 679-690. https://doi.org/10.1007/s10114-010-7625-6

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  In the paper we extend and generalize some results of complete moment convergenceresults(or the refinement of complete convergence)obtained by Chow[On the rate of moment completeconvergence of sample sums and extremes.Bull.Inst.Math.Academia Sinica,16,177–201(1988)]andLi&Sp?taru[Refinement of convergence rates for tail probabilities.J.Theor.Probab.,18,933–947(2005)]to sequences of identically distributed φ-mixing random variables.
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  The Support of a Refinable Vector Satisfying anInhomogeneous Refinement Equation

  Song LI, Yi SHEN

  Acta Mathematica Sinica. 2010, 26(4): 691-698. https://doi.org/10.1007/s10114-010-7598-5

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  In this paper,we investigate the support of a refinable vector satisfying an inhomoge-neous refinement equation.By using some methods introduced by So and Wang,an estimate is givenfor the support of each component function of a compactly supported refinable vector satisfying aninhomogeneous matrix refinement equation with finitely supported masks.
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  Hamiltonicity of 4-connected Graphs

  Hao LI, Feng TIAN, Zhi Xia XU

  Acta Mathematica Sinica. 2010, 26(4): 699-710. https://doi.org/10.1007/s10114-010-7514-z

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  Let σ_k(G)denote the minimum degree sum of k independent vertices in G andα(G)denotethe number of the vertices of a maximum independent set of G.In this paper we prove that if G is a4-connected graph of order n andσ_{5(G)≥n+3α(G)+11,then G is Hamiltonian.}
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  Dimensions of Weighted Besicovitch Sets

  Yu Qiong XIE, Zhi Xiong WEN, Min YU

  Acta Mathematica Sinica. 2010, 26(4): 711-716. https://doi.org/10.1007/s10114-010-7499-7

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  We consider weighted Besicovitch sets which are defined in terms of the weighted frequencyof digit 1 in the dyadic expansion of real numbers.Explicit formulas for their Hausdor?dimensionsare given.
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  On Rough Marcinkiewicz Integrals along Surfaces

  Huo Xiong WU, Jian Kai XU

  Acta Mathematica Sinica. 2010, 26(4): 717-730. https://doi.org/10.1007/s10114-010-7457-4

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  In this paper,the authors establish the L^p<>/i-mapping properties of certain classes ofMarcinkiewicz integral operators along surfaces with rough kernels.The results in this paper essentiallyextend as well as improve previously known results.
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  The Hausdor?Dimension of SetsRelated to the General Sierpinski Carpets

  Yong Xin GUI, Wen Xia LI

  Acta Mathematica Sinica. 2010, 26(4): 731-742. https://doi.org/10.1007/s10114-010-7453-8

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  In this paper we study a class of subsets of the general Sierpinski carpets for which theallowed two digits in the expansions occur with proportional frequency.We calculate the Hausdorffand box dimensions of these subsets and give necessary and su?cient conditions for the correspondingHausdor?measure to be positive finite.
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  Convergence Theorems forλ-strict Pseudo-contractionsin q-uniformly Smooth Banach Spaces

  Hai Yun ZHOU

  Acta Mathematica Sinica. 2010, 26(4): 743-758. https://doi.org/10.1007/s10114-010-7341-2

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  In this paper,we continue to discuss the properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space.The results presented in this paper are interesting extensions and improvements upon those knownones of Marino and Xu[Marino,G.,Xu,H.K.:Weak and strong convergence theorems for strictpseudo-contractions in Hilbert spaces.J.Math.Anal.Appl.,324,336–349(2007)].In order to geta strong convergence theorem,we modify the normal Mann’s iterative algorithm by using a suitableconvex combination of a fixed vector and a sequence in C.This result extends a recent result of Kimand Xu[Kim,T.H.,Xu,H.K.:Strong convergence of modified Mann iterations.Nonl.Anal.,61,51–60(2005)]both from nonexpansive mappings toλ-strict pseudo-contractions and from Hilbert spacesto q-uniformly smooth Banach spaces.
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  On Pseudo-Holomorphic Curves from Two-Spheresinto a Complex Grassmannian G(2,5)

  Xiao Xiang JIAO

  Acta Mathematica Sinica. 2010, 26(4): 759-762. https://doi.org/10.1007/s10114-010-7262-0

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  Let s:S²→G(2,5)be a linearly full totally unramified pseudo-holomorphic curve withconstant Gaussian curvature K in a complex Grassmann manifold G(2,5).It is prove that K is either1/2 or 4/5 if s is non-±holomorphic.Furthermore, K=1/2 if and only if s is totally real. We also provethat the Gaussian curvature K is either 1 or 4/3 if s is a non-degenerate holomorphic curve under someconditions.
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  Bootstrap Approximation of Wavelet Estimatesin a Semiparametric Regression Model

  Liu Gen XUE, Qiang LIU

  Acta Mathematica Sinica. 2010, 26(4): 763-778. https://doi.org/10.1007/s10114-010-7236-2

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  and the bootstrap methods.The bootstrap statistics are constructed by using Efron'sresampling technique,and the strong uniform convergence of the bootstrap approximation is proved.Our results can be used to construct the large sample confidence intervals for the parameters of interest.A simulation study is conducted to evaluate the finite-sample performance of the bootstrap methodand to compare it with the normal approximation-based method.
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  Bifurcations of Limit Cycles in a Z₄-EquivariantQuintic Planar Vector Field

  Yu Hai WU, Xue Di WANG, Li Xin TIAN

  Acta Mathematica Sinica. 2010, 26(4): 779-798. https://doi.org/10.1007/s10114-010-6487-2

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  In this paper,a Z₄-equivariant quintic planar vector field is studied.The Hopf bifurcationmethod and polycycle bifurcation method are combined to study the limit cycles bifurcated fromthe compounded cycle with 4 hyperbolic saddle points.It is found that this special quintic planarpolynomial system has at least four large limit cycles which surround all singular points.By applyingthe double homoclinic loops bifurcation method and Hopf bifurcation method,we conclude that 28limit cycles with two different configurations exist in this special planar polynomial system.Theresults acquired in this paper are useful for studying the weakened 16th Hilbert's Problem.