15 May 2009, Volume 25 Issue 5

    

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  Rank-One Cross Commutators on Backward Shift Invariant Subspaces on the Bidisk

  Kei Ji IZUCHI, Kou Hei IZUCHI

  Acta Mathematica Sinica. 2009, 25(5): 693-714. https://doi.org/10.1007/s10114-009-7215-7

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  For a backward shift invariant subspace N in H²(Γ²),the operators S_z and S_w on N aredefined by S_z=P_N T_z|N and S_w=P_N T_w|_N,where P_N is the orthogonal projection from L²(Γ²)onto N.We give a characterization of N satisfying rank[S_z,S^*_w]=1.
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  On the Existence of Feller Semigroups with Discontinuous Coeffcients II

  Kazuaki TAIRA

  Acta Mathematica Sinica. 2009, 25(5): 715-740. https://doi.org/10.1007/s10114-008-6564-y

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  This paper is devoted to the functional analytic approach to the problem of existenceof Markov processes with Dirichlet boundary condition,oblique derivative boundary condition andfirst-order Wentzell boundary condition for second-order,uniformly elliptic differential operators withdiscontinuous coeffcients.More precisely,we construct Feller semigroups associated with absorption,reflection,drift and sticking phenomena at the boundary.The approach here is distinguished by theextensive use of the ideas and techniques characteristic of the recent developments in the Calder on–Zygmund theory of singular integral operators with non-smooth kernels.
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  Stochastic Comparison and Preservation of Positive Correlations for Levy-type Processes

  Jie Ming WANG

  Acta Mathematica Sinica. 2009, 25(5): 741-758. https://doi.org/10.1007/s10114-009-7670-1

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  The stochastic comparison and preservation of positive correlations for Levy-type pro-cesses on R^d are studied under the condition that Levy measureνsatisfies∫_{0<|z|≤1}|z||ν(x,dz)-ν(x,d(-z))|<∞,x∈R^d,while the suffcient conditions and necessary ones for them are obtained.Insome cases the conditions for stochastic comparison are not only suffcient but also necessary.
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  A Construction of Weakly Inverse Semigroups

  Bing Jun YU, Bing Jun YU

  Acta Mathematica Sinica. 2009, 25(5): 759-784. https://doi.org/10.1007/s10114-009-7480-5

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  Let S^obe an inverse semigroup with semilattice biordered set E^oof idempotents and E aweakly inverse biordered set with a subsemilattice EP={e∈E|^of∈E,S(f,e)^oω(e)}isomorphicto E^obyθ:EP^o→E^o.In this paper,it is proved that if^of,g∈E,f←→g=^of^oθD S^og^oθandthere exists a mappingφfrom EP into the symmetric weakly inverse semigroup P T(E∪S^o)satisfyingsix appropriate conditions,then a weakly inverse semigroupΣcan be constructed in P T(S^o),calledthe weakly inverse hull of a weakly inverse system(S^o,E,θ,φ)with I(Σ)=～S^o,E(Σ)E.Conversely,every weakly inverse semigroup can be constructed in this way.Furthermore,a su^ocient and necessarycondition for two weakly inverse hulls to be isomorphic is also given.
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  Monotonicity and Best Approximation in Banach Lattices

  Shu Tao CHEN, Xin HE, H.HUDZIK

  Acta Mathematica Sinica. 2009, 25(5): 785-794. https://doi.org/10.1007/s10114-009-7391-5

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  Hudzik and Kurc discussed some best approximation problems in Banach lattices by meansof monotonicities.This paper deals with more general best approximation problems in Banach lattices.Existence,uniqueness,stability and continuity for such best approximation problems are discussed.
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  A Variation of a Conjecture Due to Erdos and Sos

  Jian Hua YIN, Jiong Sheng LI

  Acta Mathematica Sinica. 2009, 25(5): 795-802. https://doi.org/10.1007/s10114-009-7260-2

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  Erdos and Sos conjectured in 1963 that every graph G on n vertices with edge numbere(G)>1/2(k-1)n contains every tree T with k edges as a subgraph.In this paper,we consider avariation of the above conjecture,that is,for n≥9/2k²+37/2k+14 and every graph G on n vertices withe(G)>1/2(k-1)n,we prove that there exists a graph G on n vertices having the same degree sequenceas G and containing every tree T with k edges as a subgraph.
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  Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L^∞(0,T;Lⁿ(Ω))

  Yan YONG, Quan Sen JIU

  Acta Mathematica Sinica. 2009, 25(5): 803-814. https://doi.org/10.1007/s10114-009-7214-8

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  In this paper,we study the energy equality and the uniqueness of weak solutions to theMHD equations in the critical space L^∞(0,T;Lⁿ(Ω)).We prove that if the velocity u belongs to thecritical space L^∞(0,T;Lⁿ(Ω)),the energy equality holds.On the basis of the energy equality,wefurther prove that the weak solution to the MHD equations is unique.
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  Commutators with Finite Spectrum II

  Nadia BOUDI

  Acta Mathematica Sinica. 2009, 25(5): 815-830. https://doi.org/10.1007/s10114-009-6697-7

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  The purpose of this paper is to study derivations d,d defined on a Banach algebra Asuch that the spectrumσ([dx,dx])is finite for all x∈A.In particular we show that if the algebrais semisimple,then there exists an element a in the socle of A such that[d,d]is the inner derivationimplemented by a.
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  The Global Dimensions of Crossed Products and Crossed Coproducts

  Teng Xia JU

  Acta Mathematica Sinica. 2009, 25(5): 831-844. https://doi.org/10.1007/s10114-009-6609-x

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  In this paper,we show that if H is a finite-dimensional Hopf algebra such that H and H^*are semisimple,then gl.dim(A#_σH)=gl.dim(A),where_σis a convolution invertible cocycle.We alsodiscuss the relationship of global dimensions between the crossed product A#_σH and the algebra A,where A is coacted by H.Dually,we give a suffcient condition for a finite dimensional coalgebra Cand a finite dimensional semisimple Hopf algebra H such that gl.dim(C_αH)=gl.dim(C).
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  Semi-empirical Likelihood Confidence Intervals for the Differences of Quantiles with Missing Data

  Yong Song QIN, Jun Chao ZHANG

  Acta Mathematica Sinica. 2009, 25(5): 845-854. https://doi.org/10.1007/s10114-009-6476-5

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  Detecting population(group)differences is useful in many applications,such as medicalresearch.In this paper,we explore the probabilistic theory for identifying the quantile differencesbetween two populations.Suppose that there are two populations x and y with missing data on bothof them,where x is nonparametric and y is parametric.We are interested in constructing confidenceintervals on the quantile differences of x and y.Random hot deck imputation is used to fill in missingdata.Semi-empirical likelihood confidence intervals on the differences are constructed.
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  Rademacher Complexity in Neyman–Pearson Classification

  Min HAN, Di Rong CHEN, Zhao Xu SUN

  Acta Mathematica Sinica. 2009, 25(5): 855-868. https://doi.org/10.1007/s10114-008-6210-8

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  Neyman–Pearson(NP)criterion is one of the most important ways in hypothesis testing.It is also a criterion for classification.This paper addresses the problem of bounding the estimationerror of NP classification,in terms of Rademacher averages.We investigate the behavior of the globaland local Rademacher averages,and present new NP classification error bounds which are based on thelocalized averages,and indicate how the estimation error can be estimated without a priori knowledgeof the class at hand.