15 August 2009, Volume 25 Issue 8

    

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  A Note on Besov–Morrey Spaces and Triebel–Lizorkin–Morrey Spaces

  Yoshihiro SAWANO

  Acta Mathematica Sinica. 2009, 25(8): 1223-1242. https://doi.org/10.1007/s10114-009-8247-8

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  In the present paper,we obtain three independent results on the Besov–Morrey spacesand the Triebel–Lizorkin–Morrey spaces.That is,we are going to obtain the characterization of localmeans,the boundedness of pseudo-Diffierential operators and the characterization of the Hardy–Morreyspaces.By using the maximal estimate and the molecular decomposition,we shall integrate and extendthe known results on these spaces.
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  On Self-Mapping Degrees of S³-Geometry Manifolds

  Xiao Ming DU

  Acta Mathematica Sinica. 2009, 25(8): 1243-1252. https://doi.org/10.1007/s10114-009-8077-8

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  In this paper we determined all of the possible self-mapping degrees of the manifolds withS³-geometry,which are supposed to be all 3-manifolds with finite fundamental groups.This is a part ofa project to determine all possible self-mapping degrees of all closed orientable 3-manifold in Thurston'spicture.
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  Domination Number in Graphs with Minimum Degree Two

  Er Fang SHAN, Moo Young SOHN, Xu Dong YUAN, Michael A.HENNING

  Acta Mathematica Sinica. 2009, 25(8): 1253-1268. https://doi.org/10.1007/s10114-009-7617-6

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  A set D of vertices of a graph G=(V,E)is called a dominating set if every vertex of Vnot in D is adjacent to a vertex of D.In 1996,Reed proved that every graph of order n with minimumdegree at least 3 has a dominating set of cardinality at most 3n/8.In this paper we generalize Reed'sresult.We show that every graph G of order n with minimum degree at least 2 has a dominating set ofcardinality at most(3n+|V₂|)/8,where V₂ denotes the set of vertices of degree 2 in G.As an applicationof the above result,we show that for k≥1,the k-restricted domination number r_k(G,γ)≤(3n+5k)/8for all graphs of order n with minimum degree at least 3.
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  On the Generalization of the D.H.Lehmer Problem

  Ya Ming LU, Yuan YI

  Acta Mathematica Sinica. 2009, 25(8): 1269-1274. https://doi.org/10.1007/s10114-009-7652-3

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  Let n≥2 be a fixed positive integer,q≥3 and c be two integers with(n,q)=(c,q)=1.We denote by rn(δ₁,δ₂,c;q)(0<δ₁,δ₂≤1)the number of all pairs of integers a,b satisfying ab=c(mod q),1≤a≤δ_1q,1≤b≤δ_2q,(a,q)=(b,q)=1 and n (a+b).The main purpose of this paperis to study the asymptotic properties of rn(δ₁,δ₂,c;q),and give a sharp asymptotic formula for it.
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  A New Class of Weights Adapted to Rough Singular Integrals and Marcinkiewicz Integrals

  Chun Jie ZHANG, Ying YOU

  Acta Mathematica Sinica. 2009, 25(8): 1275-1288. https://doi.org/10.1007/s10114-009-7537-5

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  In this pA_per,we introduce a new class of weightsA_p(Rⁿ)which retains many fine propertiesof the classical Muchenhoupt weights A_p(Rⁿ).While A_p(Rⁿ)is too big a class to obtain the weightednorm inequalities for rough singular integrals and Marcinkiewicz integrals,our new classA_p(Rⁿ)adA_ptswell to these rough operators.As A_pplications,we improve some known weighted estimates.
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  Some Remarks on Teichmuller Spaces and Modular Groups

  Yun HU

  Acta Mathematica Sinica. 2009, 25(8): 1289-1296. https://doi.org/10.1007/s10114-009-7443-x

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  We study the fixed point sets of subgroups of modular groups acting on Teichmuller spaces.
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  Interpolation on Weak Martingale Hardy Space

  Yong JIAO, Wei CHEN, Pei De LIU

  Acta Mathematica Sinica. 2009, 25(8): 1297-1304. https://doi.org/10.1007/s10114-009-7415-1

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  By the weak atomic decompositions of weak martingale Hardy spaces,we investigate theinterpolation spaces between weak martingale Hardy spaces and martingale Hardy spaces.
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  L^pand L^∞Norm Estimates of the Cost of the Controllability for Heat Equations

  Pei Dong LEI, Xu LIU, Hang GAO

  Acta Mathematica Sinica. 2009, 25(8): 1305-1324. https://doi.org/10.1007/s10114-009-7272-y

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  This paper is concerned with the bound of the cost of approximate controllability and nullcontrollability of heat equations,i.e.,the minimal L^p norm and L^∞norm of a control needed to controlthe system approximately or a control needed to steer the state of the system to zero.The methodswe use combine observability inequalities,energy estimates for heat equations and the dual theory.
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  Existence of Three HMOLS of Type 2ⁿu¹

  Yun Qing XU

  Acta Mathematica Sinica. 2009, 25(8): 1325-1336. https://doi.org/10.1007/s10114-009-7093-z

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  A Latin squares of order v with ni missing sub-Latin squares(holes)of order hi(1≤i≤k),which are disjoint and spanning(i.e.−^k_i=1 n_ih_i=v),is called a partitioned incomplete Latin squaresand denoted by PILS.The type of PILS is defined by(hⁿ¹₁hⁿ²₂···h^nK_K).If any two PILS in a set oft PILS of type T are orthogonal,then we denote the set by t-HMOLS(T).It has been proved that3-HMOLS(2ⁿ3₁)exist for n≥6 with 11 possible exceptions.In this paper,we investigate the existenceof 3-HMOLS(2ⁿu¹)with u≥4,and prove that 3-HMOLS(2ⁿu¹)exist if n≥54 and n≥7/4u+7
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  Construct Weak Hopf Algebras by Using Borcherds Matrix

  Zhi Xiang WU

  Acta Mathematica Sinica. 2009, 25(8): 1337-1352. https://doi.org/10.1007/s10114-009-7065-3

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  We define a new kind quantized enveloping algebra of a generalized Kac–Moody algebraG by adding a new generator J satisfying J m=J for some integer m.We denote this algebra by_wU^τ_q(G).This algebra is a weak Hopf algebra if and only if m=2,3.In general,it is a bialgebra,and contains a Hopf subalgebra.This Hopf subalgebra is isomorphic to the usual quantum envelopealgebra U_q(G)of a generalized Kac–Moody algebra G.
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  The Affine Complete Hypersurfaces of Constant Gauss–Kronecker Curvature

  Bao Fu WANG

  Acta Mathematica Sinica. 2009, 25(8): 1353-1362. https://doi.org/10.1007/s10114-009-7063-5

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  Given a bounded convex domainΩwith C^∞boundary and a functionφ∈C^∞(∂Ω),Li–Simon–Chen can construct an Euclidean complete and W-complete convex hypersurface M withconstant affine Gauss–Kronecker curvature,and they guess the M is also affine complete.In this paper,we give a confirmation answer.
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  Two-Dimensional Invariant Tori in the Neighborhood of an Elliptic Equilibrium of Hamiltonian Systems

  Hui LU, Jian Gong YOU

  Acta Mathematica Sinica. 2009, 25(8): 1363-1378. https://doi.org/10.1007/s10114-009-7053-7

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  In this paper,we prove that a Hamiltonian system possesses either a four-dimensionalinvariant disc or an invariant Cantor set with positive(n+2)-dimensional Lebesgue measure in theneighborhood of an elliptic equilibrium provided that its linearized system at the equilibrium satisfiessome small divisor conditions.Both of the invariant sets are foliated by two-dimensional invariant toricarrying quasi-periodic solutions.
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  A Fluctuation Type Limit Theorem for Jirina Processes with Immigration

  Yu Qiang LI

  Acta Mathematica Sinica. 2009, 25(8): 1379-1388. https://doi.org/10.1007/s10114-009-6629-6

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  It is proved by the theory of semigroup that the Ornstein–Uhlenbeck type process withjumps can arise from the fluctuation limit of a sequence of Jirina processes with immigration undersuitable moments conditions.
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  Conditions for Dominated Splittings

  Geng LIU, Chao LIANG

  Acta Mathematica Sinica. 2009, 25(8): 1389-1398. https://doi.org/10.1007/s10114-009-6568-2

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  In this paper,we solve the problem proposed by Lan Wen for the case of dimM=3.Roughly speaking,we prove that for fixed i,f has C¹ persistently no small angles of index i if andonly if f has a dominated splitting of index i on the C¹ i-preperiodic set Pⁱ_*(f).