15 June 2012, Volume 28 Issue 6

    

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  Further Bounds for Hardy Type Differences

  Kristina KRULIĆ, Josip PEČARIĆ, Dora POKAZ

  Acta Mathematica Sinica. 2012, 28(6): 1091-1102. https://doi.org/10.1007/s10114-012-0592-3

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  In this paper we define a functional as a difference between the right-hand side and lefthand side of the refined Boas type inequality using the notation of superquadratic and subquadratic functions and study its properties, such as exponential and logarithmic convexity. We also, state and prove improvements and reverses of new weighted Boas type inequalities. As a special case of our result we obtain improvements and reverses of the Hardy inequality and its dual inequality. We introduce new Cauchy type mean and prove monotonicity property of this mean.
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  Laws of Large Numbers for Cesãro alpha-integrable Random Variables under Dependence Condition AANA or AQSI

  De Mei YUAN, Jun AN

  Acta Mathematica Sinica. 2012, 28(6): 1103-1118. https://doi.org/10.1007/s10114-012-0033-3

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  Both residual Cesãro alpha-integrability (RCI(α)) and strongly residual Cesãro alphaintegrability (SRCI(α)) are two special kinds of extensions to uniform integrability, and both asymptotically almost negative association (AANA) and asymptotically quadrant sub-independence (AQSI) are two special kinds of dependence structures. By relating the RCI(α) property as well as the SRCI(α) property with dependence condition AANA or AQSI, we formulate some tail-integrability conditions under which for appropriate α the RCI(α) property yields L1-convergence results and the SRCI(α) property yields strong laws of large numbers, which is the continuation of the corresponding literature.
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  Approximation of Integrable Functions by General Linear Operators of Their Fourier Series

  Włodzimierz ŁENSKI, Bogdan SZAL

  Acta Mathematica Sinica. 2012, 28(6): 1119-1134. https://doi.org/10.1007/s10114-011-0568-8

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  The pointwise estimates of the deviation T_n,A,Bf(·)-f(·) in terms of moduli of continuity w ·f and w·f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.: J. Math. Anal. Appl., 220, 434-450 (1998) Theorem 1, p. 437].
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  Existence and Regularity of Solution to the Generalized Tricomi Equation with a Singular Initial Datum at a Point

  Kang Qun ZHANG

  Acta Mathematica Sinica. 2012, 28(6): 1135-1154. https://doi.org/10.1007/s10114-011-0487-8

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  In this paper, we study the existence and regularity of a solution to the initial datum problem of a semilinear generalized Tricomi equation in mixed-type domain. We suppose that an initial datum on the degenerate plane is smooth away from the origin, and has a conormal singularity at this point, then we show that in some mixed-type domain, the solution exists and is conormal with respect to the characteristic conic surface which is issued from the origin and has a cusp singularity.
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  Light Subgraphs in the Family of 1-Planar Graphs with High Minimum Degree

  Xin ZHANG, Gui Zhen LIU, Jian Liang WU

  Acta Mathematica Sinica. 2012, 28(6): 1155-1168. https://doi.org/10.1007/s10114-011-0439-3

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  A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K₂∨(K₁∪K₂) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K₁ ∨ (K₁ ∪K₂) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.
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  Bases of the Quantum Cluster Algebra of the Kronecker Quiver

  Ming DING, Fan XU

  Acta Mathematica Sinica. 2012, 28(6): 1169-1178. https://doi.org/10.1007/s10114-011-0344-9

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  We construct bar-invariant Z[q^±12]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the corresponding cluster algebra. As a byproduct, we prove positivity of the elements in these bases.
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  Laguerre Isopararmetric Hypersurfaces in R⁴

  Tong Zhu LI, Hua Fei SUN

  Acta Mathematica Sinica. 2012, 28(6): 1179-1186. https://doi.org/10.1007/s10114-011-0322-2

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  Let x : M → Rⁿ be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R⁴.
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  New Upper Bounds on Linear Coloring of Planar Graphs

  Bin LIU, Gui Zhen LIU

  Acta Mathematica Sinica. 2012, 28(6): 1187-1196. https://doi.org/10.1007/s10114-011-0317-z

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  A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has (1) lc(G) ≤ Δ+21 if Δ ≥ 9; (2) lc(G) ≤ [Δ/2] + 7 if g ≥ 5; (3) lc(G) ≤ [Δ/2] + 2 if g ≥ 7 and Δ≥ 7.
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  Extension of Isometries on the Unit Sphere of L^p Spaces

  Dong Ni TAN

  Acta Mathematica Sinica. 2012, 28(6): 1197-1208. https://doi.org/10.1007/s10114-011-0302-6

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  In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p(μ) (1 < p < ∞, p ≠ 2) and a Banach space E can be extended to a linear isometry from L^p(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of L^p(μ), then E is linearly isometric to L^p(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of L^p(μ₁,H₁) and L^p(μ₂,H₂) must be an isometry and can be extended to a linear isometry from L^p(μ₁,H₁) to L^p(μ₂,H₂), where H₁ and H₂ are Hilbert spaces.
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  Some Results on Metric n-Lie Algebras

  Rui Pu BAI, Wan Qing WU, Zhen Heng LI

  Acta Mathematica Sinica. 2012, 28(6): 1209-1220. https://doi.org/10.1007/s10114-011-0231-4

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  We study the structure of a metric n-Lie algebra G over the complex field C. Let G = S⊕R be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R^⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R^⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G)+1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R^⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R^⊥ ⊆ R.
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  Strong Approximation of Locally Square-Integrable Martingales

  Zheng Yan LIN, Han Chao WANG

  Acta Mathematica Sinica. 2012, 28(6): 1221-1232. https://doi.org/10.1007/s10114-011-0196-3

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  In this paper, we consider the strong approximation for locally square-integrable martingales. In our results, the limit process may be a process with jumps. This is an extension of the former results.
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  Univalent Functions and Diff(S¹)/S¹

  Simon DAVIS

  Acta Mathematica Sinica. 2012, 28(6): 1233-1236. https://doi.org/10.1007/s10114-011-0125-5

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  The relation between Diff(S¹)/S¹ and the space of univalent analytic functions on the disk is elucidated and shown to provide upper bounds for the volumes of exhaustive approximations to an analytic submanifold of an infinite-dimensional space. The maximum magnitudes of the coefficients in the series expansions of univalent superanalytic functions on the superdisk are inferred.
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  The Boundedness of Calderón-Zygmund Operators by Wavelet Characterization

  Cheng-Cong HUNG, Ming-Yi LEE

  Acta Mathematica Sinica. 2012, 28(6): 1237-1248. https://doi.org/10.1007/s10114-011-0065-0

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  This article deals with the boundedness properties of Calderón-Zygmund operators on Hardy spaces H^p(Rⁿ). We use wavelet characterization of H^p(Rⁿ) to show that a Calderón-Zygmund operator T with T^*1 = 0 is bounded on H^p(Rⁿ), n/n+ε < p ≤ 1, where ε is the regular exponent of kernel of T. This approach can be applied to the boundedness of operators on certain Hardy spaces without atomic decomposition or molecular characterization.
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  Dirichlet Problem for Hermitian-Einstein Equation over Almost Hermitian Manifold

  Yue WANG, Xi ZHANG

  Acta Mathematica Sinica. 2012, 28(6): 1249-1260. https://doi.org/10.1007/s10114-011-0018-7

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  In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.
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  Existence of Homoclinic Solutions for a Class of Neutral Functional Differential Equations

  Shi Ping LU

  Acta Mathematica Sinica. 2012, 28(6): 1261-1274. https://doi.org/10.1007/s10114-011-9259-8

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  By means of Mawhin’s continuation theorem and some analysis methods, the existence of 2kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above periodic solutions set.
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  Moderate Deviations for M-estimators in Linear Models with φ-mixing Errors

  Jun FAN

  Acta Mathematica Sinica. 2012, 28(6): 1275-1294. https://doi.org/10.1007/s10114-011-9188-6

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  In this paper, the moderate deviations for the M-estimators of regression parameter in a linear model are obtained when the errors form a strictly stationary φ-mixing sequence. The results are applied to study many different types of M-estimators such as Huber’s estimator, L^p-regression estimator, least squares estimator and least absolute deviation estimator.