10 October 2010, Volume 26 Issue 10

    

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  The ρ-variation of the Hermitian Riesz Transform

  MBENI, Francisco Javier MARTÍN-REYES, Alberto DE LA TORRE, José L.TORREA

  Acta Mathematica Sinica. 2010, 26(10): 1827-1838. https://doi.org/10.1007/s10114-010-9122-3

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  In this paper we prove the behaviour in weighted L^p spaces of the oscillation and variation of the Hilbert transform and the Riesz transform associated with the Hermite operator of dimension 1. We prove that this operator maps L^p(R,W(x)dx) into itself when w is a weight in the Ap class for 1 < p < ∞. For p = 1 we get weak type for the A₁ class. Weighted estimated are also obtained in the extreme case p = ∞.
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  Positively Expansive Differentiable Maps

  Kazuhiro SAKAI

  Acta Mathematica Sinica. 2010, 26(10): 1839-1846. https://doi.org/10.1007/s10114-010-9149-5

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  We study the space of positively expansive differentiable maps of a compact connected C^∞ Riemannian manifold without boundary. It is proved that (i) the C¹-interior of the set of positively expansive differentiable maps coincides with the set of expanding maps, and (ii) C¹-generically, a differentiable map is positively expansive if and only if it is expanding.
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  Maximal Operators and Singular Integrals with Non-Isotropic Dilation on Product Domains

  Zhong Kai LI, Bo Lin MA, Huo Xiong WU

  Acta Mathematica Sinica. 2010, 26(10): 1847-1864. https://doi.org/10.1007/s10114-010-8400-4

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  In this paper, the authors study the L^p-mapping properties of certain maximal operators with non-isotropic dilation on product domains. As an application, the L^p-boundedness of the corresponding non-isotropic multiple singular integral operator is also obtained. Here the integral kernelfunctions Ω belong to the spaces L(logL)^α(Σ₁ × Σ₂) for some α > 0, which is optimal.
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  On the Restricted Arc-connectivity of s-geodetic Digraphs

  Camino BALBUENA, Pedro GARCÍA-VÁZQUEZ

  Acta Mathematica Sinica. 2010, 26(10): 1865-1876. https://doi.org/10.1007/s10114-010-9313-y

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  For a strongly connected digraph D the restricted arc-connectivity λ' (D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D ? S has a non-trivial strong component D1 such that D?V(D₁) contains an arc. Let S be a subset of vertices of D. We denote by ω⁺(S) the set of arcs uv with u ∈ S and v∉S, and by ω^?(S) the set of arcs uv with u∉S and v ∈ S. A digraph D = (V,A) is said to be λ' -optimal if λ' (D) = ξ' (D), where ξ' (D) is the minimum arc-degree of D defined as ξ (D) = min{em>ξ (xy) : xy∈ A}, and ξ (xy) = min{|ω +({x, y})|, |ω-({x, y|, |ω+(x) ∪ ω-(y)|, |ω-(x)∪ω+(y)|}. In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ'-optimal is given in terms of its diameter. Furthermore we see that the h-iterated line digraph L^h(D) of a s-geodetic digraph is λ'-optimal for certain iteration h.
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  Properties on Solutions of Some q-Difference Equations

  Bao Qin CHEN, Zong Xuan CHEN, Sheng LI

  Acta Mathematica Sinica. 2010, 26(10): 1877-1886. https://doi.org/10.1007/s10114-010-8339-5

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  We consider the properties on solutions of some q-difference equations of the form (?)where a₀(z), . . . , a_n+1(z) are meromorphic functions, a₀(z)a_n(z) ≡ 0 and q ∈ C such that 0 < |q| ≤ 1.We give estimates on the upper bound for the length of the gap in the power series of entire solutions of (?) when the coefficients a₀(z), . . . , a_n+1(z) are polynomials and 0 < |q| < 1. For some special cases,we give estimates of growth of f(z). And we also show that the case 0 < |q| < 1 is different from the case |q| = 1.
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  The Frattini p-subsystem of a Solvable Restricted Lie Triple System

  Liang Yun CHEN, Dong LIU

  Acta Mathematica Sinica. 2010, 26(10): 1887-1898. https://doi.org/10.1007/s10114-010-9251-8

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  As a natural generalization of a restricted Lie algebra, a restricted Lie triple system was defined by Hodge. In this paper, we develop initially the Frattini theory for restricted Lie triple systems, generalize some results of Frattini p-subalgebra for restricted Lie algebras, obtain some properties of the Frattini p_{-subsystem and give the relationship between φp(T) and φ(T) for solvable Lie triple systems.}
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  Large Sets of Pure Directed Triple Systems with Indexλ

  Bing Li FAN, Jun Ling ZHOU

  Acta Mathematica Sinica. 2010, 26(10): 1899-1914. https://doi.org/10.1007/s10114-010-9249-2

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  A directed triple system of order v with index λ, briefly by DTS(v, λ), is a pair (X, B)where X is a v-set and B is a collection of transitive triples (blocks) on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS(v, λ) is a DTS(v, λ) without repeated blocks. A simple DTS(v, λ) is called pure and denoted by PDTS(v, λ) if (x, y, z) ∈ B implies z, y, x), (z, x, y), (y, x, z),(y, z, x), (x, z, y) ∈ B. A large set of disjoint PDTS(v, λ), denoted by LPDTS(v, λ), is a collection of 3(v ? 2)/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS(v, λ) are presented. Especially, we determine the spectrum of LPDTS(v, 2).
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  Cohomogeneity One de Sitter Space Sⁿ₁

  P.AHMADI, S.M.B.KASHANI, H.ABEDI

  Acta Mathematica Sinica. 2010, 26(10): 1915-1926. https://doi.org/10.1007/s10114-010-8142-3

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  In this paper we study the cohomogeneity one de Sitter space Sⁿ_{1 .} We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.
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  Estimations of Parametric Functions under a System of Linear Regression Equations with Correlated Errors

  Yong Ge TIAN

  Acta Mathematica Sinica. 2010, 26(10): 1927-1942. https://doi.org/10.1007/s10114-010-8434-7

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  Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool — the matrix rank method was utilized to simplifyvarious complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators (BLUEs) of parametric functions under the system. In particular, we give thenecessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.
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  The High-order SPDEs Driven by Multi-parameter Fractional Noises

  Ting Ting WEI

  Acta Mathematica Sinica. 2010, 26(10): 1943-1960. https://doi.org/10.1007/s10114-010-8179-3

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  The existence and uniqueness of the solutions are proved for a class of fourth-order stochastic heat equations driven by multi-parameter fractional noises. Furthermore the regularity of the solutions is studied for the stochastic equations and the existence of the density of the law of the solutionis obtained.
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  The Nonexistence of Sensitive Commutative Group Actions on Dendrites

  En Hui SHI, Bin Yong SUN, Li Zhen ZHOU

  Acta Mathematica Sinica. 2010, 26(10): 1961-1968. https://doi.org/10.1007/s10114-010-7199-3

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  In this paper, using the integral method observed by Mai Jiehua recently, we show that no dendrite admits a sensitive commutative group action.
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  On the Projections of Convex Lattice Sets

  Jing ZHOU

  Acta Mathematica Sinica. 2010, 26(10): 1969-1980. https://doi.org/10.1007/s10114-010-7164-1

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  As a discrete analogue of Aleksandrov’s projection theorem, it is natural to ask the following question: Can an o-symmetric convex lattice set of the integer lattice Zⁿ be uniquely determined by its lattice projection counts? In 2005, Gardner, Gronchi and Zong discovered a counterexample with cardinality 11 in the plane. In this paper, we will show that it is the only counterexample in Z² up tounimodular transformations and with cardinality not larger than 17.
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  Uniform Convergence of Wavelet Solution to the Sideways Heat Equation

  Jin Ru WANG

  Acta Mathematica Sinica. 2010, 26(10): 1981-1992. https://doi.org/10.1007/s10114-010-7242-4

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  We consider the problem u_xx(x, t) = u_t(x, t), 0 ≤ x < 1, t≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space V_j . In the previous papers, the theoretical results concerning the error estimate are L²-norm and the solutions aren’t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ [0, 1].
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  The Reduction Square Root and Perturbation for a Class of Strongly Continuous Operator Families

  Chuang CHEN, Xiao Qiu SONG, Hui Min LI

  Acta Mathematica Sinica. 2010, 26(10): 1993-2002. https://doi.org/10.1007/s10114-010-7644-3

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  We introduce the concept of α times C-second resolvent families and present the relationship between α times C-resolvent families and α times C-second resolvent families. Moreover,the perturbation and square root for α times C-resolvent families are considered in this paper which generalize the counterparts of C-Cosine operator functions.
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  On the Variation of a Metric and Its Application

  Fa En WU

  Acta Mathematica Sinica. 2010, 26(10): 2003-2014. https://doi.org/10.1007/s10114-010-7470-7

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  Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system. In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a ritical point of the third heat invariant functional for a compact 4-dimensional manifold M, then (M,g) is either scalar flat or a space form.
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  Remarks on One-Dimensional Compressible Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum

  Zhen Hua GUO, Chang Jiang ZHU

  Acta Mathematica Sinica. 2010, 26(10): 2015-2030. https://doi.org/10.1007/s10114-009-7559-z

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  The Navier–Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper.When the viscosity coefficient μ is proportional to ρ^θ with 0 < θ < 1, the global existence and the uniqueness of weak solutions are proved which improves the previous results in [Vong, S. W., Yang,T., Zhu, C. J.: Compressible Navier–Stokes equations with degenerate viscosity coefficient and vacuum II. J. Differential Equations, 192(2), 475–501 (2003)]. Here ρ is the density. Moreover, a stabilization rate estimate for the density as t → +∞ for any θ > 0 is also given.