15 September 2008, Volume 24 Issue 9

    

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  Oscillation Theorems for Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

  Samir H. SAKER, Donal O'REGAN, Ravi P. AGARWAL

  Acta Mathematica Sinica. 2008, 24(9): 1409-1432. https://doi.org/10.1007/s10114-008-7090-7

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  By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation
  [r(t)[y(t)+p(t)y(τ(t))]^Δ]^Δ+q(t)f(y(δ(t)))=0,
  on a time scale T. The results improve some oscillation results for neutral delay dynamic equations and in the special case when T=R our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math., 48 (1996), 871-886]. When T=N, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36 (1998), 123-132]. When T=hN, T={t:t=q^k,k∈N,q>1},T=N²={t²:t∈N},T=T_n={t_n=∑_k=1ⁿ 1/k,n∈N₀},T={t²:t∈N},T={√n:n∈N₀} and T={√n1/3:n∈N₀} our results are essentially new. Some examples illustrating our main results are given.
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  Projective Blaschke Manifolds

  An Min LI, Guo Song ZHAO

  Acta Mathematica Sinica. 2008, 24(9): 1433-1448. https://doi.org/10.1007/s10114-008-7099-y

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  In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds.
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  New Hardy Spaces Associated with Some Anisotropic Herz Spaces and Their Applications

  Yong DING, Sen Hua LAN, Shan Zhen LU

  Acta Mathematica Sinica. 2008, 24(9): 1449-1470. https://doi.org/10.1007/s10114-007-7085-9

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  In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on Rⁿ are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderón-Zygmund operators on the anisotropic Herz-type Hardy spaces.
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  On Hyperinvariant Subspaces of Contraction Operators on a Banach Space Whose Spectrum Contains the Unit Circle

  Ming Xue LIU

  Acta Mathematica Sinica. 2008, 24(9): 1471-1474. https://doi.org/10.1007/s10114-008-7012-8

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  In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace.
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  The Isometric Extension of the Into Mapping from the Unit Sphere S₁(E) to S₁(l^∞(Γ)

  Xiao Hong FU

  Acta Mathematica Sinica. 2008, 24(9): 1475-1482. https://doi.org/10.1007/s10114-008-7286-x

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  This paper considers the isometric extension problem concerning the mapping from the unit sphere S₁(E) of the normed space E into the unit sphere S₁(l^∞(Γ)). We find a condition under which an isometry from S₁(E) into S₁(l^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V₀: S₁(E)→S₁(F) is a surjective isometry, where c₀₀(Γ)⊆F⊆l^∞(Γ), then V₀ can be extended to be an isometric operator defined on the whole space.
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  Nonparametric Inference in a Simple Change-point Model

  Zhan Feng WANG, Yao Hua WU, Lin Cheng ZHAO

  Acta Mathematica Sinica. 2008, 24(9): 1483-1496. https://doi.org/10.1007/s10114-008-6070-2

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  In this paper, we consider a change point model allowing at most one change, X(i/n)=f(i/n)+e(i/n), where f(t)=α+θI_(t0,1)(t), 0≤t≤1, {e(1/n),..., e(n/n)} is a sequence of i.i.d. random variables distributed as e with 0 being the median of e. For this change point model, hypothesis test problem about the change-point t₀ is studied and a test statistic is constructed. Furthermore, a nonparametric estimator of t₀ is proposed and shown to be strongly consistent. Finally, we give an estimator of jump θ and obtain it's asymptotic property. Performance of the proposed approach is investigated by extensive simulation studies.
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  Limsup Results and LIL for Partial Sum Processes of a Gaussian Random Field

  Yong-Kab CHOI, Miklós CSÖRGÖ

  Acta Mathematica Sinica. 2008, 24(9): 1497-1506. https://doi.org/10.1007/s10114-008-6205-5

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  Let {ξ _j ; j∈Z₊^d be a centered stationary Gaussian random field, where Z₊^d is the d-dimensional lattice of all points in d-dimensional Euclidean space R^d, having nonnegative integer coordinates. For each j=(j₁,..., j_d) in Z₊^d, we denote |j|=j₁... j_d and for m, n∈Z₊^d, define S(m, n]=Σ _m<j≤n ζ_j, σ²(|n-m|)=ES²(m, n], S_n=S(0, n] and S₀=0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t>0, which is nondecreasing and regularly varying with exponent α at b≥0 for some 0<α<1. Under some additional conditions, we study limsup results for increments of partial sum processes {S_n}_n∈Z+^d and prove as well the law of the iterated logarithm for such partial sum processes.
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  A Four-filtrated May Spectral Sequence and Its Applications

  Xiu Gui LIU, Xiang Jun WANG

  Acta Mathematica Sinica. 2008, 24(9): 1507-1524. https://doi.org/10.1007/s10114-008-6219-z

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  In this paper, we introduce a four-filtrated version of the May spectral sequence (MSS), from which we study the general properties of the spectral sequence and give a collapse theorem. We also give an efficient method to detect generators of May E₁-term E₁^s,t,b,* for a given (s, t, b, *). As an application, we give a method to prove the non-triviality of some compositions of the known homotopy elements in the classical Adams spectral sequence (ASS).
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  Blow-up for Doubly Degenerate Nonlinear Parabolic Equations

  Ming Yu CHEN

  Acta Mathematica Sinica. 2008, 24(9): 1525-1532. https://doi.org/10.1007/s10114-008-6389-8

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  In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations.
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  L^p Approximation Capability of RBF Neural Networks

  Dong NAN, Wei WU, Jin Ling LONG, Yu Mei MA, Lin Jun SUN

  Acta Mathematica Sinica. 2008, 24(9): 1533-1540. https://doi.org/10.1007/s10114-008-6423-x

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  L^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R₊¹→R¹ and g(‖x‖_Rⁿ)∈L_loc^p(Rⁿ) with 1≤p<∞, then the RBF neural networks with g as the activation function can approximate any given function in L^p(K) with any accuracy for any compact set K in Rⁿ, if and only if g(x) is not an even polynomial.
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  Best Simultaneous Approximation to Totally Bounded Sequences in Banach Spaces

  Xian Fa LUO, Chong LI, Genaro LOPEZ

  Acta Mathematica Sinica. 2008, 24(9): 1541-1554. https://doi.org/10.1007/s10114-007-6575-0

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  This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth.
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  Bi-Lipschitz Maps in Q-regular Loewner Spaces

  Ke Ying CHEN, Ai Nong FANG

  Acta Mathematica Sinica. 2008, 24(9): 1555-1568. https://doi.org/10.1007/s10114-007-6630-x

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  By applying the theory of quasiconformal maps in measure metric spaces that was introduced by Heinonen-Koskela, we characterize bi-Lipschitz maps by modulus inequalities of rings and maximal, minimal derivatives in Q-regular Loewner spaces. Meanwhile the sufficient and necessary conditions for quasiconformal maps to become bi-Lipschitz maps are also obtained. These results generalize Rohde's theorem in Rⁿ and improve Balogh's corresponding results in Carnot groups.
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  Center and Isochronous Center Problems for Quasi Analytic Systems

  Yi Rong LIU, Ji Bin LI

  Acta Mathematica Sinica. 2008, 24(9): 1569-1582. https://doi.org/10.1007/s10114-008-6701-7

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  This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved.