15 March 2013, Volume 56 Issue 2

    

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  Global Classical Solutions to the Goursat Problem for First-Order Quasilinear Hyperbolic Systems

  Cun Ming LIU, Jian Li LIU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 145-154. https://doi.org/10.12386/A2013sxxb0015

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  We consider the Goursat problem for first-order quasilinear hyperbolic systems. Under the assumptions that the system is weakly linearly degenerate and the boundary conditions on the characteristics are small and decaying, we obtain the existence of global C¹ solutions and give a pointwise estimate to classical solutions. As an important example, we apply this result to the equation for timelike extremal surface in Minkowski space.
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  On Classification of Higher-Dimensional Algebraic Varieties with Ample Vector Bundles

  Zhi Yong CHEN, Fang Fang DENG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 155-162. https://doi.org/10.12386/A2013sxxb0016

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  Let X be a smooth projective variety of dimension n(n≥3) and ε an ample vector bundle with rank r=n-k(k≥1) over X. We denote Λ(ε,K_X)=max{(-K_X-c₁(ε))·C|R=R+[C]∈Ω, and l(R)=-K_X·C}, where K_X is the canonical bundle of X, c₁(ε) means the first Chern class of ε, Ω denotes the set of extremal rays R such that (K_X+c₁(ε))·C≤0, R₊ is the set of positive real number, and l(R) is the length of R. The classification of (X, ε) will ba given when Λ(ε,K_X)≥k-1.
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  Boundedness for Toeplitz Type Operator Related to Singular Integral Operators Satisfying a Variant of Hörmander's Condition

  Run Sheng YANG, Lan Zhe LIU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 163-174. https://doi.org/10.12386/A2013sxxb0017

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  We establish some sharp maximal function inequalities for the Toeplitz type operator related to certain singular integral operators satisfying a variant of Hörmander's condition. As an application, we obtain the boundedness of the operator on Lebesgue, Morrey and Triebel-Lizorkin spaces.
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  Boundedness of Fractional Integral Operators with Rough Kernels on Weighted Morrey Spaces

  Hua WANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 175-186. https://doi.org/10.12386/A2013sxxb0018

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  Let M_Ω,α and T_Ω,α be the fractional maximal and integral operators with rough kernels, where 0<α<n. In this paper, we shall study the continuity properties of M_Ω,α and T_Ω,α on the weighted Morrey spaces L^p,κ(w). The boundedness of their commutators with BMO functions is also obtained.
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  Quasi-Lipschitz Equivalence of Moran Fractals

  Qin WANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 187-196. https://doi.org/10.12386/A2013sxxb0019

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  We consider the class of Moran sets satisfying the conditions c_*>0 and s_*=s_*∈(0, 1). We obtain that two Moran fractals in the class are quasi-Lipschitz equivalent if and only if they have the same Hausdorff dimension.
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  A Note on Blow-up Estimates for a System of Heat Equations Coupled Boundary Flux

  Zhong Ping LI, Chun Lai MU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 197-202. https://doi.org/10.12386/A2013sxxb0020

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  In this note, we adapt the scaling method and give the blow-up estimates for a system of heat equations coupled boundary flux.
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  A Note on the Diagonal Theorem of Bivariate Rational Formal Power Series

  Xiao Li WU, Shao Shi CHEN

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 203-210. https://doi.org/10.12386/A2013sxxb0021

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  Special functions that satisfy linear differential equations with polynomial coefficients appear ubiquitously in combinatorics and mathematical physics. Such kind of special functions are called D-finite functions by Stanley. In the early 1980's, many combinatorists, such as Gessel, Stanley, Zeilberger etc., conjectured that the diagonal of rational power series in several variables is D-finite. Gessel and Zeilberger proved this conjecture in their papers, respectively. Later, Lipshitz pointed out that their proofs are not complete and he gave a proof by basing on a different idea. Zeilberger completed his proof with the theory of holonomic D-modules. In this note, we follow the spirit of Gessel's proof strategy and fix the gap in his proof in the case of bivariate rational formal power series. The key ingredients we used are some basic properties of the diagonal operation.
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  Decycling Number and Upper-Embeddibility of Generalized Petersen Graphs

  Er Ling WEI, Yi Fan LI

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 211-216. https://doi.org/10.12386/A2013sxxb0022

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  Given a graph G=(V,E), S⊆V, the minimal number of S whose removal eliminates all the cycles in G is called the decycling number, denoted as ∇(G). The problem of deternimating the decycling number of a graph is NP-complete. Bau and Beineke proposed the following question: which cubic graphs of order 2n have decycling number [n+1/2]? In this paper we proved that generalized Petersen graph P(n, k) is such a class of graphs. Based on this result, we proved that P(n, k) is upper-embeddable which generalized a result of Ren.
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  On the Coarse Embeddability of l_p-Spaces Into l₂

  Zheng Hua LUO, Wen ZHANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 217-222. https://doi.org/10.12386/A2013sxxb0023

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  We made some further research on the coarse embeddability between l_p spaces and l₂ and we proved that a metric space X can be coarsely embedded into l₂ if and only if there exists some p₀>0 such that X can be coarsely embedded into l_p, 2<p≤p₀} uniformly. This gave an equivalent condition for the coarse embeddability of metric spaces into l₂.
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  A Variant of Hörmander's Condition and Weighted Estimates for Singular Integrals

  Pu ZHANG, Dai Qing ZHANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 223-232. https://doi.org/10.12386/A2013sxxb0024

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  Let 1<r<∞, B_j and φ_j (j=1,...,m) are suitable functions. We say a kernel K satisfies a variant of the classical L^r-Hörmander's condition, if there exist constants c^r>0 and C^r>0, such that for any y∈Rⁿ and R>c^r|y|,
  
  . In this paper, the authors establish the weighted norm inequalities for singular integral operators with kernel satisfying the above variant of the classical L^r-Hörmander's condition.
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  Toeplitz Operators with L¹ Symbols on Weighted Bergman Spaces of the Unit Ball

  Li HE, Guang Fu CAO, Zhong Hua HE

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 233-244. https://doi.org/10.12386/A2013sxxb0025

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  We investigate the boundedness and compactness of Toeplitz operators with L¹ symbols on weighted Bergman spaces of the unit ball A_α^p(B_n, dV_α) (1<p<∞). We describe the boundedness and compactness of Toeplitz operators with L¹ symbols on A_α^p(B_n, dV_α) equivalently by using the Berezin transform of the Toeplitz operators, and promote the conclusion given by Agbor about the boundedness and compactness of Toeplitz operators with L¹ symbols on Bergman space of the unit disk L_a²(D).
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  The Existence of Weak and Strong Solutions for Oseen Equations

  Wan Min ZHANG

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 245-256. https://doi.org/10.12386/A2013sxxb0026

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  We deal with the existence of weak and strong solutions for Oseen equations in the case that the divergence of convective vector is nonzero. Using Lax-Milgram Theorem, we first prove the existence of weak solution in H¹(Ω), and then on the basis of this result, we use methods of iterative and dual principle to prove the existence of weak and strong solutions in generic Sobolev spaces and obtain corresponding estimates for these solutions.
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  Quasirecognition of the Simple Group G₂(3ⁿ)

  Li Li LI

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 257-262. https://doi.org/10.12386/A2013sxxb0027

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  Let G be finite group such that M(G)=M(G₂(3ⁿ)), where n≥1. Then G has a normal subgroup isomorphic to G₂(3ⁿ). Especially, if |G|=|G₂(3ⁿ)|, then G≌G₂(3ⁿ).
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  On the Busemann-Petty Problem of λ-Intersection Bodies

  Tong Yi MA

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 263-278. https://doi.org/10.12386/A2013sxxb0028

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  For λ<n and λ=0, Rubin introduced the concept of the λ-intersection body IB_λ(K) of an origin-symmetric star body K in Rⁿ. In this paper, we consider Busemann-Petty's problem of whether IB_λ(K)⊆IB_λ(L) implies vol_n(K)≤vol_n(L) (or vol_n(K)≥vol_n(L)). We proved that Busemann-Petty's problem of λ-intersection body in Rⁿ has a positive answer if and only if any origin-symmetric star body in Rⁿ is a λ-intersection body. Our results generalize the specific affirmative answer of classic intersection bodies to Busemann-Petty problem.
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  Some Inequalities for L_p-Mixed Affine Surface Area

  Xian Yang ZHU

  Acta Mathematica Sinica, Chinese Series. 2013, 56(2): 279-288. https://doi.org/10.12386/A2013sxxb0029

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  In this paper, the concepts of the i th L_p-mixed affine surface area and L_p-polar curvature images are introduced, some new inequalities connecting these new notions with L_p-centroid bodies and p-Blaschke bodies are established. Moreover, a Blaschke-Santaló type inequality for L_p-mixed affine surface area is established. Our results also imply a generalization of the class dual Urysohn inequalities.