15 March 2001, Volume 17 Issue 1

    

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  Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

  Qi Yu SUN

  Acta Mathematica Sinica. 2001, 17(1): 1-14. https://doi.org/10.1007/s101140000092

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  Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z0 be a subset of Z such than n∈Z0 implies n+1∈Z0. Denote the space of all compactly supported distributions by D', and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence Gn and Hn, n∈Z0, in D', define the corresponding nonstationary nonhomogeneous refinement equation
  Φ_n = Hn ∗ Φ_n+1(A·) + G_n for all n ∈ Z₀, (*)
  where Φn, n∈Z0, is in a bounded set of D'. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φn, n∈Z0, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations
  
  where the matrices Sn and the vectors , n∈Z0, can be constructed explicitly from Hn and Gn respectively. The results above are still new even for stationary nonhomogeneous refinement equations.
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  Hausdorff Measure of Homogeneous Cantor Set

  Cheng Qin QU, Hui RAO, Wei Yi SU

  Acta Mathematica Sinica. 2001, 17(1): 15-20. https://doi.org/10.1007/s101140000089

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  This paper gives the Hausdorff measure of a class of homogeneous Cantor sets.
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  Embeddings of Kleinian Groups with Torsion

  Shi Cheng WANG, Qing ZHOU

  Acta Mathematica Sinica. 2001, 17(1): 21-34. https://doi.org/10.1007/s101140000093

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  Rips conjectured that a non-clementary word hyperbolic group is cohopfian if and only if it is freely indecomposable. The results and examples in this paper show that cohopficity phenomenon in the case of word hyperbolic group with torsion is much more complicated than the conjecture. In particular, the cohopficity of such groups is not determined by the numbers of their ends and the cohopficity is not preserved by finite index subgroups. Our results and examples arise from Kleinian groups. Orbifold structures and orbifold maps are the new tools in our discussions.
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  The Blowup Mechanism of Small Data Solutions for the Quasilinear Wave Equations in Three Space Dimensions

  Hui Cheng YIN

  Acta Mathematica Sinica. 2001, 17(1): 35-76. https://doi.org/10.1007/s101140000094

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  For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and Hörmander. As an application of our result, we show that the solution of three-dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.
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  A Note on Hermitian-Einstein Metrics on Parabolic Stable Bundles

  Jia Yu LI, M. S. NARASIMHAN

  Acta Mathematica Sinica. 2001, 17(1): 77-80. https://doi.org/10.1007/s101140000091

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  Let be a compact complex manifold of complex dimension two with a smooth Köhler metric and D a smooth divisor on . If E is a rank 2 holomorphic vector bundle on with a stable parabolic structure along D, we prove that there exista a Hermitian-Einstein metric on compatible with the parabolic structure, whose curvature is square integrable.
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  On the Multi-Dimensional Duality Principle of Sawyer Type

  Yves RAKOTONDRATSIMBA

  Acta Mathematica Sinica. 2001, 17(1): 81-88. https://doi.org/10.1007/s101140000090

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  A multi-dimensional version of the duality principle of Sawyer type [1] is obtained whenever the corresponding weight satisfies some doubling property.
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  Fuzzy Topology Based on Residuated Lattice-Valued Logic

  Ming Sheng YING

  Acta Mathematica Sinica. 2001, 17(1): 89-102. https://doi.org/10.1007/s101140000079

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  We use a semantical method of complete residuated lattice-valued logic to give a generalization of fuzzy topology as a partial answer to a problem by Roser and Turquette.
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  Approximation Theorems and Fixed Point Theorems for Various Classes of 1-set-contractive Mappings in Banach Spaces

  Li Shan LIU

  Acta Mathematica Sinica. 2001, 17(1): 103-112. https://doi.org/10.1007/s101140000095

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  In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234-240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with int K≠Ø. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors.
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  On Graded Selfduality

  Sheng Gui ZHANG

  Acta Mathematica Sinica. 2001, 17(1): 113-118. https://doi.org/10.1007/s101140000096

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  In this paper, we discuss the connection between rigid graded selfduality over some noncommutative graded ring and the selfduality over its initial subring.
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  A Proof of the Jacobian Conjecture on Global Asymptotic Stability

  Peng Nian CHEN, Jian Xun HE, Hua Shu QIN

  Acta Mathematica Sinica. 2001, 17(1): 119-132. https://doi.org/10.1007/s101140000098

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  Let f∈C¹ (R², R²), f(0)=0. The Jacobian Conjecture states that if for any x∈R², the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true.
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  Two Theorems on the Commutativity of Arbitrary Rings

  Chang Lin FU, Xin Song YANG

  Acta Mathematica Sinica. 2001, 17(1): 133-140. https://doi.org/10.1007/s101149900018

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  We prove a theorem on the commutativity of the PI ring and use it to generalize a famous theorem of Herstein.
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  Uniqueness of the Infinite Open Cluster for High-density Percolation on Lattice Sierpinski Carpet

  Xian Yuan WU

  Acta Mathematica Sinica. 2001, 17(1): 141-146. https://doi.org/10.1007/s101140000100

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  We prove the uniqueness of infinite open cluster for high-density bond percolation on lattice Sierpinski Carpet; forthermore, an alternative proof of the existence of phase transition of the model is given. A rescaling technique is developed and used as the main tool of our proofs.
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  A Family of Nonnormal Cayley Digraphs

  Yan Quan FENG, Dian Jun WANG, Jing Lin CHEN

  Acta Mathematica Sinica. 2001, 17(1): 147-152. https://doi.org/10.1007/s101140000097

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  We call a Cayley digraph Γ=Cay(G, S) normal for G if G_R, the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found.
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  Cubic Diophantine Inequalities

  Hong Ze LI

  Acta Mathematica Sinica. 2001, 17(1): 153-160. https://doi.org/10.1007/s101140000101

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  Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ_i≥1. In this paper, we prove there are integers x₁,..., x₇ such that the inequalities and hold simultaneously.
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  Explicit Expressions of Smith’s Determinant on a Poset

  Bo Ying WANG

  Acta Mathematica Sinica. 2001, 17(1): 161-168. https://doi.org/10.1007/s101149900019

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  We introduce the meet-function and join-function on a subset of a poset and use them to give some explicit expressions of Smith's determinant.
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  A Berry-Essen Inequality for the Kaplan-Meier L-Estimator

  Qi Hua WANG, Li Xing ZHU

  Acta Mathematica Sinica. 2001, 17(1): 169-180. https://doi.org/10.1007/s101149900013

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  Let be the Kaplan-Meier estimator of distribution function F. Let J(·) be a measurable real-valued function. In this paper, a U-statistic representation for the Kaplan-Meier L-estimator, T()=∫xJ((x))d(x), is derived. Furthermore, the representation is also used to establish a Berry-Essen inequality for .