中国科学院数学与系统科学研究院期刊网

15 December 1996, Volume 12 Issue 4
    

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  • Chen Mufa
    Acta Mathematica Sinica. 1996, 12(4): 337-360. https://doi.org/10.1007/BF02106789
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    The study of the convergent rate (spectral gap) in the L2-sense is motivated from several different fields: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in[7] for the estimate of the convergent rate and as a continuation of[4],[5],[7-9],[23] and[24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper[10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings[4] is also extended in the paper.
  • Jiang Jifa
    Acta Mathematica Sinica. 1996, 12(4): 361-371. https://doi.org/10.1007/BF02106790
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    This paper establishes criteria for the existence and non-existence of nonzero periodic solutions of the generalized Liénard equation +g(x)=0. The main goal is to study to what extent the dampingf can be small so as to guarantee the existence of nonzero periodic solutions of such a system. With some standard additional assumptions we prove that if for a small |x|,∫±∞|f(x,y)|-1 dy=±∞, then the system has at least one nonzero periodic solution, otherwise, the system has no nonzero periodic solution. Many classical and well-known results can be proved as corollaries to ours.
  • Zhu Deming
    Acta Mathematica Sinica. 1996, 12(4): 372-378. https://doi.org/10.1007/BF02106791
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    Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in[3-5].
  • Ma Renyi
    Acta Mathematica Sinica. 1996, 12(4): 379-384. https://doi.org/10.1007/BF02106792
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    In this note, we prove that the symplectic blow-up or blow-down in the dimension 4 is rigid, i.e. the symplectic area of the divisor does not exceed the symplectic radius of the neighbourhood on which we do the blow-up or blow-down.
  • He Shuyuan
    Acta Mathematica Sinica. 1996, 12(4): 385-397. https://doi.org/10.1007/BF02106793
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    Let Y={Yn;nN2} be a stationary linear random field generated by a twodimensional martingale difference. Where N2 denotes the two dimensional integer lattice. The main purpose of this paper is to obtain the LIL convergence for the partial-sums of Y.
  • Zheng Zhiming
    Acta Mathematica Sinica. 1996, 12(4): 398-412. https://doi.org/10.1007/BF02106794
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    In this paper, we want to show the abundance of chaotic systems with absolutely continuous probability measures in the generic regular family with perturbable points. More precisely, we prove that if fa:I → I, a P is a regular family satisfying some conditions described in the next section, then there exists a Borel set Ω⊂P of positive Lebesgue measure such that for every a ∈ Ω,fa admits an absolutely continuous invariant probability measure w.r.t. the Lebesgue measure. The idea of proof in this paper, as compared with that shown in[1] and[7], follows a similar line.

  • Zhou Caijun
    Acta Mathematica Sinica. 1996, 12(4): 413-414. https://doi.org/10.1007/BF02106795
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    In this short paper, we prove that if R is a regular local ring of unequal characteristic, then there exists an additive covariant functor G from the category of abelian sheaves on Spec R to the category of abelian groups such that id R (G(R))>dim G(R). This result shows that the answer to the question 3.8 (ii) in[3] may be negative.
  • Xiong Jianfei
    Acta Mathematica Sinica. 1996, 12(4): 415-432. https://doi.org/10.1007/BF02106796
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    In this paper we classify (2,2)-stable unfoldings of map germs with codimensions ≤ 4 by means of the algorithm given in[4] by Wassermann.
  • Jia Chaohua
    Acta Mathematica Sinica. 1996, 12(4): 433-445. https://doi.org/10.1007/BF02106797
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    Let P(x) denote the greatest prime factor of n. In this paper, we shall prove that P(x)>x0.728 holds true for sufficiently largex.