中国科学院数学与系统科学研究院期刊网

15 September 2000, Volume 16 Issue 3
    

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  • Mufa Chen
    Acta Mathematica Sinica. 2000, 16(3): 361-368. https://doi.org/10.1007/s101140000067
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    A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented. The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions.
  • Liming Wu
    Acta Mathematica Sinica. 2000, 16(3): 369-394. https://doi.org/10.1007/PL00011549
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    In this paper we shall characterize the large deviation principles (abbreviated to LDP) of Donsker-Varadhan of a Markov process both for the weak convergence topology and for the τ-topology, by means of a hyper-exponential recurrence property. A Lyapunov criterion for this type of recurrence property is presented. These results are applied to countable Markov chains, unidimensional diffusions, elliptic or hypoelliptic diffusions on Rienmannian manifolds. Several counter-examples are equally presented.
  • Yaochen Zhu
    Acta Mathematica Sinica. 2000, 16(3): 395-398. https://doi.org/10.1007/s101140000076
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    Let f(x) be a continued fraction with elements anx, where coefficients an are positive algebraic numbers. Using the criterion of [1] for any nonzero real algebraic numbers α1,..., αs with distinct absolute values the algebraic independence of the values f(α1),...,f(αs) is proved under certain assumption concerning only with an. For some transcendental numbers ξ the algebraic independence of values f(ξj)(j∈Z) is also established.
  • Yan Xu
    Acta Mathematica Sinica. 2000, 16(3): 399-404. https://doi.org/10.1007/s101140000041
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    This paper has studied two open questions about normal functions due to Lappan, and obtained two corresponding results for α-normal functions.
  • Guozhen Lu
    Acta Mathematica Sinica. 2000, 16(3): 405-444. https://doi.org/10.1007/PL00011552
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    This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inequalities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on G. Some very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (ε, δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.
  • Min Dai
    Acta Mathematica Sinica. 2000, 16(3): 445-454. https://doi.org/10.1007/s101140000068
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    The binomial tree method is the most popular numerical approach to pricing options. However, for currency lookback options, this method is not consistent with the corresponding continuous models, which leads to slow speed of convergence. On the basis of the PDE approach, we develop a consistent numerical scheme called the modified binomial tree method. It possesses one order of accuracy and its efficiency is demonstrated by numerical experiments. The convergence proofs are also produced in terms of numerical analysis and the notion of viscosity solution.
  • Shenghong Li, Xiangao Liu
    Acta Mathematica Sinica. 2000, 16(3): 455-468. https://doi.org/10.1007/s101140000061
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    In this paper, we introduce the concept of the G class of functions of the parabolic class, and show the Hölder continuity of the G class of functions. The introduction of this concept contributes to the proof of the regularity and existence of the solution for the first boundary problem of parabolic equation in divergence form.
  • Bingren Li, Pingkwan Tam
    Acta Mathematica Sinica. 2000, 16(3): 469-486. https://doi.org/10.1007/s101140000062
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    In this paper, we study real Banach * algebras systematically. We present the right form of Pták's inequality [1,4] in the real case, and generalize the results of Vukman in [3] to the general case (algebras with or without an identity). Moreover, this paper is a real analogue of Pták's work [1] in the complex case.
  • Peter Y. H. Pang, Hongyu Wang, Youde Wang
    Acta Mathematica Sinica. 2000, 16(3): 487-504. https://doi.org/10.1007/s101140000060
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    In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schrödinger flow for maps from a compact Riemannian manifold M with dim (M)≤3 into a compact Kähler manifold (N, J) with nonpositive Riemannian sectional curvature.
  • Zhaoli Liu
    Acta Mathematica Sinica. 2000, 16(3): 505-514. https://doi.org/10.1007/s101140000063
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    This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations. We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions.
  • Boling Guo, Bixiang Wang
    Acta Mathematica Sinica. 2000, 16(3): 515-526. https://doi.org/10.1007/s101140000064
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    In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation.
  • Yunbo Zeng
    Acta Mathematica Sinica. 2000, 16(3): 527-534. https://doi.org/10.1007/s101140000045
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    A hierarchy of multidimensional Hénon-Heiles (M-H-H) systems are constructed via the x- and tn-higher-order-constrained flows of KdV hierarchy. The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy. By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.
  • Chunlei Liu
    Acta Mathematica Sinica. 2000, 16(3): 535-540. https://doi.org/10.1007/s101140000074
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    Hua's estimate is established for character sums in a number field. A relationship between liftings of a character sum in a local field is also studied.