15 December 1987, Volume 3 Issue 4

    

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  Asymptotic Behavior of Solutions of Linear Delay Equations

  F. V. Atkinson, Zhang Shunian

  Acta Mathematica Sinica. 1987, 3(4): 289-300. https://doi.org/10.1007/BF02559909

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  The delay differential equations of the form
  x'(t)=-a(t)x(t-1),t≥0 are considered, where a(t)≥0
  is locally integrable on [0,∞). The main result: Let 0<c(t)≤a(t)≤k(t) for large t ∫^∞, and c(t)≤Mc(t') for t, t'≥T,|t-t'|≤l with some constants l>0,M>1,T≥0. Then the condition with some constant α>0 dependent on l, M, ensures that all solutions of (*) tend to zero as t→∞.
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  On Marcus-Wang Conjecture

  Hu Shu'an

  Acta Mathematica Sinica. 1987, 3(4): 301-304. https://doi.org/10.1007/BF02559910

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  Le A_m ,B_m ,m=1,...,p, be linear operators on an n-dimensional unitary space V.L is a linear operator on ⊕²V, the tensor product space with the customarily induced inner product.
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  Canonical Reducing Curves of Surface Homeomorphism

  Wu Yingqing

  Acta Mathematica Sinica. 1987, 3(4): 305-313. https://doi.org/10.1007/BF02559911

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  In this paper we will introduce the concept of canonical reducing set of a surface homeomorphism, and prove that it is unique up to an isotopy. As an application, we will give a simple proof of Thurston's theorem on classifying mappings on non-orientable surface, using the techniques of quasiconformal mappings and some known results in the orientable case, especially the Thurston theorem on orientable surfaces.
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  Monotone Polynomial Approximation in L_p Space

  Yu Xiangming

  Acta Mathematica Sinica. 1987, 3(4): 315-326. https://doi.org/10.1007/BF02559912

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  We prove that for monotone function f(x)∈L_p [-1, 1], 1≤p<∞, there exists a monotone algebraic polynomial L_n (f,x) of degree n such that , where and ω_2,Ψ, is the second-order modulus of smoothness which weighs differently the behavior of f(x) in the middle of the interval and near the endpoints. This estimate improves the known result of Shvedov.
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  Qualitative Analysis of a Mathematical Model for Tissue Inflammation Dynamics

  Jing Zhujun

  Acta Mathematica Sinica. 1987, 3(4): 327-339. https://doi.org/10.1007/BF02559913

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  The non-linear differential eguation
  
  is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.
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  On the Maximum Modulus of Complete Trigonometric Sums

  Zhang Mingyao, Hong Yi

  Acta Mathematica Sinica. 1987, 3(4): 341-350. https://doi.org/10.1007/BF02559914

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  Let Q_k (p) be a set consisting of all polynomials of degree k with integral coefficients
  f(x)=a_kx^k+...+a₁x,
  where p×a_k.
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  A Note on Nonlinear Oscillations at Resonance

  Pierpaolo Omari, Fabio Zanolin

  Acta Mathematica Sinica. 1987, 3(4): 351-361. https://doi.org/10.1007/BF02559915

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  Existence of 2π-periodic solutions to the equation is proved, under sharp nonresonance conditions on the interaction, of g(x)/x with two consecutive eigenvalues m² and(m+1)²: touch with the eigenvalues is allowed.
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  Local Index Theorem for Signature Operators

  Yu Yanlin

  Acta Mathematica Sinica. 1987, 3(4): 363-372. https://doi.org/10.1007/BF02559916

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  This article gives a direct proof of the local index theorem for the signature operators.
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  Condition for Generating Limit Cycles by Bifurcation of the Loop of a Saddle-point Separatrix

  Feng Beiye

  Acta Mathematica Sinica. 1987, 3(4): 373-382. https://doi.org/10.1007/BF02559917

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  In paper[1] (On the stability of a saddle-point separatrix loop and analytical criterion for its bifurcation limit cycles Acta Mathematica Sinica Vol. 28. No. 1, 55-70, Bejing China 1985), we considered the problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-point separatrix. We gave for the first time a criterion for the stability of the loop as following:L₀ is stable (unstable) if wherex=-(t),y=-(t) then a sufficient condition for the bifurcation which generates limit cycles. This paper generalizes the result of[1] to the case where the loop contains a center or the loop tends to an infinite saddle-point, and removes the restriction that the saddle-point should be an elementary singular point. Applying the results of this paper, the author studies a two-parameter system
  
  The results obtained by the author in this in real field coincides with the results given by Prof. Qin Yuanxun by means of the complex qualitative theory in complex field.