15 September 1989, Volume 5 Issue 3

    

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  A Geometrical index for the Group Z^p

  Liu Jiaquan

  Acta Mathematica Sinica. 1989, 5(3): 193-196. https://doi.org/10.1007/BF02107545

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  A geometrical index for group Z^p as well as its applications is given,where p is a prime integer.
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  Multiple Periodic Solutions for a Class of Nonlinear Nonautonomous Wave Equations^*

  Wang Zhiqiang

  Acta Mathematica Sinica. 1989, 5(3): 197-213. https://doi.org/10.1007/BF02107546

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  In this paper,by means of our Z_p index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations.All previous known results rely either on the oddness of nonlinear terms,or on autonomous systems,and the best result for the general case is the existence of two nontrivial periodic solutions (Amann,Zehnder,K.C.Chang,S.P.Wu,S.J.Li,Z.Q.Wang).In this paper,under the assumption that the nonlinear term is T/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.
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  On the LaSalle’s Invarisnt Set for Five-Dimensional Lotka-Volterra Prey-Predator Chain Systems

  Lu Zhengyi

  Acta Mathematica Sinica. 1989, 5(3): 214-218. https://doi.org/10.1007/BF02107547

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  In this paper,we discuss n-dimensional Lotka-Volterra prey-predator chain systems and prove that the locally,asymptotically stable equilibrium point of a five-dimensional system is a globally stable one.Namely,we prove that Chen Lansun's conjecture is true for the case n=5.
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  The Uniform Convergence of the Sum of Independent Random Variables

  Zheng Zukang

  Acta Mathematica Sinica. 1989, 5(3): 219-222. https://doi.org/10.1007/BF02107548

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  Let X_i (i=1,2,...) be the independent random variables on the probability space (Ω,π,P).In this paper we will show that the necessary and sufficient condition of the uniform convergence of Σ X_i is the convergence of Σ m_i⁽¹⁾ and Σ m_i⁽⁰⁾,where m_i⁽¹⁾,m_i⁽⁰⁾ denote the 1-quantile and 0-quantile of X_i respectively.
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  Necessary and Sufficient Condition of the Existence of Global Hopf Bifurcation for Hamiltonian Systems

  Deng Shitao

  Acta Mathematica Sinica. 1989, 5(3): 223-234. https://doi.org/10.1007/BF02107549

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  For H∈C² (Ω,R) where 0∈ΩR²ⁿ,Hⁿ (0)=0 and det H"(0)≠0,the paper proves that there is a global Hopf bifurcation from x=0 for Hamiltonian system x=JH(x) iff JH(0) possesses purely imaginary eigenvalues.The work improves the corresponding result of J.C.Alexander and J.Yorke (Amer.J.Math.,100 (1978),263-292).
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  An a Priori Estimate for Douglas Problem in Riemannian Manifolds^*

  Ji Min

  Acta Mathematica Sinica. 1989, 5(3): 235-249. https://doi.org/10.1007/BF02107550

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  This paper is concerned with the Douglas problem in general Riemannian manifolds.For its perturbed problem we establish a uniform a priori estimate under the energy level min {d^*,m^*+s₀} (see Theorem 3.3).This level seems to be the best possible since in case of Rⁿ it is just the Douglas condition.
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  Morse Theory on Banach Manifolds

  Wang Tixiang

  Acta Mathematica Sinica. 1989, 5(3): 250-262. https://doi.org/10.1007/BF02107551

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  Let M be a C²-Finsler manifold modeled on a Banach space,and let f be a C²-real-valued function defined on M.Using the A-gradient vector field which was introduced in [31] we give a suitable definition for nondegenegacy of critical points of f,then generalize the Morse handle-body decomposition theorem and the Morse inequalities to a kind of Banach manifolds.A generalization in the reflexive case has been done in [31].
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  Amalgamss of Rank 2 in Characteristic 3 Involving L₂ (5)^*

  Huang Jianhua, Ma Jixin, Bernd Stellmacher

  Acta Mathematica Sinica. 1989, 5(3): 263-270. https://doi.org/10.1007/BF02107552

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  In this paper we treat a variation of the (B,N)-Pair Theorem.We investigate the amalgam (S₅,Chev (3)),whose structure is similar to that of a Weak (B,N)-Pair,and thus give a description of the 3-local subgroups of M^c-L and U₅ (2).
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  Solutions of Eqnations Involving Analytic Functions in a Banach Algebra

  Li Bingren, Yang Yiming

  Acta Mathematica Sinica. 1989, 5(3): 271-280. https://doi.org/10.1007/BF02107553

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  In this paper,we discuss the existence and uniqueness of solution of an equation f(x)=a in a Banach algebra.We have the following fundamental lemma:

  Let A be a Banach algebra with identity,U be an open subset of complex plane C,f be an univalent function in U,V=f(U),a∈A.If the spectral set of a,σ (a)⊂V,then there exists an unique x∈A,such that
  σ(x)⊂U,and f (x)=a.
  Using this lemma,we can get generalizations of corresponding results in [1-8].Moreover,we also give a generalization of the fundamental theorem on special solution in [3].

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  A Note on Unit and Class Number of Real Quadratic Fields

  Takashi Agoh

  Acta Mathematica Sinica. 1989, 5(3): 281-288. https://doi.org/10.1007/BF02107554

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  Let p be an odd prime with p≡1 (mod 4) and Q(√p) be the real quadratic field.Also let ε and h denote the fundamental unit and the class number of Q(√p),respectively.The main purpose of this paper is to study the "explicit" expressions of ε^h and ε^2h,and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.