15 June 1989, Volume 5 Issue 2

    

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  On the Existence of Density of the Law of a Wiener Functional^*

  Yan Jiaan

  Acta Mathematica Sinica. 1989, 5(2): 97-100. https://doi.org/10.1007/BF02107663

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  In this note we improve a recent result of Nualart and Zakai [1] on the existence of density of the law of a Wiener functional.
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  Multiple Solutions for Indefinite functionals and applications to asymptotically linear problems^*

  Wang Zhiqiang

  Acta Mathematica Sinica. 1989, 5(2): 101-113. https://doi.org/10.1007/BF02107664

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  In this paper,by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann,Conley,Zehnder and K.C.Chang.As applications,the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated.And our results generalize some recent ones due to Coti-Zelati,J.Q.Liu,S.Li,etc.
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  Existence, Uniqueness and Convergence as Vanishing Viscosity for a Reaction-Diffusion-Convection System

  Teng Zhenhuan, Ying Lungan

  Acta Mathematica Sinica. 1989, 5(2): 114-135. https://doi.org/10.1007/BF02107665

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  A simple qualitative model of dynamic combustion
  (1)
  is considered,where constant q>0 represents the binding energy.The existence and uniqueness of classical solution for the initial value problem and for the first boundary value problem of system (1) are proved and then it is proved that as ε→0 classical solutions of the initial value problem of system (1) converge to a pair of limit functions,which is an admissible solution of the initial value problem of system (1) with ε=0.
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  Coordinate Gap and Bidualily

  Xu Mingwei

  Acta Mathematica Sinica. 1989, 5(2): 136-145. https://doi.org/10.1007/BF02107666

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  For a variety X in Pⁿ,we define a number b₂,called coordinate gap number,and prove that b₂>2 only if X is not reflexive.Then,for a smooth surface in P³,we obtain a concrete sufficient and necessary condition for b₂>2,which enables us to discuss the biduality of surfaces in P³.
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  Isomorphisms of Steinberg Groups over Commutative Rings^*

  Li Fuan

  Acta Mathematica Sinica. 1989, 5(2): 146-158. https://doi.org/10.1007/BF02107667

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  Let A and R be commutative rings,and m and n be integers≥3.It is proved that,if Λ:St_m(A)→St_n(R) is an isomorphism,then m=n.When n≥4,we have: (1) Every isomorphism Λ:St_n(A)→St_n(R) induces an isomorphism λ:E_n(A)→E_n(R),and Λ is uniquely determined by if St λ; (2) If St_n(A)≡St_n(R) then K_2.n(A)≡K_2.n(R); (3) Every isomorphism E_n(A)→E_n(R) can be lifted to an isomorphismSt_n(A)→St_n(R); (4)St_n(A)≡St_n(R) if and only if A≡R.For the case n=3,if St₃(A) and St₃(R) are respectively central extensions of E₃(A) and E₃(R),then the above (1) and (2) hold.
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  An extension of the Massera Theorem^*

  Ding Tongren

  Acta Mathematica Sinica. 1989, 5(2): 159-164. https://doi.org/10.1007/BF02107668

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  This paper deals with the implications of the boundedness of a solution for the periodic dynamical system of second order.
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  Inner Amenable Groups and the InducedC^*-Algebra Properties

  Yuan Chuankuan

  Acta Mathematica Sinica. 1989, 5(2): 165-169. https://doi.org/10.1007/BF02107669

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  A discrete group can generate a C^*-algebra,denoted by C_β^*(G),by considering the so called conjugation regular representation β of the group G.Based on this treatment,the inner amenability of G shall be characterized by the existence of the states on C_β^* (G) satisfying some certain conditions.
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  Bifurcation of Homoclinics in a Nnonlinear Oscillation

  Zhang Weinian

  Acta Mathematica Sinica. 1989, 5(2): 170-184. https://doi.org/10.1007/BF02107670

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  In this paper we discuss the bifurcation of homoclinics of the equation
  (*)
  whereg (x) is such that the unperturbed equation x"+g (x)=0 has homoclinic orbits through zero.We give the bifurcation graph of small parameters μ and λ,and that of small functions g₁ and f₁.Then we give a criterion to determine the codimensions of bifurcation manifolds of small functions g₁ and f₁.Thus we generalize the conclusions of J.K.Hale.
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  Laws of Large Numbers for Weighted Sums of Random Variables and Random Elements^*

  Lin Zhengyan

  Acta Mathematica Sinica. 1989, 5(2): 185-192. https://doi.org/10.1007/BF02107671

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  Consider the weighted sums of a sequence {X_n} of independent random variables or random elements inD [0,1].For convergence of S_n in probability and with probability one,in [2],[3] etc.,the following stronger condition is required: {X_n} is uniformly bounded by a random variable X,i.e.P(|X_n|≥x)≤P(|X|≥x) for allx>0.Our paper aims at trying to drop this restriction.