15 September 1991, Volume 7 Issue 3

    

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  Maxim um Principle of Stochastic Controlled Systems of Functional Type^*

  Zhou Xunyu

  Acta Mathematica Sinica. 1991, 7(3): 193-204. https://doi.org/10.1007/BF02582996

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  This paper studies the optimal controls of stochastic systems of functional type with end constraints. The systems considered may be degenerate and the control region may be nonconvex. A stochastic maximum principle is derived. The method is based on the idea that stochastic systems are essentially infinite dimensional systems.
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  Normal Extensions of Semigroups of Operators to Krein Spaces

  Wu Jingbo

  Acta Mathematica Sinica. 1991, 7(3): 205-215. https://doi.org/10.1007/BF02582997

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  In this paper, we prove that every strongly continuous semigroup of bounded operators on a Hilbert space may be extended to a strongly continous semigroup of normal operators on a larger Krein space. Several equivalent formulations for the case where the extension space is a Pontrjagin space are given.
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  A Classical Integrable System and the Involutive Representation of Solutions of the KdV Equation^*

  Cao Cewen

  Acta Mathematica Sinica. 1991, 7(3): 216-223. https://doi.org/10.1007/BF02582998

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  Under the constrained condition induced by the eigenfunction expression of the reflectionless potentialu=-2<q,q>≡f(q), the Schr?dinger equation, the spatial part of the Lax pair of theKdV equation, is nonlinearized to be a completely integrable system (R^2N ,dpΛdq, H=F₀):
  ,
  while the nonlinearization of the time part leads to itsN-involutive system {F _m}. The involutive solutions of the compatible system (F₀), (F_m) are mapped byf into the solutions of the m-thKdV equation.
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  Decomposition of a Class of Functionals^*

  Fan Ruzong

  Acta Mathematica Sinica. 1991, 7(3): 224-240. https://doi.org/10.1007/BF02582999

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  Let X(t) be a symmetric conservative Hunt process. In this paper, we decompose the functional u(X(t), t) into a sum of a square integrable martingale and a regular 0-quadratic variation process.
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  Some Results on Unstable Polynomial Aigebras over the Steenrod Algebra^*

  Shen Xinyao, Ying Kefeng

  Acta Mathematica Sinica. 1991, 7(3): 241-246. https://doi.org/10.1007/BF02583000

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  In this paper we give a condition under which an unstable polynomial A_p -algebra can be obtained from the modular polynomial algebra. From this we car deduce both the Aguades theorem^[2] and Clark's theorem^[4].
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  All Retraction Operators on a Complate Lattice Form a Complete Lattice^*

  Li Boyu

  Acta Mathematica Sinica. 1991, 7(3): 247-251. https://doi.org/10.1007/BF02583001

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  H. Crapo raised in 1982 the following problem: If P is a complete lattice, is Retr (P) a complete lattice? here Retr (P) denotes the set of all retraction operators on P with the pointwise order, that is,f≤g in Retr (P) iff f(x)≤g(x) for every x in P. An affirmative answer will be given in the present paper.
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  Uniformly Contrastive Semigroups and Submarkovian Semigroups(I)

  Yan Shaozong, Zhang Yinnan

  Acta Mathematica Sinica. 1991, 7(3): 252-258. https://doi.org/10.1007/BF02583002

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  Let (M, s,d, x) be a σ-finite measure space with a σ-field s countably generated. We call a linear map T uniformly contractive if which maps measurable functions on M to measurable functions and
  ‖Tf‖p≤‖f‖p,for p,1≤p≤+∞.
  If a linear map T which maps measurable functions on M to measurable functions has positivity property, namely,Tf≧0 forf≧0, we call it a submarkovian operator. In this article we prove.
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  On the Representation oof Large Odd integer as a Sum of Three Almost Equal Primes^*

  Zhan Tao

  Acta Mathematica Sinica. 1991, 7(3): 259-272. https://doi.org/10.1007/BF02583003

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  The well-known Goldback-Vinogradov theorem states that every large odd integer is a sum of three primes. In the present paper it is further proved that every large odd integer N can be represented as
  
  wher c>0 is an absolute constant, and p_i(1≦i≦3) are primes. The result is obtained by means of Hardy-Littlewood circle method, Heath-Brown's identity and power moments of Dirichlet L-functions over short intervals.
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  Multiple Criticsal Points of Periodic Functional and Some Applications

  Li Shujie

  Acta Mathematica Sinica. 1991, 7(3): 273-278. https://doi.org/10.1007/BF02583004

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  In this paper we will study some existence theorems of critical points for periodic functional. As applications, we consider the multiple solutions of conservative pendulum-like equation with periodic boundary conditions and forcing term of mean value zero.
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  On the Least Prime in an Arithmetic Progression

  Wang Wei

  Acta Mathematica Sinica. 1991, 7(3): 279-288. https://doi.org/10.1007/BF02583005

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  Let q, k be positive integers, (q, k)=1, and P(k, q) be the smallest prime number satisfying p≡k (mod q) In this paper, we prove that forc q sufficiently large, one has P(k,q)<<q⁸.