15 December 1988, Volume 4 Issue 4

    

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  On Chern Characters^* of Algebraic Hypersurface with Arbitrary Singularities

  Shi He

  Acta Mathematica Sinica. 1988, 4(4): 289-300. https://doi.org/10.1007/BF02560632

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  In this paper,we discuss the Chern characters of hypersurfaces with arbitrary singularities.When the hypersurface is smooth,the Chern characters are just the usual Chern numbers.For any given dimension,we prove that the Chern characters satisfy a kind of inequalities.And we discover that the Chern characters satisfy some kind of equalities when the dimension is greater than 3.Therefore we obtain some more inequalities which are satisfied by the Chern characters of hypersurfaces with arbitrary singularities.
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  On the Weak Compactness in Banach Lattices

  Chen Zili

  Acta Mathematica Sinica. 1988, 4(4): 301-308. https://doi.org/10.1007/BF02560633

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  The Grothendieck's criterion of weak compactness in space C(S) has been extended by C.P.Niculescu for weakly sequentially complete Banach lattices.This paper first provides some sufficient and necessary conditions for weakly sequentially complete Banach lattices,with the result that the criterion C.P.Niculescu obtained of the weak compactness has actually characterized the weakly sequentially Banach lattices.At the same time we extend C.P.Niculescu's important results.

  Then we solve negatively Problem 1.11 which was posed by C.P.Niculescu,and a sufficient condition for this is given which generalizes Pelczyński's result.

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  A Case of the Jacobian Conjecture^*

  Xu Xiaolong

  Acta Mathematica Sinica. 1988, 4(4): 309-315. https://doi.org/10.1007/BF02560634

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  H.Bass,E.H.Connell and D.Wright^[1] settled a case for the Jacobian Conjecture.In the present paper we will give another proof of the case.
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  Boolean Planarity Characterization of Graphs^*

  Liu Yanpei

  Acta Mathematica Sinica. 1988, 4(4): 316-329. https://doi.org/10.1007/BF02560635

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  Although many criteria for testing the planarity of a graph have been found since the beginning of the thirties,this paper presents a new criterion described by Boolean technique which is proved in an independent way without any use of the criteria obtained before.
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  The Topological Markov Chain^*

  Zhou Zuoling

  Acta Mathematica Sinica. 1988, 4(4): 330-337. https://doi.org/10.1007/BF02560636

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  The topological Markov chain or the subshift of finite type is a restriction of the shift on an invariant subset determined by a 0,1-matrix,which has some important applications in the theory of dynamical systems.

  In this paper,the topological Markov chain has been discussed.First,we introduce a structure of the directed gragh on a 0,1-matrix,and then by using it as a tool,we give some equivalent conditions with respect to the relationship among topological entropy,chaos,the nonwandering set,the set of periodic points and the 0,1-matrix involved.

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  Unique Ergodicity of Flows on the Torus and the Rotation Number

  Han Maoan

  Acta Mathematica Sinica. 1988, 4(4): 338-342. https://doi.org/10.1007/BF02560637

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  We prove in this paper that any flow on the 2-torus with no singular points and periodic orbits which is generated by a vector field V=(P,Q) satisfying V∈C¹ or V∈C⁰ and P≠0 is uniquely ergodic.Then we give an expression of the rotation number by using an invariant measure of a flow.
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  Laws of Iterated Logarithm for State Variables with Applications^*

  Chen Hanfu, Guo Lei

  Acta Mathematica Sinica. 1988, 4(4): 343-355. https://doi.org/10.1007/BF02560638

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  Let x_k be the state variable (solution) of a stochastic difference equation.This paper gives the laws of iterated logarithm for {x_k} and {x_kx_k^τ},which being strongly correlated,are neither stationary nor ergodic.The results obtained are then applied to Kalman filter and LQG control problem.
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  Fixed Point Theorems for Convex Sets^*

  Jiang Jiahe

  Acta Mathematica Sinica. 1988, 4(4): 356-363. https://doi.org/10.1007/BF02560639

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  A conjecture of the author is verified by setting up a fundamental fixed point theorem so that some earlier results are unified and strengthened.
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  A Note on Certain Fields of Finite Transcendence Type^*

  Zhu Yaochen

  Acta Mathematica Sinica. 1988, 4(4): 364-371. https://doi.org/10.1007/BF02560640

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  Let be a finitely generated extension field of Q,andα_i,β_j(1≤i≤m,1≤j≤n) be some complex numbers.Let (k=1,2,3) be fields obtained by adjoining to the numbers {α_i,β_j exp(α_iβ_j)},{α_i,exp(α_iβ_j)},and {exp(α_iβ_j)},respectively.In the present note the relation between the transcendental degree of over and the transcendence type of over Q is given.
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  On the Extended Genus^*

  Peter Hilton

  Acta Mathematica Sinica. 1988, 4(4): 372-382. https://doi.org/10.1007/BF02560641

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