15 March 1989, Volume 5 Issue 1

    

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  A Direct Attempt to Goldbach Problem

  Hua Loo-keng

  Acta Mathematica Sinica. 1989, 5(1): 1-8. https://doi.org/10.1007/BF02107617

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  The so-called "principal term" of Goldbach problem, i.e., the principal order of the number of representations of an even number as a sum of two primes, is evaluated here in a way quite different from the classical circle method. It is amazing that the author did this with the knowledge just at the PNT (Prime Number Theorem) level, even without using high-level theorems in analytic number theory as usual, e.g., Siegel-Walfisz Theorem.
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  The Dirichlet Problem of a Discontinuous Markov Process

  Liao Ming

  Acta Mathematica Sinica. 1989, 5(1): 9-15. https://doi.org/10.1007/BF02107618

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  Given a Markov process satisfying certain general type conditions, whose paths are not assumed to be continuous. Let D be an open subset of the state space E. Any bounded function defined on the complement of D extends to be a function on E such that it is harmonic in D and satisfies the Dirichlet boundary condition at any regular boundary point of D. The relation between harmonic functions and the characteristic operator of the given process is discussed.
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  Surface Subshifts and Subshift Data^*

  Zhu Deming

  Acta Mathematica Sinica. 1989, 5(1): 16-26. https://doi.org/10.1007/BF02107619

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  This paper is concerned with Smale diffeomorphisms on compact oriented surfaces. We determine with which genus a closed oriented surface can carry a Smale diffeomorphism associated with a subshift of finite type corresponding to a given reciprocal polynomial with constant term one, and determine what a combination of subshifts and period imformations of sinks and sources can be realized on a given oriented surface by a Smale diffeomorphism in a definite isotopic class. Necessary and sufficient conditions are given.
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  PL Embeddings of PL Manifolds into Some Euclidean Spaces

  Guo Jingmei

  Acta Mathematica Sinica. 1989, 5(1): 27-36. https://doi.org/10.1007/BF02107620

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  The main result in this paper is the following:
  Theorem. Assume that W is a k-connected compact PL n-manifold with boundary, BdW is (k-1)-connected, k≥1(BdW is 1-connected for k=1), 0≤ h≤2k, 2 n-h>5 and there exists a normal block (n-h-1)- bundle v over W, then(1) There is a neat PL embedding W→ D{D^2n-h} which normal block bundle is isomorphic to v⊕ε.(2) There is a PL embedding W→S^2n-h which normal block bundle is isomorphic to v.
  Where ε denotes the trivial block 1-boundle, D^2n-h={x=(x₁, x₂,…, x_2n-h)∈ R^2n-h||x_i|≤1} and S^2n-h-1=BdD^2n-h.
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  Bombieri’s Theorem in Short Intervals

  Zhan Tao

  Acta Mathematica Sinica. 1989, 5(1): 37-47. https://doi.org/10.1007/BF02107621

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  The well-known Bombieri-A. I. Vinogradov theorem states that
  (1)
  where A is an arbitrary positive constant, B=B(A)>0, and as usual,
  
  Λ being the Von Mangoldt's function.
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  Generalized Monotone Approximation in L_p Space^*

  Yu Xiangming, Ma Yongpei

  Acta Mathematica Sinica. 1989, 5(1): 48-56. https://doi.org/10.1007/BF02107622

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  Let f(x)∈ L_p[0,1], 1≤ p≤∞. We shall say that function f(x)∈Δ^k (integer k-1) if for any h∈[0, 1/k] and x∈[0,1-kh], we have Δ_h^k f(x)≥0. Denote by ∏ _n the space of algebraic polynomials of degree not exceeding n and define
  
  We prove that for any positive integer k, if f(x)∈Δ ^k ∩L_p[0, 1], 1≤p≤∞, then we have
  
  where C is a constant only depending on k.
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  New Phenomena on Bounded Nonhomogeneous Domains

  Yin Weiping

  Acta Mathematica Sinica. 1989, 5(1): 57-63. https://doi.org/10.1007/BF02107623

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  In this paper we get the invariant functions and the invariant harmonic functions under Aut (D) for certain Reinhardt domains D. By using the invariant functions, we get much more invariant Kahler metrics. And the Ricci curvatures, scalar curvatures and holomorpic sectional curvatures are also obtained, which are very different for the bounded homogeneous domains.
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  Boolean Approach to Planar Embeddings of a Graph¹

  Liu Yanpei

  Acta Mathematica Sinica. 1989, 5(1): 64-79. https://doi.org/10.1007/BF02107624

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  The purpose of this paper which is a sequel of "Boolean planarity characterization of graphs" ^[9] is to show the following results.(1)Both of the problems of testing the planarity of graphs and embedding a planar graph into the plane are equivalent to finding a spanning tree in another graph whose order and size are bounded by a linear function of the order and the size of the original graph, respectively.(2)The number of topologically non-equivalent planar embeddings of a Hamiltonian planar graph G is
  τ(G)=2^c(H)-1,
  where c (H) is the number of the components of the graph H which is related to G.
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  On the Divisibility of the Class Number of the Imaginary Quadratic Field Q(a²-4kⁿ)

  Le Maohua

  Acta Mathematica Sinica. 1989, 5(1): 80-86. https://doi.org/10.1007/BF02107625

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  Let D' be a positive integer which is square free, and let h' (-D') denote the class number of the imaginary quadratic field 2e (√-D'). In this paper we prove that if D'> exp exp exp 1000, D'=4k″-a², where a, k and n are integers with a>0, k>1, n>1, then
  
  except possibly for n=3, D'=3(2 a₁+1)²±4, k=(2 a₁+1)²±1, a=2 a₁+1, where a₁ is an integer.
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  Random Weighting Method for T Statistic

  Zheng Zhongguo

  Acta Mathematica Sinica. 1989, 5(1): 87-94. https://doi.org/10.1007/BF02107626

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  In this paper we deal with the measure model with independently distributed errors. For estimating the error distribution of the sample mean, random weighting method is utilized. It is proved that the accuracy of the approximation is of the order of o (1/√n), which is the ideal rate.
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  Fixed-point-free Automorphisms of Lie Algebras

  Zha Jianguo

  Acta Mathematica Sinica. 1989, 5(1): 95-96. https://doi.org/10.1007/BF02107627

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  In this paper we have proved that the Lie algebra L over a field F of characteristic 0 which admits a fixed-point-free automorphism of finite order is solvable, and L is nilpotent if L admits a fixed-point-free automorphism of prime order. The problem seems to be open if the field F has prime characteristic.