15 January 1998, Volume 14 Issue 1

    

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  The Eigen Functions of the Complex Projective Space

  Lu Qikeng

  Acta Mathematica Sinica. 1998, 14(1): 1-8. https://doi.org/10.1007/BF02563877

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  In the complex n-dimensional projective space CPⁿ, let λ_p(=4 p(p+n)) be the eigen value of the Laplace-Beltrami operator and H_p be the space of all eigen functions of eigen value λ_p. The reproducing kernel h_p(z, w) of H_p is constructed explicitly in this paper, and a system of complete orthogohal functions of H_p is constructed from h_p(z,w)(p=1,2, …).
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  Estimate of Exponential Convergence Ratein Total Variation by Spectral Gap

  Chen Mufa

  Acta Mathematica Sinica. 1998, 14(1): 9-16. https://doi.org/10.1007/BF02563878

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  This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap.
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  Picard Values and Normal Families of Meromorphic Functions with Multiple Zeros

  Wang Yuefei, Fang Mingliang

  Acta Mathematica Sinica. 1998, 14(1): 17-26. https://doi.org/10.1007/BF02563879

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  In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.
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  On the Cauchy Problem for Certain System of Semilinear Parabolic Equations

  Jian Huaiyu, Chen Donggui

  Acta Mathematica Sinica. 1998, 14(1): 27-34. https://doi.org/10.1007/BF02563880

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  In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations:
  
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  Small Random Perturbations of One-Dimensional Diffusion Processes

  Xi Fubao

  Acta Mathematica Sinica. 1998, 14(1): 35-40. https://doi.org/10.1007/BF02563881

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  In the present paper we consider the small random perturbations of one-dimensional diffusion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero.
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  On the Estimation of Solutions for Some Linear Systems of Differential Equations

  L. R. Awad, E. M. El-Kholy, Sh. El-Bendary

  Acta Mathematica Sinica. 1998, 14(1): 41-46. https://doi.org/10.1007/BF02563882

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  In this paper we estimate the solutions of homogeneous linear system of differential equations with unbounded coefficients on the real line R. We also give a necessary and sufficient condition in order that the linear differential operator with unbounded coefficients has a bounded inverse in the scalar case.
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  Convex Functions, Subdifferentiability and Renormings

  Cheng Lixin, Wu Congxin, Xue Xiaoping, Yao Xiaobo

  Acta Mathematica Sinica. 1998, 14(1): 47-56. https://doi.org/10.1007/BF02563883

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  This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.
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  Small Zeros of Additive Formsin Several Variables

  J. S. Hwang

  Acta Mathematica Sinica. 1998, 14(1): 57-66. https://doi.org/10.1007/BF02563884

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  Let f(X) be an additive form defined by
  
  where a_#em/em#≠0 is integer, #em/em#=1,2…, s. In 1979, Schmidt proved that if ε>0 then there is a large constant C(k,ε) such that for s>C(k,ε) the equation f(X)=0 has a nontrivial, integer solution in σ₁, σ₂, …, σ_s, x₁, x₂, …, x_s satisfying
  
  Schmidt did not estimate this constant C(k,∈) since it would be extremely large. In this paper, we prove the following result.
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  Strongly Singular Convolution Operators on the Weighted Herz-Type Hardy Spaces

  Li Xiaochun, Lu Shanzhen

  Acta Mathematica Sinica. 1998, 14(1): 67-76. https://doi.org/10.1007/BF02563885

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  Let 0<p≤1<q<0, and w₁, w₂∈A₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spaces HK_q^{α, p}(w₁; w₂) to the homogeneous weighted Herz spaces K_q^{α, p}(w₁; w₂), provided α=n(1-1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spaces HK_q^{α, p} (w₁; w₂) is also investigated.
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  A Series of Convex Functionson a Banach Space

  Zheng Xiyin

  Acta Mathematica Sinica. 1998, 14(1): 77-84. https://doi.org/10.1007/BF02563886

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  A series of convex functions on a Banach space is studied. The generalizations of the Moreau-Rockafellar theorem are given. As its application, the partial generalization of the Kuhn-Tucker theorem is obtained.
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  Abelian real W^*-algebras

  Li Minli, Li Bingren

  Acta Mathematica Sinica. 1998, 14(1): 85-90. https://doi.org/10.1007/BF02563887

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  In this paper, we point out that most results on abelian (complex) W^*-algebras hold in the real case. Of course, there are differences in homeomorphisms of period 2. Moreover, an abelian real Von Neumann algebra not containing any minimal projection on a separable real Hilbert space is * isomorphic to L_τ^∞ ([0, 1]) (all real functions in L^∞([0, 1])), or L^∞([0, 1]) (as a real W^*-algebra), or L_τ^∞ ([0, 1])⊕L^∞ ([0, 1]) (as a real W^*-algebra), and it is different from the complex case.
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  The Quaternion Matrix Equation ΣA^#em/em#XB_#em/em#=E

  Huang Liping

  Acta Mathematica Sinica. 1998, 14(1): 91-98. https://doi.org/10.1007/BF02563888

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  Let H_F be the generalized quaternion division algebra over a field F with char F≠2. In this paper, the adjoint matrix of any n×n matrix over H_F[λ] is defined and its properties is discussed. By using the adjoint matrix and the method of representation matrix, this paper obtains several necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix equation Σ_#em/em#=0^kA^#em/em#XB_#em/em#=E over H_F, and gives some explicit formulas of solutions.
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  Fixed Point Separating Congruences on Double MS-Algebras

  Fang Jie

  Acta Mathematica Sinica. 1998, 14(1): 99-104. https://doi.org/10.1007/BF02563889

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  Here we show that there exists a greatest fixed point separating congruence denoted by Ψ on a double MS-algebra (L;^0,+). Some properties of Ψ under certain conditions on fixed point complete and fixed point distributive have also been described.
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  Derivations and Isomorphisms of Certain Reflexive Operator Algebras

  Hou Chengjun, Han Deguang

  Acta Mathematica Sinica. 1998, 14(1): 105-112. https://doi.org/10.1007/BF02563890

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  In this paper, we will prove that every derivation of completely distributive subspace lattice (CDS) algebras on Banach space is automatically continuous. This is new even in the Hilbert space case. As an application of this result, we obtain that every additive derivation of nest algebras on Banach spaces is inne. We will also prove that every isomorphism between nest algebras on Banach space is automatically continuous, and in addition, is spatial.
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  Sufficient and Necessary Conditions for Limit Laws on Lag Sums of I. I. D. R. V. S

  Zhang Lixin

  Acta Mathematica Sinica. 1998, 14(1): 113-124. https://doi.org/10.1007/BF02563891

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  Let {X, X_n; n≥1} be a sequence of i.i.d.r.v.s with EX=0 and EX²=σ²(0 < σ < ∞). we obtain some sufficient and necessary conditions for
  
  to hold, get the widest range of k's and answer a question of Hanson and Russo (1983).
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  Bifurcation of Nonlinear Problems Modeling Flows Through Porous Media

  Liu Xiyu

  Acta Mathematica Sinica. 1998, 14(1): 125-134. https://doi.org/10.1007/BF02563892

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  This paper deals with a class of nonlinear boundary value problems which appears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems.
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  Embeddings of Simple Directed Triple Systems

  Chu Wensong

  Acta Mathematica Sinica. 1998, 14(1): 135-138. https://doi.org/10.1007/BF02563893

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  In this paper, we completely solve the embedding problem of simple directed triple systems by proving that the necessary conditions for the embeddings of directed triple systems are also sufficient.
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  On the Iteration of Holomorphic Self-Maps of C^*

  Fang Liping

  Acta Mathematica Sinica. 1998, 14(1): 139-144. https://doi.org/10.1007/BF02563894

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  Let f be a holomorphic self-map of the punctured plane C^*=C\{0} with essentially singular points 0 and ∞. In this note, we discuss the sets I₀(f)={z∈C^*: fⁿ(z)→0, n→∞} and I_∞(f)={z∈C^*: fⁿ(z)→0, n→∞}. We try to find the relation between I₀(f), I_∞(t) and J(f). It is proved that both the boundary of I₀(f) and the boundary of I_∞(f) equal to J(f), I₀(f)∩J(f)≠θ and I_∞(f)∩J(f)≠θ. As a consequence of these results, we find both I₀(f) and I_∞(f) are not doubly-bounded.