15 March 1992, Volume 8 Issue 1

    

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  On Harmonic Maps between Generalized Flag Manifolds¹

  Hou Zixin

  Acta Mathematica Sinica. 1992, 8(1): 1-16. https://doi.org/10.1007/BF02595015

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  The theory of harmonic maps has been developed since the 1960's (see [2]). In recent years, some authors discussed the harmonicity of "homogeneous" maps between Riemannian homogeneous spaces using the theory of Lie groups. Let G and G' be compact Lie groups, H and H' their closed subgroups respectively. Assume that a homomorphism θ: G→G' maps H into H'; then there exists an induced map f_θ: G/H→G'/H'. M.A. Guest gave a necessary and sufficient condition for such a map to be harmonic, when G/H and G'/H' are generalized flag manifolds, H=T is a maximal torus and G' is a unitary group; and he gave some interesting examples (see [3]). We generalize his results to the case of general generalized flag manifolds G/H, i.e. H is a centralizer of a torus, and give some new examples of harmonic maps.
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  Cohomology of Graded Lie Algehras of Cartan Type of Characteristic 0¹

  Chiu Sen

  Acta Mathematica Sinica. 1992, 8(1): 17-25. https://doi.org/10.1007/BF02595016

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  In [6], D. Fux determined the structures of cohomology of formal vector fields Lie algebras with coefficients in trivial modules or dual adjoint modules. We determine the structure of H¹ (L, V), where L=W (n), S(n) or H(n) is an infinite dimensional Lie algebra of characteristic 0 and V is a graded L-module V₀ (mixed product) or a graded irreducible L-module.
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  Hausdorff Dimension of the sample Path of the Westwater Process¹

  Zhou Xian-Yin

  Acta Mathematica Sinica. 1992, 8(1): 26-45. https://doi.org/10.1007/BF02595017

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  In [6], D. Fux determined the structures of cohomology of formal vector fields Lie algebras with coefficients in trivial modules or dual adjoint modules. We determine the structure of H¹ (L, V), where L=W (n), S(n) or H(n) is an infinite dimensional Lie algebra of characteristic 0 and V is a graded L-module V₀ (mixed product) or a graded irreducible L-module.
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  The Best Uniform Convergence Rate of Linear Empirical Bayer Estimators

  Tao Bo

  Acta Mathematica Sinica. 1992, 8(1): 46-59. https://doi.org/10.1007/BF02595018

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  In this paper the thought on the uniform convergence of an empirical Bayes estimator or linear empirical Bayes (l.e.B.) estimator is advanced. Under two different models the l.e.B. estimators of the parameter are constructed respectively. It is proved that the convergence rates and uniform convergence rates of these l.e.B. estimators are all one with respect to the corresponding prior families. It is shown that the uniform convergence rate one is the best, under mild assumptions imposed on the conditional density of the sample.
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  The Construction of Isolated Reducible SU(2)-Connections over S²×S²¹

  Wang Hongyu

  Acta Mathematica Sinica. 1992, 8(1): 60-77. https://doi.org/10.1007/BF02595019

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  In this paper, we construct a double indexing family of reducible SU(2)-connctions over S²×S² in a geometric way. By separation of variables, we compute the spectrum of the Hessian of the Yang-Mills functional at these connections. Using the implicit function theorem, we prove that these connections are isolated non-minimal solutions to the Yang-Mills equations on S²×S².
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  Involutive Solutions for Toda and Langmuir Lattices Thrvugh Nanlinearization of Lax Pairs

  Geng Xianguo

  Acta Mathematica Sinica. 1992, 8(1): 78-90. https://doi.org/10.1007/BF02595020

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  Under the constraint determined by a relation (a_n, b_n)^T={f(Φ)}_n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution variety N of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented. f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.
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  Boundedness for Solutions of Nonlinear Periodic Differential Equations Via Moser’s Twist Theorem

  Liu Bin

  Acta Mathematica Sinica. 1992, 8(1): 91-98. https://doi.org/10.1007/BF02595021

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  We consider the nonlinear periodic differential equation
  
  where a(t) and p(t) are continuous and 1-periodic, β is a positive constant. The purpose of this paper is to prove that all solutions of the above-mentioned equation are bounded for t∈R and there are infinitely many quasi-periodic solutions and an infinity of periodic solutions of minimal period m, for each positive integer m.
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  New Relationships Betweew Steenrad operations for Certain Spaces¹

  John Klippenstein

  Acta Mathematica Sinica. 1992, 8(1): 99-111. https://doi.org/10.1007/BF02595022

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  In this paper, we show that under a certain technical condition, if a space has no 2-torsion, then either Sq²ⁿ x≠0 or there exists some y with Sq²ⁿy=Sq²ⁿ⁺¹ x, if for some n≥3 Sq²ⁿ⁺¹ x≠0. The proof uses relations between Steenrod operations and operations in connective real K-Theory.