1992年, 第5卷, 第3期　刊出日期：1992-08-15

  

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  NONTRIVIAL PERIODIC SOLUTIONS OF A SEMILINEAR WAVE EQUATION WITHOUT ASSUMPTION OF MONOTONICITY

  LI Shujie

  Journal of Systems Science and Complexity. 1992, 5(3): 193-204.

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  In references, some authors have succeeded in obtaining existence results of the superlinear wave equation without the assumption of monotonicity for the nonlinear term. In this paper, we present existence results for this problem byusing local linking idea and improve some previous results.
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  ON THE EQUIVARIANT HARMONIC MAPS BETWEEN SURFACES IN R~3

  YANG Xuefeng

  Journal of Systems Science and Complexity. 1992, 5(3): 205-212.

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  In this paper,we study the existence and regularity of G-equivariant harmonic maps between surfaces with a finite group G\subset O(3)action. We discoverthat the energy level of compactness in the G-equivariant perturbed problems is higher than that in general case. This implies the multiplicity of harmonic mapsfor some examples.
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  ADAPTIVE STOCHASTIC APPROXIMATION FOR MEASUREMENT WITH CORRELATED NOISE

  GAO Aijun;CHEN Hanfu(H.F.Chen)

  Journal of Systems Science and Complexity. 1992, 5(3): 213-226.

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  A multidimensional adaptive stochastic approximation procedure truncated at randomly varying bounds is studied for the regression function observed with correlated noise. The algorithm is proved to be asymptotically efficient. The result of the paper generalizes the one shown in[1] for the observation noise being i.i.d. random vectors and under other restrictive conditions imposed on the regression function.
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  ON COMPLETE CONVERGENCE OF NONPARAMETRIC REGRESSION M-QUANTILES

  CAI Zongwu

  Journal of Systems Science and Complexity. 1992, 5(3): 227-232.

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  Consider the nonparametric regression model Y_i=m(x_i)+ε_i,i=1,…,n,where m(.) is an unknown function, and the design points x_i are knownand nonrandom. The robust nonparametric estimators were introduced by Hardleand Gasser in 1984. These estimators can be viewed as regression M-quantiles. We then establish complete convergence for such quantiles under only the finite moment condition.
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  BASIC PROPERTIES OF K-SPACES

  LI Ronglu;CHARLES Swartz;CHO Minhyung

  Journal of Systems Science and Complexity. 1992, 5(3): 233-238.

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  Every complete metric linear space is a K-space. A non-complete normed space can be a K-space. And non-metrizable K-spaces exist. A series of basic results on complete metric linear spaces Can be generalized to K-spaces. Forexample, every bounded linear operator from a K-space into a locally convex spaceis sequentially continuous; every bounded semi-norm on a K-space is sequentially continuous and, therefore, piecewise separable; a. C-sequeatial K-space is both barrelled and bornological; the family of sequentially continuous linear functionals ona K-space is weak sequentially complete.
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  OPTIMAL RATE OF CONVERGENCE FOR ESTIMATORS IN THE NONPARAMETRIC MEDIAN MODEL

  ZHENG Zhongguo

  Journal of Systems Science and Complexity. 1992, 5(3): 239-250.

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  Let (Y,X) be a random vector with its value in R~1×R~d,d\le 1. Let \Theta be the collection of real valued functions θ(x) on R~d which is p times differ-entiable at x=0 and p-1 times differentiable on an open neighborhood U of the origin of R~d. The conditional distribution of Y is assumed to be of the form of f(y|x,θ(x))dy where θ(x)∈\Theta is called the parameter of the family. (Y,X) is called a nonparametric median model if furthermore the conditional median of Y given X=x is θ(x). In this paper, the optimal rate of convergence for estimators of T(θ)=θ(0)is discussed. Under certain conditions, it is proved that for the nonparametric median model the optimal rate of convergence is r=p/(2p+d). A sequence of estimators, which is asymptotically normal with the optimal rate of covergence, is constructed.
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  EXISTENCE OF CONSISTENT ESTIMATES OF LINEAR REGRESSION COEFFICIENTS WHEN THE ERROR VARIANCES ARE UNEQUAL

  JIN Mingzhong;CHEN Xiru

  Journal of Systems Science and Complexity. 1992, 5(3): 251-259.

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  Consider the linear regression model Y_i=x_i′β+σ_ie_i,i=1,…,n,…, where E(e_i)=0, E(e_ie_j)=δ_(ij), 0<σ_1~2\le σ_2~2\le … and {x_i} satisfies A_1\le sum from i=1 to n x_ix_i′/n\le A_2, for n large, A_1>0,A_2>0. This paper shows that (i) if σ_i~2,i=1,2…, are known, then the necessary and sufficient condition for the consistency of the best unbiased linear estimate of β is sum from i=1 to ∞ σ_i~(-2)=∞;(ii) if {σ_i~2} is only known to belong to the set {(h_1,h_2,…)∶ 0
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  TRANSFORMATION THEOREMS AMONG CAYLEY-KLEIN GEOMETRIES

  GAO Xiaoshan

  Journal of Systems Science and Complexity. 1992, 5(3): 260-273.

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  A theorem of transformation between two geometries is proved on the basis of Wu's method of mechanical theorem proving. By using this theorem for Cayley-Klein geometries of dimension two, we prove that the nine Cayley-Klein geometries can be divided into three groups within which the geometries are mutually equivalent in the sense that certain geometry statements are correct inone geometry if and only if they are correct in the other geometries of the same group. This means that for each group we only need to choose a model geometry to study and the theorems of other geometries in the same group can be obtained from the model geometry automatically. The three model geometries chosen forthe nine Cayley-Klein geometries are: Euclidean geometry, Riemann geometry, and Galilean geometry.
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  THE TAIL ESTIMATION FOR THE DISTRIBUTION OF THE SUPREMA OF A P-BRIDGE

  ZHANG Jian

  Journal of Systems Science and Complexity. 1992, 5(3): 274-286.

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  Let X_1,X_2,… be a sequence of independent random vectors which take values in IR~2 and X_i～P. By the projection pursuit (PP) method, we canconstruct PP type Kolmogrov-Smirnov statistics: K_n=\sup_{\|a\|=1,z\in R}\sup_{|t<+\infty|}1/\sqre{n} | \sum_{i=1}^{n} (I_[a~rX_i\leq t]-P(a~rX_j\leq t))|, where I_[·] is the indicator function of the set [·] and P is a probability distribution. The limit distribution of K_n is the distribution for the suprema of a P-bridgewhose index is the set of all half spaces in IR~2. Let{W_p([0,\[t_{\tilde}\]])∶\[t_{\tilde}\]∈[0,1]~2} be a two parameter Brownish bridge. The main result of this paper is P(?)for all λ, provided P is the uniform distribution on [0,k_1]×[0,k_2], 0
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  DEFECT CORRECTION FOR FINITE ELEMENT GRADIENT

  LIN Qun;ZHOU Aihui

  Journal of Systems Science and Complexity. 1992, 5(3): 287-288.

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  A defect correction scheme of fourth order accuracy for linear finiteelement gradient is proposed in this note.This extends the Rannacher's scheme forfinite element solution to gradient and can compete with superconvergence fromquadratic elements.