1991年, 第4卷, 第2期　刊出日期：1991-05-15

  

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  HECKE OPERATOR AND PELLIAN EQUATION CONJECTURE Ⅲ

  Lu Hongwen

  Journal of Systems Science and Complexity. 1991, 4(2): 97-103.

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  In this note, we give a relationship between the Mordell conjecture on the Pellian equation and the action of the Hecke operator on the Hirzebruch sum.
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  A CRYPTOGRAPHIC STUDY ON S—BOXES OF DES TYPE Ⅰ AN INTEGRATED ANALYSIS OF THE DESIGN CRITERIA FOR S-BOXES

  Yang Junhui;Dai Zongduo;Zeng Kencheng

  Journal of Systems Science and Complexity. 1991, 4(2): 104-110.

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  An integrated analysis of the design criteria for the S-boxes used in the DES is made. Among others, the exclusion principle between the degree of nonlinearity and the degree of I/O—correlation immunity for partial variables is established.
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  MODELLING RANDOM CONVEX SETS

  Lee Tzao-Lin;Chen Keh-Wei;Rohan Attele

  Journal of Systems Science and Complexity. 1991, 4(2): 111-127.

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  A model is developed for convex-set valued data, which are the Minkowski sum of aconvex parameter set and a convex noise set. This model is generalized to include location and scale parameters. A further generalization results in a model for convex sets which is analogous to aone-way analysis of variance models. In each case, estimators of the parameters are given which, under certain conditions, are shown to be consistent. Data generated from the model and the estimators are studied.An outlier detection method is examined.
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  THE SECOND-ORDER CONDITIONS OF NONDOMINATED SOLUTIONS FOR C~(1,1) GENERALIZED MULTIOBJECTIVE MATHEMATICAL PROGRAMMING

  Liu Liping

  Journal of Systems Science and Complexity. 1991, 4(2): 128-138.

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  In this paper, the generalized Hessian matrix and the generalized second-order directional derivative for C~(1,1) vector functions are defined.The extension of the vector second-order Taylor expansion is derived. The second-order necessary and sufficient conditions for the local nondominated solutions associated with the given convex cone and polyhedral convex cone of the generalized multiobjective mathematical programming problem with C~(1,1) constrained functions are discussed.
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  ADMISSIBILITY OF LINEAR ESTIMATORS OF REGRESSION COEFFICIENTS IN THE CLASS OF ALL ESTIMATORS UNDER THE MATRIX LOSS FUNCTION

  Chen Jianbao;Zhan Jinlong

  Journal of Systems Science and Complexity. 1991, 4(2): 139-147.

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  For the general fixed effects linear model: Y=Xτ+ε,ε～N(0,V)(or N(0,σ~2V)), V≥0, we obtain the necessary and sufficient condition for LY+α to be admissible for Sτ in the class of all estimators under matrix loss function (d-Sτ)(d-Sτ)′; for the general random effects linear model: Y=Xβ+ε,(?)where (?)=XV_(11)X′+XV_(12)+V_(21)X′+V_(22)≥0, we also obtain the necessary and sufficient condition for LY+α to be admissible for Sα+Qβ in the class of all estimators under matrix loss function (d-Sα-Qβ)(d-Sα-Qβ)′.
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  PERIODIC SOLUTIONS AND PERIOD-SIMILARITY OF A GROWTH MODEL WITH AGE-STRUCTURED POPULATION

  Fan Yinshui;Chen Lansun

  Journal of Systems Science and Complexity. 1991, 4(2): 148-158.

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  The present paper deals with the periodic behaviours of an age-structured population model. The period-similarity is proposed, which reveals a certain similar structure between those population models with distinct age structure. In addition, other results show that the fluctuations of an age-structured population are closely related with the age structure.
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  BOOTSTRAPPING PP L-STATISTICS FOR TESTS

  Shi Peide;Li Guoying

  Journal of Systems Science and Complexity. 1991, 4(2): 158-172.

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  Some bootstrap approximations of the asymptotic distributions of PP L-statistics whichwere proposed by the authors in[7]are discusse. Bootstrap statistics and their asymptotic propertiesare established. Two examples and the numerical results are given, by which one can actually implement the proposed tests under the condition of normal population.
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  STRONG CONSISTENCY OF THE PERIODIC SMOOTHING SPLINES

  Liu Kejian

  Journal of Systems Science and Complexity. 1991, 4(2): 173-185.

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  Consider the regression model Y_i=u(x_i)+ε_i,i=1,2,…,n, where u(x)∈W_(2,per)~2[0,1],x_i=(i-1)/n,{ε_i}_1^nare i.i.d, random variables. We use the periodic smoothing spline u_(λp)(x)to estimate u(x). Under certain conditions, strong consistency results of u_(λp)(x) are obtained, i.e., forall 0
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  OBSERVING THE HEAT EQUATION ON A TORUS ALONG A DENSE GEODESIC

  Lance D.Drager;Robert L.Foote;Clyde F.Martin

  Journal of Systems Science and Complexity. 1991, 4(2): 186-192.

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  Several authors have considered observability problems for the heat equation and relatedpartial differential equations.A basic problem is to determine what kinds of sampling provide sufficient information to uniquely determine the initial heat distribution. We address the case where the temperature is measured while travelling along a curve. We consider the special case where the space is a flat torus (of arbitrary dimension) and the curve is a geodesic. It is shown that, in this case, the observed temperature is sufficient information to uniquely determine the initial heat distribution if and only if the geodesic is dense in the torus. In the case of a torus, Fourier analysis techniques can be used to write down the solution of the heat equation. This allows us to derive an explicit representation of the observed temperature interms of the initial distribution. We use this representation and some ideas from the theory of almost periodic functions to show that the Fourier coefficients of the initial distribution can be recovered from the observation.