1993年, 第6卷, 第3期　刊出日期：1993-07-15

  

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  THE ALGEBRAIC CRITERIA FOR THE ASYMPTOTIC BEHAVIOR OF COOPERATIVE SYSTEMS WITH CONCAVE NONLINEARITIES

  JIANG Jifa

  Journal of Systems Science and Complexity. 1993, 6(3): 193-208.

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  For the discrete dynamical system on the nonnegative orthant generated bya cooperative and concave map T, we present an algebraic criterion of its asymptotic behavior. The global behavior of such a system is completely determined by the sign of all principal minors of the matrix I-DT(0). This criterion applies to the cooperative systemwhich is concave, time-dependent and periodic in t. We give the sufficient conditions that the zero solution of such a system is globally asymptotically stable and that it possesses anonzero periodic solution which attracts all initial conditions in the nonnegative orthant, except at the origin. The results of Smith under weaker conditions and some applications are included.
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  OPTIMAL REPLACEMENT FOR DISCRETE SHOCK MODELS

  CHENG Shixue

  Journal of Systems Science and Complexity. 1993, 6(3): 209-216.

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  A device is subject to damage. The damage occurs randomly in discrete time according to an increasing homogeneous Markov chain, where the incremental damage caused by a shock depends on the magnitude of the shock. The device has a threshold, and it fails once the damage level exceeds the threshold. Upon failure the device is replaced by a new and identical one and a cost is incurred. If the device is replaced before failure, a smaller cost is incurred. In this paper we study the problem of specifying a replace mentrule which minimizes the long-run (expected) average cost per unit time. Under mild conditions we show that the optimal replacement policy exists and it possesses the control limit property.
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  ON THE BOUNDEDNESS OF SOLUTIONS,EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF CUBIC DIFFERENTIAL SYSTEM

  WANG Chengwen

  Journal of Systems Science and Complexity. 1993, 6(3): 217-226.

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  In this paper, we discuss the boundedness of the solutions, the existence and the uniqueness of the limit cycle of the following cubic differential system: x'=y, y'=-x+δy+a_2xy+a_4x+a_5x~2y. (*) We obtain the following results: (1) System (*) is bounded if and only if (i) a_5<0, a_4=0; or (ii) a_5=0, a_4<0, δ≤0,-(-8a_4)~(1/2)
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  ON THE CONNECTIVITY OF CAYLEY COLOR GRAPHS

  HUANG Qiongxiang;LIU Xin

  Journal of Systems Science and Complexity. 1993, 6(3): 227-230.

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  In this paper we obtain the necessary and sufficient condition for the connec-tivity of the Cayley color graphs or the Cayley graphs to be equal to their minimum degree. The sharp lower bounds of connectivity of Cayley color graphs and the Cayley graphs arealso obtained. Our results generalize the previous results obtained in [1] to [3].
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  THE RANKINGS OF CONTROL SYSTEMS AND APPLICATIONS

  ZHENG Yufan;CAO Li

  Journal of Systems Science and Complexity. 1993, 6(3): 231-238.

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  The notion of ranking was initiated into control theory by Glad a few yearsago. In this paper we mainly discuss one kind of rankings and study its invariance properties. This kind of rankings is closely related to the structure algorithm. The relations between invariants of this kind of rankings and the invariants given by other studies, e.g.the invertibility indices of a system and so on, are examined. As applications of rankings, we deal with some well-known problems in "ranking technique". In this paper we mainlyintroduce a new method rather than solve new problems.
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  SOME THEOREMS OF THE NORDHAUS-GADDUM CLASS

  WANG Zhijian

  Journal of Systems Science and Complexity. 1993, 6(3): 239-244.

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  Let the chromatic number of G, the edge chromatic number of G and thetotal chromatic number of G be denoted by x(G), x_1(G) and x_2(G), respectively. For any simple graph G of order p and its complement G, the following inequalities of the Nordhaus-Gaddum class are obtained: (i)|2p~(1/2)|-ε_1≤x(G)+x_1(G)≤2p-2 and 0≤x(G)·x_1(G)≤(p-1)~2 for p≥2,(ii)|2p~(1/2)|+ε_1≤x(G)+x_2(G)≤2p-1 and 0≤x(G)·x_2(G)≤p(p-1) for p≥3,(iii)p≤x_1(G)+x_2(G)≤2p-1 and 0≤x_1(G)·x_2(G)≤p(p-1) for p≥3, where ε_1=0, if p~(1/2) is an odd integer, 1, otherwise, ε_2=1, if p~(1/2) is an even integer, 0, otherwise, and [x] denotes the ceiling of x. We also show that these bounds are sharp for every positive integer p.
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  PROPER EFFICIENT SOLUTIONS AND PROPER SADDLE POINTS

  LI Shizheng

  Journal of Systems Science and Complexity. 1993, 6(3): 245-251.

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  This paper offers a characterization method for the proper efficient solution ofthe multiobjective programming problem in terms of the proper saddle point and proves a necessary and sufficient condition for the equivalence between the proper efficient solutionand the associated proper saddle point by using the duality of a pair of problems.
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  NECESSARY AND SUFFICIENT CONDITIONS FOR G-STRUCTURES WITH LOGICAL RELATIONS TO FORM BOUNDED LATTICES IN RECONSTRUCTABILITY ANALYSIS

  SHU Guangfu

  Journal of Systems Science and Complexity. 1993, 6(3): 252-258.

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  After a brief review of the basic notions the paper defines the G~L-structures and their partial ordering. On this basis, the paper proves the necessary and sufficient conditions for G~L-structures to form bounded lattices.
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  GRADUAL CHANGE AND SUDDEN CHANGE

  ZHANG Weitao;FENG Dexing

  Journal of Systems Science and Complexity. 1993, 6(3): 259-268.

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  In this paper, we prove that, if 0
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  ON THE LEAST EIGENVALUE OF A GRAPH

  HONG Yuan

  Journal of Systems Science and Complexity. 1993, 6(3): 269-272.

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  Let G be a simple graph with n vertices and λ_n(G) be the least eigenvalue of G. In this paper, we show that, if G is connected but not complete, then λ_n(G)≤λ_n(K_(n-1)~1) and the equality holds if and only if G K_(n-1)~1, where K_(n-1)~1, is the graph obtained by the coalescence of a complete graph K_(n-1) of n-1 vertices with a path P_2 of length one of its vertices.
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  ON SIMULTANEOUS L~P-APPROXIMATION (0 < p ≤ +∞) OF QUASI-HERMITE INTERPOLATION AND ITS DERIVATIVE

  MIN Guohua

  Journal of Systems Science and Complexity. 1993, 6(3): 273-283.

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  In this paper, simultaneous uniform approximation and mean convergence of quasi-Hermite interpolation and its derivative based on the zeros of Jacobi polynomials are considered separately. The degrees of the corresponding approximations are respectively given also. Some known results are improved aud extended.
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  THE COMPARISON-NUMBER METHOD FOR FINDING MAJOR OPTIMAL SOLUTIONS AND MAJOR EFFICIENT SOLUTIONS OF THE DISCRETE MULTIOBJECTIVE PROGRAMMING

  HU Yuda;SUN Erjiang

  Journal of Systems Science and Complexity. 1993, 6(3): 284-288.

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  In this paper, we introduce the comparison number for one feasible solutioncompared with another. With the help of it the comparison-number method for finding the major optimal solutions aud major efficient solutions to discrete multiobjective programming is given.