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26 June 1981, Volume 1 Issue 2 Previous Issue    Next Issue

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ON THE RANDOM FIXED POINT THEOREMS IN ABSTRACT SPACE Chen Shaozhong, Liu Zuoshu Acta mathematica scientia,Series A. 1981, 1 (2):  133-144.  Abstract ( 52 )   RICH HTML PDF (605KB) ( 28 )   Save In the present paper, we studied the random fixed point theorems in the abstract metric space X, which is metricized by the elements of the partially ordered set G. In §3, we also studied some fundamental properties of the abstract topological measure space X,. In §4, we generalized the classical random fixed point theorems. Related Articles | Metrics

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FUZZY MAPPING Lou Shih-pe, Cheng Lichun Acta mathematica scientia,Series A. 1981, 1 (2):  145-155.  Abstract ( 64 )   RICH HTML PDF (479KB) ( 75 )   Save Fuzzy controllers can be regarded as a fuzzy mappinq from the set of linguistic variables describing the observed object to that of linguistic variables describing the controlled objects. Therefore, it is of great importance, both theoretically and practically, to the research of fuzzy mapping. The extension principle developed by Zadeh as a definition can be deduced from the definition of fuzzy mapping discussed in this paper. In the paper, the α level relation of fuzzy mapping is also defined, and three resolution theorems are proved on the basis of α level set and α level relation. Given a fuzzy mapping T:X→Y and a fuzzy set B in Y, we raise the problem:Under what Condition can we find such a fuzzy set A in X that T(A)=B? The two sufficient and necessary conditions given in the paper are the answer to the problem. Related Articles | Metrics

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ON THE PERIODIC BOUNDARY PROBLEMS FOR SEMILINEAR DEGENERATE EVOLUTION SYSTEMS OF HIGHER ORDER Zhou Yulin, Fu Hongyuan Acta mathematica scientia,Series A. 1981, 1 (2):  156-164.  Abstract ( 55 )   RICH HTML PDF (445KB) ( 20 )   Save It is the purpose of this paper to consider some semilinear evolution systems of partial differential equations, which are found in physics, chemical reactions and biology. The existence, uniqueness and regularity of solutions for periodic boundary problems in global are proved for following degenerate parabolic and hyperbolic systems of higher order ∂u/∂t=Σm-1M(-1)m+1Am(x)∂2mu/∂x2m+Στ=0RBτ(t)∂2r+1u/∂x2r+1+f(u), where the matrices A(t), m=1,…, M are nonnegative definite and the matrices B(t), r=1.., M are symmetric. One can show that the nonlinear Schrödinger equation or system, which may be treated as a real degenerate parabolic system, the Sine-Gordon equation, Klein-Gordon equation, some simultaneous equations of Schrödinger equation and wave equation etc. may be changed into the above type. Related Articles | Metrics

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ASYMPTOTIC SOLUTION OF HELMHOLTZ'S EQUATION NEAR A CAUSTIC Li Zhishen Acta mathematica scientia,Series A. 1981, 1 (2):  165-176.  Abstract ( 53 )   RICH HTML PDF (619KB) ( 55 )   Save In this paper we obtained the asymptotic solution of Helmholtz's ēquation(▽2+k2a(x))u=0 near a smooth convex caustic by means of the approximate method of short-wave and discussed the physical significance of the problems related to the asymptotic solution. Related Articles | Metrics

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CONSISTENCY OF LEAST SQUARES IDENTIFICATION Chen Hanfu Acta mathematica scientia,Series A. 1981, 1 (2):  177-183.  Abstract ( 58 )   RICH HTML PDF (334KB) ( 276 )   Save In this paper the least squares identification (LSI) of unknown coefficient matrices in the multi-dimensional dynamic system with uncorrelated noise is considered. Various conditions guaranteeing strong consistency of the LSI are put into comparison and the conditions proposed here and weaker than the previous one's are verified to be sufficient for the consistency of the LSI for which the asymptotic behaviors are also shown in the present paper. The main result is proved for the discrete-time system and its continuous-time analogue is also demonstrated. Related Articles | Metrics

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ON BIFURCATION OF NONLINEAR EQUATIONS Fang Dexing, Liu Jiaquan Acta mathematica scientia,Series A. 1981, 1 (2):  184-194.  Abstract ( 49 )   RICH HTML PDF (564KB) ( 21 )   Save In this paper we discuss in the Banach space the Banach space the bifurcation problem of the nonlinear equation F(λ,χ)=0 with trivial solution(λ, o). The sufficient conditions are given for (λ0, o) to be a bifurcation point of this equation, and the stability of the corresponding branching solutions is studied. Related Articles | Metrics

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THE STATIONARY DISTRIBUTIONS AND THE STEADY PROBABILITY CURRENTS OF ONEDIMENSIONAL DIFFUSION PROCESSES UNDER NON-LOCAL BOUNDARY CONDITIONS Wu Chengxun, Guo Maozheng Acta mathematica scientia,Series A. 1981, 1 (2):  195-206.  Abstract ( 42 )   RICH HTML PDF (625KB) ( 22 )   Save In the present paper we study the limit distributions of one-dimensional diffusion processes under non-local boundary conditions. We obtain the existence conditions for the non-trivial stationary distributions, get their concrete formulas in all cases according to boundary classification, and give the results of some probability explanations. One-dimensional diffusion processes, which provide a simple mathematical model for studying irreversible steady systems, show that the basic characteristic of an irreversible stationary state is the occurrence of the probability current, i. e. the circulation phenomenon. Related Articles | Metrics

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ON THE PROBLEMS OF BENDING OF ELASTIC PLATES Zhou Huan wen Acta mathematica scientia,Series A. 1981, 1 (2):  207-217.  Abstract ( 40 )   RICH HTML PDF (548KB) ( 35 )   Save In this paper, a mathematical model for the problems of bending of elastic of elastic plates is established. In this mathematical model. we presented three arbitrary functions to express the bending couples、twisting couple、shear forces and angular rotation and transuerse deflection etc. mechanical quantities which satisfied equiliber ium equations. For example, considered Reissner model. If let the sign A. ψ and R to note above three arbitrary functions respectively, then functions A, ψ and R need to satisfy the following equations: △△A=q/D, △ψ-10/h2ψ=0, and △R=-(1-2μ)/10μ q/D, and where μ is Poisson's ratio. The above first two equations are known, while the third is new. Related Articles | Metrics

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SOME PROPERTIES IN FUZZY CONVEX SETS Liu Yingming Acta mathematica scientia,Series A. 1981, 1 (2):  218-226.  Abstract ( 51 )   RICH HTML PDF (523KB) ( 48 )   Save Let E be an n-dimensioal Euclidean space, T the induced fuzzy topology for E[8], and λ and μ fuzzy sets in E. In[10], where L. Zadeh introduced the notion of the shadow of fuzzy set λ on a hyperplane H (denoted by SH(λ)), the following statement was given:If λ and μ be fuzzy convex sets E and SH(λ)=SH(μ) for all hyperplanes H, then λ=μ. Related Articles | Metrics

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SPACES BBp1q(r)(φ) AND DIFEFRENCE METHOD Luo Peizhu, Ding Xiaxi Acta mathematica scientia,Series A. 1981, 1 (2):  227-239.  Abstract ( 44 )   RICH HTML PDF (539KB) ( 18 )   Save In we introduced spaces Lρ (φ), Eρ(φ). In this paper we shall take the Lρ(φ)-norm in stead of Lρ-norm in the Spaces Bp,q(r), then we get the spaces Bp,q(r)(φ). we prove a trace theorem of the spaces Bp,q(r)(φ) and we can apply this method to the estimate of difference method of the initial problem. Related Articles | Metrics

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INFORMATION FLOW OF TREE (1) Tree Flow Theorems Li Chuanxiang Acta mathematica scientia,Series A. 1981, 1 (2):  240-256.  Abstract ( 41 )   RICH HTML PDF (782KB) ( 14 )   Save This paper is divided into two parts, where the general regularities of information flow in the tree graph are discussed. In the first part we discuss the equivalent theorems of the tree flow, the main sections are as follows:1. fundamental definitions; 2. equivalent mapping structure; 3. equivalent partition of vector set F*; 4. theorems of tree flow. And in the second part we put stress on the establishment of tree flow equations. Related Articles | Metrics

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GAUGE THEORY AND FUNCTIONAL HARMONIC MAPPING Guo Hanying Acta mathematica scientia,Series A. 1981, 1 (2):  257-259.  Abstract ( 43 )   RICH HTML PDF (171KB) ( 30 )   Save It is shown that the loop integral phase factors in the gauge theory (i. e. the elements of holonomy group of the gauge potentials) as the functional mappings from loops passing a given point in the space-time to the holonomy group HG of the gauge group G can be regarded as a harmonic mapping in the functional sense. The corresponding functional harmonic equations are equivalent to the sourceless Yang-Mills equatios. Related Articles | Metrics