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26 June 1982, Volume 2 Issue 2 Previous Issue    Next Issue

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INVARIANT SURROGATION IN SCIENTIFIC DIALOGUE P. D. Finch Acta mathematica scientia,Series A. 1982, 2 (2):  145-174.  Abstract ( 48 )   RICH HTML PDF (1657KB) ( 22 )   Save The term surrogation is used instead of approximation, estimation etc, to separate rol playing from how well a role is Played. To obtain a surrogate for an actual but unknown object one needs an algorithm to compute it from input data giving what is known about that object. Invariance arguments are used to show that surrogates must be fixed points of certain mappings associated with the mathematical structure characterizing practical context. The corresponding surrogation algorithms are said to he macroinvariant. Some special cases where there is essentially only one such algorithm are then considered. These results are derived without reference to proximity to actuality, that is to how good the surrogate is. Finally it is shown that if one does not use macroinvariant surrogation, then he could, in principle, do better, in any given metric sense, by means of such an invariant procedure. Related Articles | Metrics

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SYMMETRIES OF EQUATIONS qtt=g(q,qx,qxx,…) AND THE FORMAL COMPLETELY INTEGRABILITY OF BOUSSINESQ EQUATION Tu Guizhang Acta mathematica scientia,Series A. 1982, 2 (2):  175-182.  Abstract ( 47 )   RICH HTML PDF (434KB) ( 48 )   Save In this paper the symmetries of equations qtt=g(q,q1, q2,…) are discussed, where q=q(x,t) and qi=∂iq/∂xi. It is shown that if g=aqs+ḡ(q,…,qr), a=const, s-r ≥ 2, then any symmetry of the equation wilt be linear with respect to the term of highest order Furthermore, if the equation can be reduced to a Hamiltonian equation, then pairs of its conserved densities are in involution. As an application of this result, the Boussinesq equation qtt=q4+6q1q2 is shown to be a formal completely integrable Hamiltonian equation. Related Articles | Metrics

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PROBABILITY DISTRIBUTIONS OF ENERGY OF ONE-DIMENSIONAL PARTICLES OF TWO KINDS IN THE ENERGY EIGENSTATES Tao Zongying, Ding Lifeng, Hu Beihua Acta mathematica scientia,Series A. 1982, 2 (2):  183-192.  Abstract ( 46 )   RICH HTML PDF (544KB) ( 19 )   Save Starting from Voe Neumann's axiomatic system of quantum mechanics, this paper calculates the probability distributions of the kinetic and potential energy of the one-dimensional harmonic oscillator and the particle in one-dimensional infinitely deep square potential well. The results are compared with the classical cases, and it is shown that their relation conforms to Bohr's correspondence principle. Then we put forward a new viewpoint:it seems that the concept of the total energy of the particle in quantum mechanics should be deliberated again; the two basic hypotheses in quantum mechanics-"Bohr's probabilistic interpretation" and "the postulate that the particle's energy eguals the eigenvalue in the energy eigenstate" ars not logically harmonious. At the same time, the limitation of von Neumann's axiomatic system is pointed out. Related Articles | Metrics

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THE CONVERGENCE THEORY FOR DISCRETE-ORDINATE APPROXIMATIONS IN HIGHER SPATIAL DIMENSIONS Yang Mingzhu, Zhu Guangtian Acta mathematica scientia,Series A. 1982, 2 (2):  193-206.  Abstract ( 44 )   RICH HTML PDF (723KB) ( 47 )   Save In this paper the approximation theory of p-order quasi-collectively compact operators established by Ref. is applied to proving that the critical parameter and critical flux,and that the fundamental mode decay constant and fundamental mode computed by discrete-ordinate do converge to the corresponding quantities for the undiscretized, three-dimensional transport equation. Related Articles | Metrics

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ON SINGULAR INTEGRALS WITH SINGULARITIES OF HIGH FRACTIONAL ORDER AND THEIR APPLICATIONS Lu Jianke Acta mathematica scientia,Series A. 1982, 2 (2):  207-224.  Abstract ( 40 )   RICH HTML PDF (865KB) ( 153 )   Save Singular integrals of high order in complex integration were lifts introduced in[1] and investigated in[2,3,4]. Using this concept, the classical residue theorem was extended to such cases and was applied to solving certain class of singular integral equations. Related Articles | Metrics

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A METHOD OF THE RECURSIVE ESTIMATION WITHOUT INITIAL VALUE Wang Yaoyuan Acta mathematica scientia,Series A. 1982, 2 (2):  225-240.  Abstract ( 37 )   RICH HTML PDF (728KB) ( 18 )   Save In Kalman filtering the initial values X0, P0 should be known, but they are usually unknown. In[1] the recursive estimation problem without initial value is discussed when the state noise is white noise. Related Articles | Metrics