This article presents a framework for studying the existence of optimal control based on fuzzy rules. The framework consists of two propositions : (1)The family of fuzzy membership functions with bounded gradients is a uniformly bounded, equicontinuous closed set. Hence the family of fuzzy membership functions is compact by Ascoli-Arzela's theorem.(2)The defuzzification function is continuous on the family of fuzzy membership functions with bounded gradients. Then the existence of fuzzy optimal control is proved.

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A class of recurrent neural networks is considered in which w_<ij>the connection weight from the jth to the ith element, is randomly generated under the condition wij=wji. The expected number of equilibrium states in the network consisting of two-state threshold elements having outputs of(-1, 1)or(0, 1)with variable thresholds, but uniform throughout the network, is derived by the method of statistical neurodynamics. It is shown that the expected number of equilibrium states and the rate of excited neurons in these states are uniquely determined by the threshold value. Applications of this network as a module for memory systems are discussed.

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We consider certain special types of single evolution equations of third and fourth order and systems of equations of third order ; both of them are linear with constant coefficients with respect to the heighest order spacial derivatives. Among these equations, we enumerate the ones which are formally linearizable in the sense of Sokolov and Shabat.

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An interpolation formula whose abscissae are the zeros of the Bessel functions of integral order is proposed along with its theoretical error analysis. Based on the interpolation formula, two kinds of quadrature formulae of interpolatory type are also proposed : one is for"symmetric integrals"∫^∞_<-∞>f(x)dx and the other for"anti-symmetric integrals"∫^∞_<-∞>sgn x f(x)dx. Both the efficiency of the former quadrature and that of the latter quadrature are studied in theory and in practice, especially compared with the efficiency of the tapezoidal formula.

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The number of successes s_n in n Bernoulli trials increases as n. But the broken line connecting points(n, s_n)shows some peculiar features due to the high correlation between neighboring points. For the special case of cointossing where the probability of success p is equal to 1/2, Feller[1] treats the subject quite impressively. This paper deals with the general case by theoretical computations. The results suggest Propositions on the asymptotic behavior of the distribution of the number of touches(crossings and other contacts)with the theoretical line, and the proportion of the points on one side of the theoretical line.

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Data reduction algorithms using wavelet transform are treated. From many Kinds of wavelets, discrete orthonormal Daubechies wavelets and symmetric bi-orthogonal wavelets are taken. Various degree of wavelets are tested for a simple spiral line to check the fundamental properties of the transform. Then they are applied to data of a shoreline. Through these computations, following conclusions are obtained : ・For high-speed data processing the algorithms using bi-orthogonal discrete wavelet transform are efficient. ・In our test range, the higher degree of wavelet generally gives the more accurate result. ・For complicated shore line, it is practical to use multi resolution method.

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The method of quantifying protein sequences by application of principal component analysis has been improved to detect the similarity by numerical criteria. The alignment of sequences that are quantified as the first principal components which reflect the physicochemical properties of constituent amino acid residues, is searched by the information on residue variety in the sequencs as well as the resemblance of the properties. The ability of newly developed method is tested by two known but obscure motifs and is also discussed with reference to usual methods for similarity search.

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The purpose of this paper are ; (1)to evaluate each posterior debugging models by P-L method, (2)to characterize each debugging model by means of it′s curve, (3)to analyze the relation between the standard model and the raw model and the standard submodel and the raw submodel, (4)to divde each model into the three stages and analyze these stage models, (5)to discuss the characterization of these debugging models and give some advantage of using these models in order to apply the posterior debugging analysis.

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The purpose of this paper is to give a ranking procedure which helps to decide if the examination is appropriate or not. In general, a ranking of the examination is determined by the scores that students made. It is very common to use this method in order to decide the ranking of students who took the examination. But it represents the only one aspect of student's knowledge. In this paper, we present the two different procedures on the scores of the examinations. In order to decide the ranking, we apply the Perron-Frobenius theorem to the ranking of the scores and estimate the ranking which is considered to be rather appropriate ranking better than one which is used in common.

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A user interface to manipulate natural mathematical notations is proposed. Mathematical notations have two-dimensional structures such as superscript, subscript, fraction, etc., therefore a two-dimensional user interface is required. To input complicated structures of equations, we have implemented the input method of multi-selection of overlay. This interface generates a formal representation, which can be used for the input of both mathematical expressions understanding and typesetting.

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