Fuzzy reasoning is widely used in various fields, in which membership functions are usually of real valued. But, when values of membership functions are not easily determined, it would be desireable to consider interval-valued membership functions. In this note, we propose two methods of Fuzzy reasoning in terms of interval-valued Fuzzy sets, and investigate some of their properties.

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Recently, many authors have been trying to obtain Hausdorff dimension of several fractal sets. However, it is only the self-similar set that Hausdorff dimension is exactly obtained. A self-similar set is constructed inductively by using a number of similarities, and then the contraction ratios are invariant at each stage of the construction. The middle third Cantor set and the von Koch curve are examples of self-similar sets. In this paper, we extend the middle third Cantor set, which we denote C, and examine Hausdorff dimension of C.C is constructed inductively by removing an open interval from each of the closed interval at the previous stage. The length of the removed open interval is arbitrary, except that the length is less than that of the closed interval at the previous stage from which the open interval is removed. Then the contraction ratios are variant at each stage of the construction of C. It is explained in Section 1 how to construct C. In Section 3, using Billingsley's Theorem, we give the lower bound for Hausdorff dimension of C under some conditions. In Section 4, we give Hausdorff dimension of C under conditions stronger than those of Section 3. Furthermore, we give a simple example of which Hausdorff dimension is 1 and Lebegue measure is 0.

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In this paper, we first give the proof of the convergence theorem of the Gauss-Seidel method using a matrix of the type(I+βU)as preconditioner. And we also show being able to transform the irreducibly diagonally dominant matrix to the strictly diagonally dominant matrix by multiplying the preconditioner.

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In the software testing phase, software reliability growth models have been used for assessing software quality/reliability. At the same time, software-testing managers are interested in the degree of testing-progress. In this paper, we propose a statistical method for software testing-progress control based on a control chart method. This method uses several instantaneous fault-detection rates derived from the software reliability growth models, i.e.the Gompertz growth curve model, the exponential and delayed S-shaped software reliability growth models. We can evaluate the testing-progress by applying a regression analysis to the observed fault-detection count data. Numerical illustrations for the testing-progress control chart are also presented.

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This paper presents an approximation method for obtaining a feasible solution of an assignment problem in which several constraints exist for each period and all assignments are adjusted according to a priority order. The proposed method is applicable to assignment problems whose cost coefficients cannot clearly be determined. In the method, the following three procedures are iteratively carried out to decrease the sum of infeasibilities ; (i)a set of integer variables to be changed is chosn by means of search trees, (ii)the trees are modified after the integer variables are changed, and (iii)the assignments are altered so that the number of deleted arcs is as small as possible.

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A parallel slit domain is the entire plane with parallel rectilinear slits. In this paper, we present a simple numerical method for computing conformal maps from domains extreior to closed Jordan curves onto the parallel slit domain. The mapping is important in problems of two-dimensional potential flow passing obstacles. The numerical method is based on the charge simulation method, in which conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials. It is shown that the method works well for typical domains.

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A new scaling theory is proposed to predict the'b-value'observed in frequency-magnitude distribution of earthquakes. The present model shows that the seismic moment released in small shallow earthquakes scale differently with rupture length that it does for large events, where the crossover between small shallow and large events is defined at a rupture dimension equal to the down-dip width of the seismological zone. Bases on the complete catalog recorded between 1900 and 1989 for the magnitude M larger than 7.0, it was found that the frequency-size distribution yields an average-value that changes from b1=1.04±0.03 for 7.0<M<7.5 to b2=1.51±0.08 for 7.6<M<8.0. The theoretically predicted b-values, namely, b1=21/20=1.05 and b2=3/2=1.5, agree well with the observed values.

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An approximate solution of orbit for long jump from takeoff to landing is obtained by considering the orbit as a projectile of a particle. The effect of air-drag is included as the linear form. Comparison with the exact solution of numerical integration shows the solution has good accuracy. Based on the solution, we can determine the optimum angle of takeoff successfully and esrimate the value from measured data.

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CG, Bi-CG methods are well-known iterative solutions of linear equation : Ax=b. CGS, Bi-CGSTAB, GPBi-CG methods have been developed as faster modifications for Bi-CG. The residual of interative methods, involving CGS, Bi-CGSTABand GPBi-CG, is represented with a product of the residual of Bi-CG method and the matrix polynomial which accelerates the convergence of the Bi-CG method. We call such iterative methods product-type, while the polynomial is said to be the accelerating polynomial. It has been recognized that the methods are useful to many practical applications. They are, however, easily influenced by rounding errors. In this paper, we analyze the effects of the accelerating polynomial on rounding errors for the product-type methods through numerical experiments. The present paper is hopefully a guideline for practical use of a the product-type methods.

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Several finite difference approximations which used only adjacent grid points for three-dimensional Poisson equation are proposed. In this paper, the optimal acceleration parameter and the optimal convergence rate of the SOR method for solving a linear system of equatins arising from those approximations with a rectangular region are investigated. We prove the red-black ordering SOR method to suited to vector processors is superior to the usual SOR method for one of those approximations, and study theoretically the optimal acceleration parameter and the optimal convergence rate of the red-black ordering SOR method for the approximation.

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The orbit of a linear Hamiltonian system with one degree of freedom is closed, divergent to infinity exponentially, or divergent as a free motion. This paper proves that every symplectic integrator reproduces a similar phase portrait in the former two cases when the step size is small enouth. This fact is considered based on a one-parameter family of conservatives admitted by a discrete and continuous systems. Furthermore, the existence of a conservative is studied for certain nonlinear systems.

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