Download e-book for kindle: Fuzzy Sets and Fuzzy Information Granulation Theory by Chongfu Huang, Da Ruan By Chongfu Huang, Da Ruan

This can be the 3rd quantity of gathered Papers by means of Lotfi A.Zadeh.

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For further analysis we define an angular parameter when A is real; the arguments are all to be taken as 0 when A = 0, and calculated thence by continuous’ variation. ) =a + 2rv. 5) Here r will run through some sequence A of consecutive integers; in the event that 03 is an eigenvalue, we may for definiteness take as representative, and incorporate the corresponding r-value in A . T h e eigenvalues thus defined are, of course, dependent on a and on m, +-= A,. 6) and an important group of topics centers on the dependence of h, on m and a, and more generally on the dependence on m and a of the spectral function, to be defined shortly.

BOUNDARY PROBLEMS FOR RATIONAL FUNCTIONS sional cases by way of a limiting process. 6. 2-3) be all finite. 7-8) follows standard lines, and is given in Appendix 111. 12), we have the dual expansion theorem, likewise purely elementary. 7. 2-3) have only finite eigenvalues. 15). It is a simple matter to deduce one of these expansion theorems from the other; the situation naturally becomes much more difficult in infinitedimensional cases, even when both expansions are discrete. 6) holds. 7-8). 5. 19) where X, is any of the finite eigenvalues.

We pursue this example to the extent of setting up an inhomogeneous problem, leading to the characteristic function. We suppose a force F cos ( w t ) applied transversely to the particle a, , the string being pinned down at both ends (Fig. 1). Assuming, if possible, the displace~ F C OwSt FIG. 1 . Inhomogeneous problem for vibrating string. 8) = 0, and finally v, = 0. 6). 8) if we arrange that + F = C-10-1 = c-l(uy-1 + /%-I) = Pc-1 , 18 0. INTRODUCTION by the initial conditions. We deduce that + /lzm= 0, we The function -z,(h)/ym(h) will be a special case of what we use in Chapters 4 and 5 as a characteristic function; it will be expedient to use a more general function, corresponding to a more general boundary condition at the upper end, with a view to limit-circle investigations in the complex plane.