Boolean Methods in Operations Research and Related Areas by Professor Dr. Peter L. Hammer, Professor Dr. Sergiu Rudeanu

By Professor Dr. Peter L. Hammer, Professor Dr. Sergiu Rudeanu (auth.)

In classical research, there's a gigantic distinction among the category of difficulties that could be dealt with through the equipment of calculus and the category of difficulties requiring combinatorial ideas. With the appearance of the electronic desktop, the excellence starts to blur, and with the expanding emphasis on difficulties concerning optimization over constructions, tIlE' contrast vanishes. what's beneficial for the analytic and computational remedy of vital questions coming up in sleek keep watch over conception, mathematical economics, scheduling conception, operations learn, bioengineering, and so on is a brand new and extra versatile mathematical concept which subsumes either the cla8sical non-stop and discrete t 19orithms. The paintings by way of HAMMER (IVANESCU) and RUDEANU on Boolean equipment represents an incredible step during this dnectlOn, and it's therefore a superb excitement to welcome it into print. it is going to definitely stimulate loads of extra examine in either idea and alertness. RICHARD BELLMAN collage of Southern California FOf(,WOl'

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N) form a particular solution of equation (15); (~) conversely, if (xt, ... , x~) E Bn is particular solution of equation (15), then there exist n elements pt, ... , p~ E B such that relations (36) hold. We say also that formulas (36) define the general solution. As it will be shown in § 4, there may exist several general solutions fOf the same equations, that is there may be several families of Boolean functions {cp,} , {cp:}, {cp;'}, . satisfying the above properties ((X, P). Lemma 3. The double inequality (37) c ::;: x ~ d is equivalent to relation (38) x-=dpvcp, where P is an arbitrary parameter.

L. SCHUBERT [A 1], W. SEMON [A 1], A. SVOBODA [A 1] (or A. SVOBODA and K. CULIK [A 1]), Y. TOHMA [A 1], R. M. TOMS [A 1], A. AR" [A 1, A 2], H. ZEMANEK [A 1]. A thorough presentation of the above works would be very useful, but, unfortunately, would augment too much the dimensions of this book. Thus, we shall describe only two methods for solving equations in B2 . The first of them is very suitable for hand computation, but also workable with an electronic digital computer. It may be called: The Method of Bifurcations.

15), thus proving (12). 23"); therefore x = a x = a x u b x. lO'); further x = a x u b x, by Remark I, hence we deduce b x = 0, as before. It follows that a x u b x = 0, and the proof is completed. Theorem 4. Assume that relation (11) holds. Then the general solution of equation (9) ax u bx= 0 ~s (14) x=apubp, where p is an arbitrary parameter varying in B (that is, relation (14) implies (9) whatever the value p E B may be, and, conversely, every solution of (9) may be written in the form (14), for a suitably chosen p).

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