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By Vassili N. Kolokoltsov, Victor P. Maslov

The first bankruptcy bargains with idempotent research in step with se . To make the pres- tation self-contained, within the first sections we outline idempotent semirings, supply a concise exposition of idempotent linear algebra, and survey a few of its functions. Idempotent linear algebra experiences the homes of the semirn- ules An , n E N , over a semiring A with idempotent addition; in different phrases, it reports platforms of equations which are linear in an idempotent semiring. Pr- ably the 1st attention-grabbing and nontrivial idempotent semiring , specifically, that of all languages over a finite alphabet, in addition to linear equations during this sern- ing, was once tested via S. Kleene [107] in 1956 . This noncommutative semiring used to be utilized in purposes to compiling and parsing (see additionally [1]) . shortly, the literature on idempotent algebra and its functions to theoretical computing device technology (linguistic difficulties, finite automata, discrete occasion structures, and Petri nets), biomathematics, good judgment , mathematical physics , mathematical economics, and optimizat ion, is huge; e. g. , see [9, 10, eleven, 12, thirteen, 15, sixteen , 17, 22, 31 , 32, 35,36,37,38,39 ,40,41,52,53 ,54,55,61,62 ,63,64,68, seventy one, seventy two, 73,74,77,78, 79,80,81,82,83,84,85,86,88,114,125 ,128,135,136, 138,139,141,159,160, 167,170,173,174,175,176,177,178,179,180,185,186 , 187, 188, 189]. In §1. 2 we current crucial evidence of the idempotent algebra formalism . The semimodules An are idempotent analogs of the finite-dimensional v- n, tor areas lR and for this reason endomorphisms of those semi modules can clearly be known as (idempotent) linear operators on An .

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3 are used in this problem. , see [13]). 5. The generalized assignment problem. A single-iteration algorithm. Numerous problems traditionally solved by multi-iteration algorithms can be solved by single-iteration algorithms under an appropriate choice of the function semimodule. As an example, let us consider the cited generalized assignment problem. Let Yk C Y and Zk C Z be subsets of cardinality k, k = 1, . , n, and let S(k, Y k, Zk) be the weight of an optimal matching of cardinality k in the subgraph of G generated by Yk U Zk.

Let K be the compact closure of some neighborhood of the point x EX, and let I be the directed set of pairs (U, V) of neighborhoods of x such that U c v c K with the partial order (U 1 , Vd ~ (U2 , V2 ) {::::::} V 2 CUt. Since the segment [a , OJ is connected in M , it follows from the Urysohn lemma that for each pair (U, V) E I there exists a function h cu,V ) E ClI(X, M) such that h(y) E [a, 0] for any y , sUPPo hE V , and hlu = a. Obviously, the net {hcu,v)} is monotone increasing and converges to g;.

LIPI) of nonnegative integers) is called a marking of the bipartite graph I' = (PUT, E). One says that the place Pi is marked with /-Li tokens. By definition, a Petri net is a bipartite graph equipped with some marking. In the standard graphic representation of Petri nets, places are drawn as circles and transitions as bars (or rectangles) . Moreover, the number of dots placed in each circle is equal to the number of tokens marking the corresponding place. The dynamics of a Petri net is defined as follows.

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