The first chapter deals with idempotent analysis per se . To make the pres- tation self-contained, in the first two sections we define idempotent semirings, give a concise exposition of idempotent linear algebra, and survey some of its applications. Idempotent linear algebra studies the properties of the semirn- ules An , n E N , over a semiring A with idempotent addition; in other words, it studies systems of equations that are linear in an idempotent semiring. Pr- ably the first interesting and nontrivial idempotent semiring , namely, that of all languages over a finite alphabet, as well as linear equations in this sern- ing, was examined by S. Kleene [107] in 1956 . This noncommutative semiring was used in applications to compiling and parsing (see also [1]) . Presently, the literature on idempotent algebra and its applications to theoretical computer science (linguistic problems, finite automata, discrete event systems, and Petri nets), biomathematics, logic , mathematical physics , mathematical economics, and optimizat ion, is immense; e. g. , see [9, 10, 11, 12, 13, 15, 16 , 17, 22, 31 , 32, 35, 36, 37, 38, 39 , 40, 41, 52, 53 , 54, 55, 61, 62 , 63, 64, 68, 71, 72, 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 present the most important facts of the idempotent algebra formalism . The semimodules An are idempotent analogs of the finite-dimensional v- n, tor spaces lR and hence endomorphisms of these semi modules can naturally be called (idempotent) linear operators on An .
Inhaltsverzeichnis
1 Idempotent Analysis. - 2 Analysis of Operators on Idempotent Semimodules. - 3 Generalized Solutions of Bellman s Differential Equation. - 4 Quantization of the Bellman Equation and Multiplicative Asymptotics. - References. - Appendix (Pierre Del Moral). Maslov Optimization Theory. Optimality versus Randomness. - 1 Maslov s Integration Theory. - 2 Performance Theory. - 3 Lebesgue-Maslov Semirings. - 4 Convergence Modes. - 5 Optimization Processes. - 6 Applications. - 7 Maslov and Markov Processes. - 8 Nonlinear Filtering and Deterministic Optimization. - 9 Monte-Carlo Principles. - 10 Particle Interpretations. - 11 Convergence. - Conclusions. - References.
