Idempotent Analysis and Its Applications
Seiten
1997
Springer (Verlag)
978-0-7923-4509-1 (ISBN)
Springer (Verlag)
978-0-7923-4509-1 (ISBN)
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 .
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.
| Erscheint lt. Verlag | 30.4.1997 |
|---|---|
| Reihe/Serie | Mathematics and Its Applications ; 401 | Mathematics and Its Applications ; 401 |
| Zusatzinfo | XII, 305 p. |
| Verlagsort | Dordrecht |
| Sprache | englisch |
| Maße | 210 x 297 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| ISBN-10 | 0-7923-4509-6 / 0792345096 |
| ISBN-13 | 978-0-7923-4509-1 / 9780792345091 |
| Zustand | Neuware |
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