Idempotent Analysis and Its Applications

1997
ISBN: 978-0-7923-4509-1
Verlag: Springer Netherlands

Buch, Englisch, Band 401, 305 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1390 g

Reihe: Mathematics and its Applications

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.

Zielgruppe

Research

Autoren/Hrsg.

Maslov, Victor P.

Kolokoltsov, Vassili N.

Fachgebiete

• Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Wirtschaftstheorie, Wirtschaftsphilosophie
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Operations Research Spieltheorie

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.