Idempotent Analysis and Its Applications

Vassili N. Kolokoltsov, Victor P. Maslov (Autoren)

Buch | Softcover

305 Seiten

2010 | Softcover reprint of the original 1st ed. 1997
Springer (Verlag)
978-90-481-4834-9 (ISBN)

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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	3.12.2010
Reihe/Serie	Mathematics and Its Applications ; 401
Zusatzinfo	XII, 305 p.
Verlagsort	Dordrecht
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
	Wirtschaft ► Allgemeines / Lexika
	Wirtschaft ► Volkswirtschaftslehre
ISBN-10	90-481-4834-0 / 9048148340
ISBN-13	978-90-481-4834-9 / 9789048148349
Zustand	Neuware