Idempotent Analysis and Its Applications

1. Front Matter

   Pages i-xii

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2. Idempotent Analysis

   • Vassili N. Kolokoltsov, Victor P. Maslov

   Pages 1-44

3. Analysis of Operators on Idempotent Semimodules

   • Vassili N. Kolokoltsov, Victor P. Maslov

   Pages 45-84

4. Generalized Solutions of Bellman’s Differential Equation

   • Vassili N. Kolokoltsov, Victor P. Maslov

   Pages 85-150

5. Quantization of the Bellman Equation and Multiplicative Asymptotics

   • Vassili N. Kolokoltsov, Victor P. Maslov

   Pages 151-231

6. Back Matter

   Pages 233-308

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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 .

• Mathematica
• calculus
• differential equation
• economics
• mathematical economics
• schrödinger equation
• partial differential equations

• Department of Mathematical Statistics, Nottingham Trent University, Nottingham, England

  Vassili N. Kolokoltsov

• Institute of New Technologies, Moscow, Russia

  Vassili N. Kolokoltsov

• Department of Physics, Moscow State University, Moscow, Russia

  Victor P. Maslov

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