Idempotent Analysis and Its Applications

  • Vassili N. Kolokoltsov
  • Victor P. Maslov
Book

Part of the Mathematics and Its Applications book series (MAIA, volume 401)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Vassili N. Kolokoltsov, Victor P. Maslov
    Pages 1-44
  3. Vassili N. Kolokoltsov, Victor P. Maslov
    Pages 45-84
  4. Vassili N. Kolokoltsov, Victor P. Maslov
    Pages 85-150
  5. Vassili N. Kolokoltsov, Victor P. Maslov
    Pages 151-231
  6. Back Matter
    Pages 233-308

About this book

Introduction

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 .

Keywords

Mathematica calculus differential equation economics mathematical economics schrödinger equation

Authors and affiliations

  • Vassili N. Kolokoltsov
    • 1
    • 2
  • Victor P. Maslov
    • 3
  1. 1.Department of Mathematical StatisticsNottingham Trent UniversityNottinghamEngland
  2. 2.Institute of New TechnologiesMoscowRussia
  3. 3.Department of PhysicsMoscow State UniversityMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-8901-7
  • Copyright Information Springer Science+Business Media B.V. 1997
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4834-9
  • Online ISBN 978-94-015-8901-7
  • About this book
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