Edited by Jeremy Gunawardena
Foreword by John M. Taylor
Foreword by Michael Atiyah
Publications of the Newton Institute (No. 11)
Publisher: Cambridge University Press
Print Publication Year: 1998
Online Publication Date:May 2010
Online ISBN:9780511662508
Hardback ISBN:9780521553445
Paperback ISBN:9780521055383
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662508.026
Subjects: Real and Complex Analysis, Differential and integral equations, dynamical systems and control theory
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Introduction
This paper is devoted to heuristic aspects of the so-called idempotent calculus. There is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semirings, in the spirit of N. Bohr's correspondence principle in Quantum Mechanics. Idempotent analogs for some basic ideas, constructions and results in Functional Analysis and Mathematical Physics are discussed from this point of view. Thus the correspondence principle is a powerful heuristic tool to apply unexpected analogies and ideas borrowed from different areas of Mathematics and Theoretical Physics.
It is very important that some problems nonlinear in the traditional sense (for example, the Bellman equation and its generalizations and the Hamilton–Jacobi equation) turn out to be linear over a suitable semiring; this linearity considerably simplifies the explicit construction of solutions. In this case we have a natural analog of the so-called superposition principle in Quantum Mechanics (see [1]–[3]).
The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis. Applications include various optimization problems such as multicriteria decision making, optimization on graphs, discrete optimization with a large parameter (asymptotic problems), optimal design of computer systems and computer media, optimal organization of parallel data processing, dynamic programming, discrete event systems, computer science, discrete mathematics, mathematical logic and so on. See, for example, [4]–[64]. Let us indicate some applications of these ideas in mathematical physics and biophysics [65]–[70].
In this paper the correspondence principle is used to develop an approach to object-oriented software and hardware design for algorithms of idempotent calculus and scientific calculations.
pp. i-iv
pp. v-vi
pp. vii-viii
pp. ix-x
List of Participants: Read PDF
pp. xi-xii
An introduction to idempotency: Read PDF
pp. 1-49
pp. 50-69
Some automata-theoretic aspects of min-max-plus semirings: Read PDF
pp. 70-79
The finite power property for rational sets of a free group: Read PDF
pp. 80-87
The topological approach to the limitedness problem on distance automata: Read PDF
pp. 88-111
Types and dynamics in partially additive categories: Read PDF
pp. 112-132
Task resource models and (max, +) automata: Read PDF
pp. 133-144
Algebraic system analysis of timed Petri nets: Read PDF
pp. 145-170
Ergodic theorems for stochastic operators and discrete event networks.: Read PDF
pp. 171-208
Computational issues in recursive stochastic systems: Read PDF
pp. 209-230
Periodic points of nonexpansive maps: Read PDF
pp. 231-241
A system-theoretic approach for discrete-event control of manufacturing systems: Read PDF
pp. 242-261
Idempotent structures in the supervisory control of discrete event systems: Read PDF
pp. 262-281
Maxpolynomials and discrete-event dynamic systems: Read PDF
pp. 282-284
The Stochastic HJB equation and WKB method: Read PDF
pp. 285-302
The Lagrange problem from the point of view of idempotent analysis: Read PDF
pp. 303-321
A new differential equation for the dynamics of the Pareto sets: Read PDF
pp. 322-330
Duality between probability and optimization: Read PDF
pp. 331-353
Maslov optimization theory: topological aspect: Read PDF
pp. 354-382
Random particle methods in (max, +) optimization problems: Read PDF
pp. 383-391
The geometry of finite dimensional pseudomodules: Read PDF
pp. 392-405
A general linear max-plus solution technique: Read PDF
pp. 406-415
Axiomatics of thermodynamics and idempotent analysis: Read PDF
pp. 416-419
The correspondence principle for idempotent calculus and some computer applications: Read PDF
pp. 420-443