Book•
Idempotent Analysis and Its Applications
30 Apr 1997-
TL;DR: In this article, a generalized solution of Bellman's Differential Equation and multiplicative additive asymptotics is presented, which is based on the Maslov Optimziation Theory.
Abstract: Preface. 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: (P. Del Moral) Maslov Optimziation Theory. Optimality versus Randomness. Index.
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08 Nov 2010
TL;DR: Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max- algebraic Subspaces.
Abstract: Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.
488 citations
Cites background from "Idempotent Analysis and Its Applica..."
...Note that idempotency of addition makes max-algebra part of idempotent mathematics [108], [110]....
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Book•
06 Sep 2010TL;DR: In this article, the authors developed the interplay between probability and analysis in nonlinear Markov evolution, and used probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior.
Abstract: A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.
283 citations
TL;DR: In this article, a nonlinear projection on subsemimodules is introduced, where the projection of a point is the maximal approximation from below of the point in the sub-semimmodule.
273 citations
Cites background from "Idempotent Analysis and Its Applica..."
...More recently, the interest for idempotent semimodules arose from the development of the max-plus algebraic approach to optimal control and asymptotic analysis (Maslov [38], Maslov and Samborskiı̆ [37], Kolokolstov and Maslov [32], Litvinov et al. [35,36]), and to discrete event systems (Cohen et al. [10], Baccelli et al. [3], Gaubert and Plus [22], Cohen et al. [13])....
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...More recently, the interest for idempotent semimodules arose from the development of the max-plus algebraic approach to optimal control and asymptotic analysis (Maslov [38], Maslov and Samborskiı̆ [37], Kolokolstov and Maslov [32], Litvinov et al....
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27 Feb 1997
TL;DR: Exotic semirings have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis.
Abstract: Exotic semirings such as the “(max, +) semiring” (ℝ ∪ {−∞},max,+), or the “tropical semiring” (ℕ ∪ {+∞}, min, +), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities).
243 citations
Cites background from "Idempotent Analysis and Its Applica..."
...Let us also mention the forthcoming book of Kolokoltsov and Maslov (an earlier version is in Russian [30])....
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TL;DR: In this article, a brief introduction to idempotent and tropical mathematics is given, which can be seen as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values.
Abstract: This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant ℏ tends to zero taking imaginary values. Bibliography: 187 titles.
242 citations