The correspondence principle for idempotent calculus and some computer applications

Litvinov, Grigori; Maslov, Victor Pavlovich

doi:10.1017/cbo9780511662508.026

Cited by 69 publications

(127 citation statements)

References 22 publications

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“…Let W = B(X, K) and V = B(Y, K) (as described in §7). where ϕ ∈ W , K f (x, y) ∈ B(X × Y, K) and the "idempotent integral" ⊕ X ψ(x) dx (e.g., see [2,[4][5][6][7][8]) is defined as sup x∈X ψ(x) for each function ψ ∈ B(X, K). We define a b-linear functional K f,y on W by setting…”

Section: §5 Tensor Products Of B-linear Mappingsmentioning

confidence: 99%

“…The present paper is one of a series of papers on idempotent functional analysis, which is an abstract version of idempotent analysis in the sense of [2][3][4][5][6][7][8][9]. In the subsequent publications, we intend to study the links between idempotent tensor products, idempotent linear integral and nuclear operators, and traces of operators.…”

Section: Introductionmentioning

confidence: 99%

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Tensor products of idempotent semimodules. An algebraic approach

Litvinov¹,

Маслов

Shpiz³

1999

Math Notes

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Section: §5 Tensor Products Of B-linear Mappingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tensor products of idempotent semimodules. An algebraic approach

Litvinov¹,

Маслов

Shpiz³

1999

Math Notes

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“…For a more copmprehensive exposition and for historical guides, the reader is reffered to the books [41], [32], [33] and to the reviews [20], [37].…”

Section: Introductionmentioning

confidence: 99%

Untitled

Kolokoltsov¹

2001

Journal of Mathematical Sciences

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Consider the set A = R ∪ {+∞} with the binary operations ⊕ = max and = + and denote by A n the set of vectors v = (v 1 , ..., v n ) with entries in A. Let the generalised sum u ⊕ v of two vectors denote the vector with entries u j ⊕ v j , and the product a v of an element a ∈ A and a vector v ∈ A n denote the vector with the entries a vj. With these operations, the set A n provides the simplest example of an idempotent semimodule. The study of idempotent semimodules and their morphisms is the subject of idempotent linear algebra, which has been developing for about 40 years already as a useful tool in a number of problems of discrete optimisation. Idempotent analysis studies infinite dimensional idempotent semimodules and is aimed at the applications to the optimisations problems with general (not necessarily finite) state spaces. We review here the main facts of idempotent analysis and its major areas of applications in optimisation theory, namely in multicriteria optimisation, in turnpike theory and mathematical economics, in the theory of generalised solutions of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games and controlled Marcov processes, in financial mathematics.

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“…The idempotent semiring R ∪ {−∞} ∪ {+∞} with the operations ⊕ = max, ⊙ = min can be obtained as a result of a "second dequantization" of R (or R + ). Dozens of interesting examples of nonisomorphic idempotent semirings may be cited as well as a number of standard methods of deriving new semirings from these (see, e.g., [2][3][4][5][6] and below).…”

mentioning

confidence: 99%

“…Obviously, if S = R min , then the standard order is opposite to the conventional order , so in this case equation (2) assumes the form…”

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confidence: 99%

Idempotent (Asymptotic) Mathematics and the Representation Theory

Litvinov¹,

Маслов

Shpiz³

2002

Asymptotic Combinatorics With Application to Mathematical Physics

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1. Introduction. Idempotent Mathematics is based on replacing the usual arithmetic operations by a new set of basic operations (e.g., such as maximum or minimum), that is on replacing numerical fields by idempotent semirings and semifields. Typical (and the most common) examples are given by the so-called (max, +) algebra R max and (min, +) algebra R min . Let R be the field of real numbers. Then R max = R ∪ {−∞} with operations x ⊕ y = max{x, y} and x ⊙ y = x + y. Similarly R min = R ∪ {+∞} with the operations ⊕ = min, ⊙ = +. The new addition ⊕ is idempotent, i.e., x ⊕ x = x for all elements x. Idempotent Mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking pure imaginary values. Some problems that are nonlinear in the traditional sense turn out to be linear over a suitable idempotent semiring (idempotent superposition principle [1]). For example, the Hamilton-Jacobi equation (which is an idempotent version of the Schrödinger equation) is linear over R min and R max .The basic paradigm is expressed in terms of an idempotent correspondence principle [2].This principle is similar to the well-known correspondence principle of N. Bohr in quantum theory (and closely related to it). Actually, there exists a heuristic correspondence between important, interesting and useful constructions and results of the traditional mathematics over fields and analogous constructions and results over idempotent semirings and semifields (i.e., semirings and semifields with idempotent addition).A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. As a result, in parallel with the traditional mathematics over rings, its "shadow", the Idempotent Mathematics, appears. This "shadow" stands approximately in the same relation to the traditional mathematics as classical physics to quantum theory. In many respects Idempotent Mathematics is simpler than the traditional one. However, the transition from traditional concepts and results to their idempotent analogs is often nontrivial.There is an idempotent version of the theory of linear representations of groups. We shall present some basic concepts and results of the idempotent representation theory. In the framework of this theory the well-known Legendre transform can be treated as an R max -version of the traditional Fourier transform (this observation

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The correspondence principle for idempotent calculus and some computer applications

Cited by 69 publications

References 22 publications

Tensor products of idempotent semimodules. An algebraic approach

Tensor products of idempotent semimodules. An algebraic approach

Untitled

Idempotent (Asymptotic) Mathematics and the Representation Theory

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