Idempotent Analysis and Its Applications

verfasst von: Vassili N. Kolokoltsov, Victor P. Maslov

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

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 .

Inhaltsverzeichnis

Frontmatter

Chapter 1. Idempotent Analysis

Abstract

The first chapter deals with idempotent analysis per se. To make the presentation 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 semimodules A ⁿ, n ∈ ℕ, over a semiring A with idempotent addition; in other words, it studies systems of equations that are linear in an idempotent semiring. Probably the first interesting and nontrivial idempotent semiring, namely, that of all languages over a finite alphabet, as well as linear equations in this semiring, 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 optimization, 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.

Vassili N. Kolokoltsov, Victor P. Maslov

Chapter 2. Analysis of Operators on Idempotent Semimodules

Abstract

In this chapter we study endomorphisms, or linear operators, on semimodules of functions that range in idempotent semirings. Here the specific nature of idempotent analysis exhibits itself in the fact that each linear operator on such a semimodule is an integral operator, that is, has the form

$$ \left( {Bh} \right)\left( x \right) = \int {^ \oplus b\left( {x,y} \right) \odot h\left( y \right)d\mu \left( y \right) = \mathop {\inf }\limits_y \left( {b\left( {x,y} \right) \odot h\left( y \right)} \right)} $$

for some idempotent integral kernel b(x,y). In §2.1 we give necessary and sufficient conditions for this function to specify a continuous operator. We give a characterization of weak and strong convergence of operator families in terms of kernels and then, in §2.2, we describe two important operator classes— invertible and compact operators—in the same terms. Here another specific feature of the semialgebra of idempotent linear operators is important—the supply of invertible operators is very small; namely, the group of invertible operators is generated by the diagonal operators and by the homomorphisms of the base. Hence, this group consists of idempotent analogs of weighted translation operators. It follows that all automorphisms of the operator semialgebra are inner automorphisms.

Vassili N. Kolokoltsov, Victor P. Maslov

Chapter 3. Generalized Solutions of Bellman’s Differential Equation

Abstract

The theory of new distributions introduced in Chapter 1 can be used to define generalized solutions of the Hamilton-Jacobi-Bellman equation just as in the conventional linear theory, by using the adjoint operator. Idempotent analysis provides a physically natural interpretation of formulas thus obtained: they are given by the convolution (in the new sense) of the initial data with the Green function (the solution whose initial value is given by the idempotent δ-function), which is obtained as the solution of the related variational problem with fixed endpoints and fixed time interval.

Vassili N. Kolokoltsov, Victor P. Maslov

Chapter 4. Quantization of the Bellman Equation and Multiplicative Asymptotics

Abstract

In the classical WKB method, local asymptotic solutions of the linear pseudodifferential equation

$$ H\left( {x,\frac{h}{i}\frac{\partial }{{\partial x}}} \right)\psi = 0 $$

(4.1)

with a small positive parameter h are sought in the form

$$ \psi \left( x \right) = A\left( x \right)\exp \left\{ {\frac{i}{h}S\left( x \right)} \right\} $$

(4.2)

Vassili N. Kolokoltsov, Victor P. Maslov

Backmatter

Titel: Idempotent Analysis and Its Applications
verfasst von: Vassili N. Kolokoltsov
Victor P. Maslov
Verlag: Springer Netherlands
Electronic ISBN: 978-94-015-8901-7
Print ISBN: 978-90-481-4834-9
DOI: https://doi.org/10.1007/978-94-015-8901-7

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Idempotent Analysis

Chapter 2. Analysis of Operators on Idempotent Semimodules

Chapter 3. Generalized Solutions of Bellman’s Differential Equation

Chapter 4. Quantization of the Bellman Equation and Multiplicative Asymptotics

Backmatter