The Maslov Dequantization, idempotent and tropical mathematics: a very brief introduction

Abstract

Abstract. This paper is a very brief introduction to idempotent mathematics and related topics. This paper is a very brief introduction, without exact theorems and proofs, to the Maslov dequantization and idempotent and tropical mathematics. Our list of references is not complete (not at all). Additional references can be found, e.g., in the electronic archive

Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach

“…Numerous examples of idempotent semimodules and spaces can be found in [8]; see also [3]- [9], [11], [14]- [16], [25]. Some examples are presented below, see, e.g., Section 3.…”

“…Maslov and his collaborators, see, e.g., [1]- [14]; in particular, important results and applications of idempotent analysis are due to V.N. Kolokoltsov (see, e.g., [3]- [6], [9], [13]- [15]). An important development of these ideas is due to the French mathematicians M. Akian, G. Cohen, S. Gaubert, J.-P. Quadrat and others, see, e.g., [9,15,16] and the survey [10].…”

“…Kolokoltsov (see, e.g., [3]- [6], [9], [13]- [15]). An important development of these ideas is due to the French mathematicians M. Akian, G. Cohen, S. Gaubert, J.-P. Quadrat and others, see, e.g., [9,15,16] and the survey [10]. A remarkable paper of O. Viro [17] was a starting point for idempotent geometry; now this subject is usually called tropical geometry.…”

Generalization of Portmanteau theorem with respect to the pseudo-weak convergence of random closed sets

“…To achieve this goal, the classical Lebesgue integral had been substituted with more general one, known as general pseudo integral [1], which is based on pseudo-addition and pseudomultiplication (as generalizations of the classical real operations) and monotone set functions (nonadditive measures), see [25], [26]. One of the basic important cases is the idempotent analysis, see [15], [16].…”

“…Section 2 contains preliminary notions, such as pseudo-addition, pseudodifference, pseudo-multiplication, and pseudo-integral (see [1], [12], [15], [16], [25], [26]). Section 3 contains results on separation functionals, which are used in the proofs in Section 5.…”