This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

Idempotent functional analysis: An algebraic approach

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 
Some problems nonlinear in the traditional sense (for example, the Bellman equation and its generalizations) turn out to be linear over a suitable semiring; this linearity considerably simplifies the explicit construction of solutions 
The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis It has a wide range of applications 
Besides a survey of the subject, in this paper the correspondence principle is used to develop an approach to object-oriented software and hardware design for algorithms of idempotent calculus

Correspondence principle for idempotent calculus and some computer applications

The word idempotency siwfies the study of semirings in wmch the addition operation is idempotent: a + a = a. The best-mown example is the max-plus semiring, consisting of the real numbers with negative infinity adjoined in which addition is defined as max(a,b) and multiplication as a+b, the latter being distnbutive over the former. Interest in such structures arose in the late 1950s through the observation that certain problems of discrete optimisation could be linearised over suitable idempotent semirings. More recently the subject has established ~triguing connections with automata . theory, ~.hscrete e~ent syste~sl nonexpanSlVe mappmgs, nonlmear partla differential equations, optunisation theory and large deviations. TIle present paper was commissioned as an introduction to the volume of proceedings for the workshop on Idempotency held at Hewlett Packard's Basic Research Institute in the Mathematical Sciences (BRIMS) in October 1994. It aims to give an introductory survey, from a coherent mathematical viewpoin1 of the recent developments in the subject. iDe major open problems are pointed out and an extensive bibliography is provided.

http://www.hpl.hp.com/techreports/96/HPL-BRIMS-96-24.pdf

Idempotency: An introduction to idempotency

(1) Models of mean field, Hamilton-Jacobi-Bellman Equations and numerical methods. Yves Achdou. (2) First-order Hamilton-Jacobi Equations and Applications. Guy Barles. (3) Basic properties of viscosity solutionsand some aspects of weak KAM theory. Hitoshi Ishii. (4) Idempotent/Tropical Analysis and the Hamilton-Jacobi-Bellman Equations. Grigory L. Litvinov. (5) Homogenization and Approximation for Hamilton-Jacobi equations. Panagiotis E. Souganidis.

/pdf/hamilton-jacobi-equations-approximations-numerical-analysis-32zui5617d.pdf

Hamilton-Jacobi equations: approximations, numerical analysis and applications

We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are expressed in purely algebraic terms. This is one of a series of papers on idempotent functional analysis.

M. Shubin

Papers

Algebraic remarks on idempotent semirings and the kernel theorem in spaces of bounded functions