Idempotent Mathematics and Its Applications in Mathematical Physics and Mathematical Economy

Dear Colleagues,

Idempotent or tropical mathematics is the family of mathematical disciplines that use idempotent semirings and semifields instead of fields. This substitution of fields by idempotent semifields and semirings could be applied to many constructions from traditional mathematics. The language of idempotent mathematics (or idempotent analysis or idempotent calculus) creates a unified theory for treating a wide class of the optimization problems via linear (in the new algebra) methods leading theoretically to a circle of result that can be referred to as the idempotent functional analysis.

Idempotent or tropical mathematics is an asymptotic version of traditional mathematics. It can be considered as the result of the application of the dequantization procedure to traditional mathematics. The Planck constant in this asymptotic analysis belongs to the imaginary axis and goes to zero. New asymptotic methods are developed based on this idea.

Methods of idempotent analysis are useful for numerical calculations of the systems of differential or algebraic equations often leading to curse-of-dimensionality-free methods.

The numeric methods provide the main asymptotic terms with interval estimations of solutions. Application of idempotent mathematics are numerous in mathematical physics (de-quantization, thermodynamics), chemical engineering and system biology (including chemical kinetics and the analysis of multiscale dynamics), the theory of dimension and entropy (including fractal and negative dimensions), in models of economics and finances, in the theory of complexity, approximation theory in high dimensions, and nonlinear problems of optimal control and optimization, among others.

Prof. Dr. Viktor Maslov
Prof. Dr. Vassili Kolokoltsov
Guest Editors

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