### Book chapter

## Выпуклый анализ, тропическая алгебра и обработка астрономических данных

### In book

The tropical arithmetic operations on R are defined as (a,b) -> min{a,b} and (a,b) -> a+b. We are interested in the concept of a semimodule, which is rather ill-behaved in tropical mathematics. In our paper we study the semimodules S in R^n having topological dimension two, and we show that any such S has always a finite weak dimension not exceeding n. For a fixed k, we construct a polynomial time algorithm deciding whether S is contained in some tropical semimodule of weak dimension k or not. The latter result provides a solution of a problem that has been open for eight years.

We give a tropical proof of a recent result of Mohindru on the CP-rank conjecture over min-max semiring.

This volume contains the proceedings of the International Workshop on Tropical and Idempotent Mathematics, held at the Independent University of Moscow, Russia, from August 26-31, 2012. The main purpose of the conference was to bring together and unite researchers and specialists in various areas of tropical and idempotent mathematics and applications. This volume contains articles on algebraic foundations of tropical mathematics as well as articles on applications of tropical mathematics in various fields as diverse as economics, electroenergetic networks, chemical reactions, representation theory, and foundations of classical thermodynamics. This volume is intended for graduate students and researchers interested in tropical and idempotent mathematics or in their applications in other areas of mathematics and in technical sciences

Properties of increasing positively homogeneous functions are studied; in particular, their representations by use of tropical inner products with coefficients chosen from tropical support sets are described. An application to a model of economic complementarityand weak links is developed. It is shown that weak links do not necessary bound total factor productivity from below but in some cases constraint it from above.

We present a number of new results on the multiplicative structure of univariate polynomials over the Boolean and tropical semirings.We answer the question asked by Kim and Roush in 2005 by proving that almost all Boolean polynomials with nonzero constant term are irreducible.We also give a lower bound for the number of reducible polynomials, and we discuss the related issues for polynomials over tropical semiring.