# Tropical and Idempotent Mathematics

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*G. L. Litvinov; S. N. Sergeev*

This volume is a collection of papers from the International
Conference on Tropical and Idempotent Mathematics, held in Moscow,
Russia in August 2007. This is a relatively new branch of
mathematical sciences that has been rapidly developing and gaining
popularity over the last decade.

Tropical mathematics can be viewed as a result of the Maslov
dequantization applied to “traditional” mathematics over fields.
Importantly, applications in econophysics and statistical mechanics lead
to an explanation of the nature of financial crises. Another original
application provides an analysis of instabilities in electrical power
networks.

Idempotent analysis, tropical algebra, and tropical geometry are the
building blocks of the subject. Contributions to idempotent analysis are
focused on the Hamilton-Jacobi semigroup, the max-plus finite element
method, and on the representations of eigenfunctions of idempotent linear
operators. Tropical algebras, consisting of plurisubharmonic functions and
their germs, are examined. The volume also contains important surveys and
research papers on tropical linear algebra and tropical convex geometry.

# Table of Contents

## Tropical and Idempotent Mathematics

- Contents v6 free
- Preface: The Maslov Dequantization and Tropical Mathematics vii8 free
- Linear Independence over Tropical Semirings and Beyond 112 free
- The Optimal Assignment Problem for a Countable State Space 3950
- Dequantization of Real Convex Projective Manifolds 6172
- Tropical Conics for the Layman 8798
- Idempotent Aspects of Hopf-Lax Type Formulas 103114
- On Some Properties of the Image Set of a Max-linear Mapping 115126
- Tropical Plücker Functions and Their Bases 127138
- A Class of Periodic Minplus Homogeneous Dynamical Systems 159170
- Basics of Linear Algebra over the Extended Tropical Semiring 173184
- Tropical Convex Hull Computations 193204
- Complex Roots of Systems of Tropical Equations and Stability of Electrical Power Networks 213224
- Dequantization, Statistical Mechanics and Econophysics 239250
- Graph Selectors and the Max-Plus Finite Element Method 281292
- Complexity Reduction, Cornices and Pruning 293304
- Tropical Analysis of Plurisubharmonic Singularities 305316
- Multiorder, Kleene Stars and Cyclic Projectors in the Geometry of Max Cones 317328
- A Tropical Version of the Schauder Fixed Point Theorem 343354
- Tropical Cones Defined by Max-linear Inequalities 351362
- Minimum Representing Measures in Idempotent Analysis 367378