Tropical Value Distribution Theory and Ultra-Discrete Equations

This is the first textbook-type presentation of tropical value distribution theory. It provides a detailed introduction of the tropical version of the Nevanlinna theory, describing growth and value distribution analysis of continuous, piecewise linear functions on the real axis. The book also includes applications of this theory to ultra-discrete equations. Three appendices are given to compare the contents of the theory with the classical counterparts in complex analysis. Detailed presentation of the proofs makes the book accessible for lecture courses and independent studies at the graduate and post-doctoral level. Contents:Tropical Polynomials, Rationals and ExponentialsTropical Entire FunctionsOne-dimensional Tropical Nevanlinna TheoryClunie and Mohon'ko Type TheoremsTropical Holomorphic CurvesRepresentations of Tropical Periodic FunctionsApplications to Ultra-Discrete EquationsAppendices:Classical Nevanlinna and Cartan TheoriesIntroduction to Ultra-Discrete Painlevé EquationsSome Operators in Complex Analysis Readership: Graduate students, post-graduates and researchers. Key Features:First monograph type presentation in this topicSome of the material is new in contents, which follows in the presentationThe contributors have been active in value distribution theory, including its tropical variantKeywords:Tropical Mathematics;Value Distribution Theory;Ultra-discrete Equations