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The General Theory of Linear Difference Equations over the Max‐Plus Semi‐Ring
Author(s) -
Joshi Nalini,
Ormerod Chris
Publication year - 2007
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2007.00364.x
Subject(s) - mathematics , ring (chemistry) , linear equation , linear system , linear algebra , system of linear equations , algebra over a field , mathematical analysis , pure mathematics , geometry , chemistry , organic chemistry
We present the mathematical theory underlying systems of linear difference equations over the max‐plus semi‐ring. The result provides an analog of isomonodromy theory for ultradiscrete Painlevé equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for linear q ‐difference equations, but stands independently of the latter. As an example, we derive linear problems in this algebra for ultradiscrete versions of the symmetric P IV equation and show how it is a necessary condition for isomonodromic deformation of a linear system.