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Geometrical proofs for the global solvability of systems
Author(s) -
Bergamasco Adalberto Panobianco,
Parmeggiani Alberto,
Zani Sérgio Luís,
Zugliani Giuliano Angelo
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700300
Subject(s) - mathematical proof , mathematics , morse code , operator (biology) , morse theory , surface (topology) , pure mathematics , genus , topology (electrical circuits) , combinatorics , geometry , computer science , telecommunications , biochemistry , chemistry , botany , repressor , biology , transcription factor , gene
We study a linear operator associated with a closed non‐exact 1‐form b defined on a smooth closed orientable surface M of genus g > 1 . Here we present two proofs that reveal the interplay between the global solvability of the operator and the global topology of the surface. The first result brings an answer for the global solvability when the system is defined by a generic Morse 1‐form. Necessary conditions for the global solvability bearing on the sublevel and superlevel sets of primitives of a smooth 1‐form b have already been established; we also present a more intuitive proof of this result.