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Pointwise convergence of gradient‐like systems
Author(s) -
Lageman Christian
Publication year - 2007
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.200410564
Subject(s) - mathematics , pointwise , pointwise convergence , riemannian manifold , manifold (fluid mechanics) , limit (mathematics) , mathematical analysis , convergence (economics) , class (philosophy) , point (geometry) , pure mathematics , geometry , mechanical engineering , approx , artificial intelligence , computer science , engineering , economics , economic growth , operating system
S. Łojasiewicz has shown that the ω ‐limit sets of the trajectories of analytic gradient systems consist of at most one point. We extend this result to the larger class of gradient‐like vector fields satisfying an angle condition. In particular, this includes gradient systems, defined by arbitrary C 1 functions from an analytic‐geometric category. Corresponding pointwise convergence results are shown for discrete gradient‐like algorithms on a Riemannian manifold. This generalizes recent results by Absil, Mahony, and Andrews to the Riemannian geometry setting. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)