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"necessary and sufficient condition" » "necessary and sufficient conditions"

1
by Atiyah, Michael Francis
Published 1970
VS Verlag für Sozialwissenschaften
... is the necessary and sufficient condition, and our results will give partial extens­ ions of Hopf's theorem...

2
by Mühlherr, Bernhard, Petersson, Holger P., Weiss, Richard M.
Published 2015
Princeton University Press
... and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form...

3
by Loi, Andrea, Zedda, Michela
Published 2018
Springer International Publishing
..., allowed him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally...