As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:
2.
In the case of weak shocks, the Hugoniot will therefore fall directly on the isentrope and can be used directly as the equivalent path.
3.
The Dalai Lama XIV states : " old translation Dzogchen and new translation " anuttarayoga tantra " offer equivalent paths that can bring the practitioner to the same resultant state of Buddhahood ".
4.
It is also possible to use the characterisations directly for the larger domain, though some problems may arise . ( 1 ), ( 2 ), and ( 4 ) all make sense for arbitrary Banach algebras . ( 3 ) presents a problem for complex numbers, because there are non-equivalent paths along which one could integrate, and ( 5 ) is not sufficient.
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