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The notion of pullback as a fiber-product ultimately leads to the very general idea of a categorical pullback, but it has important special cases: inverse image (and pullback) sheaves in algebraic geometry, and pullback bundles in algebraic topology and differential geometry. See also: Pullback (category theory) Fibred category; Inverse image sheaf
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map :, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.
More generally, any covariant tensor field – in particular any differential form – on may be pulled back to using . When the map ϕ {\displaystyle \phi } is a diffeomorphism , then the pullback, together with the pushforward , can be used to transform any tensor field from N {\displaystyle N} to M {\displaystyle M} or vice versa.
Any section s of E over B induces a section of f * E, called the pullback section f * s, simply by defining (′):= (′, ((′)) ) for all ′ ′.If the bundle E → B has structure group G with transition functions t ij (with respect to a family of local trivializations {(U i, φ i)}) then the pullback bundle f * E also has structure group G.
In this case E 1 × E 2 is a fiber bundle over B × B, and pulling back along the diagonal map B → B × B gives a space homeomorphic (diffeomorphic) to E 1 × B E 2, which is a fiber bundle over B. The pullback of two smooth transverse maps into the same differentiable manifold is also a differentiable manifold, and the tangent space of the ...
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Each sender can then back off before attempting to retransmit the same message again. A deterministic exponential backoff algorithm is unsuitable for this use case since each sender would back off for the same time period, leading them to retransmit simultaneously and cause another collision.
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