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Pullback (differential geometry) Let be a smooth map between smooth manifolds and . Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on . This linear map is known as the pullback (by ), and is frequently denoted by .
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the ...
Pullback (category theory) In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written.
Pullback (cohomology) In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R -algebra homomorphism: from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature ...
Inverse image functor. 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.
Pullback motor. A pullback motor (also pull back, pull back and go or pull-back) is a simple clockwork motor used in toy cars. A patent for them was granted to Bertrand 'Fred' Francis in 1952 as a keyless clockwork motor. [1][2] Pulling the car backward (hence the name) winds up an internal spiral spring; a flat spiral rather than a helical ...
Nilpotent group. In mathematics, specifically group theory, a nilpotent groupG is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}. Intuitively, a nilpotent group is a group that is "almost abelian ".
Definition. The pullback attractor (or random global attractor) for a random dynamical system is a - almost surely unique random set such that. almost surely. There is a slight abuse of notation in the above: the first use of "dist" refers to the Hausdorff semi-distance from a point to a set, whereas the second use of "dist" refers to the ...