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For X a topological space, the presheaf F of Abelian groups (rings, ...) on X is defined such that 1. For every open subset U subset= X, an Abelian group (ring, ...) F(U), ...

A principal bundle is a special case of a fiber bundle where the fiber is a group G. More specifically, G is usually a Lie group. A principal bundle is a total space E along ...

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map 0->1 correspond to young rabbits growing old, and 1->10 ...

The Radon transform is an integral transform whose inverse is used to reconstruct images from medical CT scans. A technique for using Radon transforms to reconstruct a map of ...

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers ...

A pathwise-connected domain is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the ...

A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between ...

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequence This commutative ...

A species of structures is a rule F which 1. Produces, for each finite set U, a finite set F[U], 2. Produces, for each bijection sigma:U->V, a function F[sigma]:F[U]->F[V]. ...

A map projection obtained by projecting points P on the surface of sphere from the sphere's north pole N to point P^' in a plane tangent to the south pole S (Coxeter 1969, p. ...