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More generally, given a vector bundle (finite-rank locally free sheaf) E on X, if R=Sym(E *) is the symmetric algebra generated by the dual of E, then the cone is the total space of E, often written just as E, and the projective cone is the projective bundle of E, which is written as ().
A subset of a vector space over an ordered field is a cone (or sometimes called a linear cone) if for each in and positive scalar in , the product is in . [2] Note that some authors define cone with the scalar α {\displaystyle \alpha } ranging over all non-negative scalars (rather than all positive scalars, which does not include 0). [ 3 ]
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
A right circular cone and an oblique circular cone A double cone (not shown infinitely extended) 3D model of a cone. A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex that is not contained in the base.
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
In mathematics, especially convex analysis, the recession cone of a set is a cone containing all vectors such that recedes in that direction. That is, the set extends ...
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example). The cone over a polygon P is a pyramid with base P. The cone over a disk is the solid cone of classical geometry (hence the concept's name). The cone over a circle given by
In mathematics, the nilpotent cone of a finite-dimensional semisimple Lie algebra is the set of elements that act nilpotently in all representations of . In other words, N = { a ∈ g : ρ ( a ) is nilpotent for all representations ρ : g → End ( V ) } . {\displaystyle {\mathcal {N}}=\{a\in {\mathfrak {g}}:\rho (a){\mbox{ is nilpotent for ...
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