MHD generator

2013-11-27T20:39:38Z

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In [[linear algebra]], a ('''linear''') '''cone''' is a [[subset]] of a [[vector space]] that is [[closed (mathematics)|closed]] under [[multiplication]] by positive [[scalar (mathematics)|scalars]]. In other words, a subset ''C'' of a real vector space ''V'' is a cone if and only if λ''x'' belongs to ''C'' for any ''x'' in ''C'' and any positive scalar λ of ''V'' (or, more succinctly, if and only if λ''C'' = ''C'' for any positive scalar λ).

A cone is said to be '''pointed''' if it includes the [[null vector (vector space)|null vector]] ([[origin (mathematics)|origin]]) '''0'''; otherwise it is said to be '''blunt'''. Some authors use "non-negative" instead of "positive" in this definition of "cone", which restricts the term to the pointed cones only. In other contexts, a cone is '''pointed''' if the only linear subspace contained in it is '''{0}'''.

The definition makes sense for any vector space ''V'' which allows the notion of "positive scalar" (i.e., where the ground field is an [[ordered field]]), such as spaces over the [[rational number|rational]], real [[algebraic number|algebraic]], or (most commonly) [[real number]]s.

The concept can also be extended for any vector space ''V'' whose scalar field is a superset of those fields (such as the [[complex number]]s, [[quaternion]]s, etc.), to the extent that such a space can be viewed as a real vector space of higher dimension.

==Related concepts==
===The cone of a set===
The ('''linear''') '''cone of''' an arbitrary subset ''X'' of '''V''' is the set ''X''<sup>*</sup> of all vectors λ''x'' where ''x'' belongs to ''X'' and λ is a positive scalar.

With this definition, the cone of ''X'' is pointed or blunt depending on whether ''X'' contains the origin '''0''' or not. If "positive" is replaced by "non-negative" in this definition, then the cone of ''X'' will be pointed, for any ''X''.

===Salient cone===
A cone ''X'' is said to be '''salient''' if it does not contain any pair of opposite nonzero vectors; that is, if and only if ''C''<math>\cap</math>(-''C'') <math>\subseteq</math> {'''0'''}.

===Convex cone===
A [[convex cone]] is a cone that is closed under [[conic combination]]s, i.e. if and only if α''x'' + β''y'' belongs to ''C'' for any non-negative scalars α, β.

===Affine cone===
If ''C'' - ''v'' is a cone for some ''v'' in ''V'',
then ''C'' is said to be an ('''affine''') '''cone with vertex''' ''v''. More commonly, in [[algebraic geometry]], the term '''affine cone''' over a [[projective variety]] ''X'' in '''P'''''V'' is the [[affine variety]] in ''V'' given as the preimage of ''X'' under the quotient map
:<math>V\setminus\{0\}\to \mathbf{P}V.</math>

===Proper cone===
The term '''proper cone''' is variously defined, depending on the context. It often means a salient and convex cone, or a cone that is contained in an open [[Half-space (geometry)|halfspace]] of ''V''.

==Properties==
===Boolean, additive and linear closure===
Linear cones are closed under [[Boolean operation]]s ([[Intersection (set theory)|set intersection]], [[set union|union]], and [[set complement|complement]]). They are also closed under addition (if ''C'' and ''D'' are cones, so is ''C'' + ''D'') and arbitrary [[linear map]]s. In particular, if ''C'' is a cone, so is its '''opposite cone''' -''C''.

===Spherical section and projection===
Let |·| be any [[norm (mathematics)|norm]] for ''V'', with the property that the norm of any vector is a scalar of ''V''. Let ''S'' be the unit-norm [[sphere]] of ''V'', that is, the set
:<math>S = \{\, x \in V\;:\; |x| = 1 \,\}</math>
By definition, a nonzero vector ''x'' belongs to a cone ''C'' of ''V'' if and only if the unit-norm vector ''x''/|''x''| belongs to ''C''. Therefore, a blunt (or pointed) cone ''C'' is completely specified by its [[central projection]] onto ''S''; that is, by the set
:<math>C' = \bigg\{\, \frac{x}{|x|} \;:\; x \in C \wedge x \neq \mathbf{0} \,\bigg\}</math>
It follows that there is a [[bijection|one-to-one correspondence]] between blunt (or pointed) cones and subsets of ''S''.
Indeed, the central projection ''C' '' is simply the '''spherical section''' of ''C'', the set ''C''<math>\cap</math>''S'' of its unit-norm elements.

A cone ''C'' is '''closed''' with respect to the norm |·| if it is a [[closed (topology)|closed set]] in the [[topology]] induced by that norm. That is the case if and only if ''C'' is pointed and its spherical section is a closed subset of ''S''.

Note that the cone ''C'' is salient if and only if its spherical section does not contain two opposite vectors; that is, ''C' ''<math>\cap</math>(-''C' '') = {}.

== See also ==
*[[Cone (disambiguation)]]
**[[Cone (geometry)]]
**[[Cone (topology)]]
**[[Convex cone]]
*[[Ordered group]] with the concept of the "positive cone"
*[[Ordered vector space]]

== References ==

{{Functional Analysis}}

[[Category:Geometric shapes]]
[[Category:Linear algebra]]

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