92.36.182.139: /* Bosnian development */

2013-11-27T20:39:38Z

Bosnian development

← Older revision		Revision as of 22:39, 27 November 2013
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	~~Adolph~~ is ~~what people call me~~ and ~~my wife doesn~~'~~t care about~~ it ~~at most~~ of. ~~My husband doesn~~'~~t like~~ it the ~~way I do but things~~ i ~~really look foward to~~ is ~~bird keeping and I~~'ll be ~~starting another thing along destinations~~. ~~Production~~ and ~~planning~~ the ~~place I support my family but~~ the ~~promotion never comes~~. He's ~~always loved living in florida~~ and ~~his parents live nearby~~. If ~~you want~~ to ~~read more away my website: http~~://[~~http://Www~~.~~thefreedictionary~~.~~com/archive archive~~].~~org/details/miley_cyrus_maxim~~<br><br>~~Take~~ a ~~look at my web page~~ - [~~http~~://~~archive~~.~~org~~/~~details/miley_cyrus_maxim Miley Cyrus strips naked again~~]		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]]

en>OgreBot: replacing File:DiskMhdGenerator.PNG with commons equivalent File:Disk MHD generator.png (report errors here)

2012-08-28T18:45:35Z

replacing File:DiskMhdGenerator.PNG with commons equivalent File:Disk MHD generator.png (report errors here)

New page

Adolph is what people call me and my wife doesn't care about it at most of. My husband doesn't like it the way I do but things i really look foward to is bird keeping and I'll be starting another thing along destinations. Production and planning the place I support my family but the promotion never comes. He's always loved living in florida and his parents live nearby. If you want to read more away my website: http://[http://Www.thefreedictionary.com/archive archive].org/details/miley_cyrus_maxim<br><br>Take a look at my web page - [http://archive.org/details/miley_cyrus_maxim Miley Cyrus strips naked again]

MHD generator - Revision history

92.36.182.139: /* Bosnian development */

en>OgreBot: replacing File:DiskMhdGenerator.PNG with commons equivalent File:Disk MHD generator.png (report errors here)