Normal basis - Revision history

Deltahedron: /* Primitive normal basis */ disambiguate

2014-06-17T20:09:45Z

Primitive normal basis: disambiguate

← Older revision		Revision as of 22:09, 17 June 2014
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	~~In [[mathematics]]~~, ~~an '''eigenplane''' is a two-[[dimension]]al [[invariant subspace]] in a given [[vector space]]~~. By analogy with the term ''[[eigenvector]]'' for a vector which, when operated on by a [[linear transformation\|linear operator]] is another vector which is a [[scalar (mathematics)\|scalar]] multiple of itself, the term '''''eigenplane''''' can be used to describe a two-dimensional [[plane (mathematics)\|plane]] (a ''2-plane''), such that the operation of a [[linear transformation\|linear operator]] on a vector in the 2-plane always yields another vector in the same 2-plane.		Bricklayer Dee Gohr from Aberdeen, loves ghost hunting, diet and architecture. Intends to give up work and take the family to a lot of the great heritage listed locales on the globe like Chongoni Rock-Art Area.

	A particular case that has been studied is that in which the linear operator is an [[isometry]] ''M'' of the [[hypersphere]] (written ''S<sup>3</sup>'') represented within four-dimensional [[Euclidean space]]:

	~~:<math>M \; [ \mathbf{s} \; \mathbf{t} ] \; = \; [ \mathbf{s} \; \mathbf{t} ] \Lambda_\theta </math>~~

	~~where '''s''' and '''t''' are four-dimensional column vectors~~ and ~~Λ<sub>θ</sub> is a two-dimensional '''eigenrotation''' within the '''eigenplane'''.~~

	In the ~~usual [[eigenvector]] problem, there is freedom to multiply an eigenvector by an arbitrary scalar; in this case there is freedom~~ to ~~multiply by an arbitrary non-zero [[rotation]].~~

	~~This case is potentially physically interesting in the case that the [[shape of the universe]] is~~ a ~~[[multiply connected]] [[3-manifold]], since finding the [[angle]]s~~ of the ~~eigenrotations of a candidate isometry for [[shape of~~ the ~~universe\|topological lensing]] is a way to [[scientific method\|falsify]] such hypotheses.~~

	~~==See also==~~
	* [[Bivector]]
	* [[Plane of rotation]]

	~~==External links==~~
	*[http://arxiv.org/abs/astro-~~ph/0409694 possible relevance of eigenplanes] in [[physical cosmology\|cosmology]]~~
	*[http://cosmo.torun.~~pl/GPLdownload/eigen/ GNU GPL software for calculating eigenplanes]~~
	~~[[Category:Linear algebra]]~~

en>Lichfielder: over GF(p)

2013-04-11T08:29:16Z

over GF(p)

← Older revision		Revision as of 10:29, 11 April 2013
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	~~I woke up another day and realized~~ - ~~I have also been single for~~ a ~~while at~~ the ~~moment and luke bryan concert videos (~~[~~http://www.banburycrossonline.com www.banburycrossonline.com~~]~~) following much bullying from pals I today locate myself opted~~ for on ~~line dating. They promised me that there are plenty~~ of ~~sweet~~, ~~standard and fun people~~ to ~~meet~~, ~~so here goes~~ the ~~pitch!<br>I strive to stay as~~ [~~http://minioasis.com luke bryan concert dates for 2014~~] ~~physically healthy as possible coming to~~ the ~~gymnasium several~~-~~times per week~~. ~~I enjoy my athletics and try to perform or view as many a potential. I~~'~~ll regularly at Hawthorn suits being winter. Note: I~~'~~ve experienced~~ the ~~carnage of fumbling matches at stocktake sales, If you would~~ [~~http~~:~~//imageshack.us/photos/contemplated+shopping contemplated shopping~~] ~~a sport I really don'~~t ~~mind.~~<br>~~My family~~ and ~~pals~~ are ~~amazing~~ and ~~spending some time together at luke brice ([http:~~/~~/lukebryantickets~~.~~iczmpbangladesh~~.~~org~~ This ~~Resource site~~]~~) bar gigs or dishes~~ is ~~definitely imperative. I have never been into clubs as I come to realize that you could not get~~ a ~~nice dialogue against~~ the ~~sound~~. ~~Additionally, I have two unquestionably cheeky~~ [http://~~www~~.~~senatorwonderling.com luke bryan tickets go on sale~~] ~~and really cute canines that are almost always~~ [http://~~www~~.~~Sharkbayte~~.~~com~~/~~keyword~~/~~enthusiastic enthusiastic~~] ~~to meet new folks.<br><br><br><br>Also visit my web page -~~ [~~http~~:~~//lukebryantickets.pyhgy.com Luke Bryan Concert Dates For 2014~~]		In [[mathematics]], an '''eigenplane''' is a two-[[dimension]]al [[invariant subspace]] in a given [[vector space]]. By analogy with the term ''[[eigenvector]]'' for a vector which, when operated on by a [[linear transformation\|linear operator]] is another vector which is a [[scalar (mathematics)\|scalar]] multiple of itself, the term '''''eigenplane''''' can be used to describe a two-dimensional [[plane (mathematics)\|plane]] (a ''2-plane''), such that the operation of a [[linear transformation\|linear operator]] on a vector in the 2-plane always yields another vector in the same 2-plane.

			A particular case that has been studied is that in which the linear operator is an [[isometry]] ''M'' of the [[hypersphere]] (written ''S<sup>3</sup>'') represented within four-dimensional [[Euclidean space]]:

			:<math>M \; [ \mathbf{s} \; \mathbf{t} ] \; = \; [ \mathbf{s} \; \mathbf{t} ] \Lambda_\theta </math>

			where '''s''' and '''t''' are four-dimensional column vectors and Λ<sub>θ</sub> is a two-dimensional '''eigenrotation''' within the '''eigenplane'''.

			In the usual [[eigenvector]] problem, there is freedom to multiply an eigenvector by an arbitrary scalar; in this case there is freedom to multiply by an arbitrary non-zero [[rotation]].

			This case is potentially physically interesting in the case that the [[shape of the universe]] is a [[multiply connected]] [[3-manifold]], since finding the [[angle]]s of the eigenrotations of a candidate isometry for [[shape of the universe\|topological lensing]] is a way to [[scientific method\|falsify]] such hypotheses.

			==See also==
			* [[Bivector]]
			* [[Plane of rotation]]

			==External links==
			*[http://arxiv.org/abs/astro-ph/0409694 possible relevance of eigenplanes] in [[physical cosmology\|cosmology]]
			*[http://cosmo.torun.pl/GPLdownload/eigen/ GNU GPL software for calculating eigenplanes]
			[[Category:Linear algebra]]

en>Helpful Pixie Bot: ISBNs (Build KH)

2012-05-11T23:28:18Z

ISBNs (Build KH)

New page

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