Strassen algorithm - Revision history

14.139.242.195 at 11:22, 31 December 2014

2014-12-31T11:22:25Z

← Older revision		Revision as of 13:22, 31 December 2014
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en>Bhaveshnande: /* Algorithm */

2014-02-27T13:35:03Z

Algorithm

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	~~In [[mathematics]],~~ a ~~'''Hurwitz quaternion''' (or '''Hurwitz integer''') is a [[quaternion]] whose components are ''either'' all [[integer]]s ''or'' all [[half~~-~~integer]]s (halves of an odd integer; a mixture of integers and half-integers is not allowed)~~. ~~The set of all Hurwitz quaternions is~~		30 seconds a rechargeable battery rrn your wireless gaming controller. You can buy re-chargeable power supplies for much controller. If you want to play video games regularly, you will be diet through a small lots of money in the batteries formerly run your controllers. A [http://www.answers.com/topic/rechargeable+battery rechargeable battery] you will save a lot of funds in the long run.<br><br>When you are locating a definite handle system tough that will use, optimize the places within your activity. The default manage podium might not be on everyone. Some different people prefer a better let you know screen, a set of more sensitive management or simply perhaps an inverted develop. In several video clip gaming, you may manage these from the setting's area.<br><br>For those who have little ones who delight in video games, then you probably know how challenging it really can be always to pull them out from the t. v.. When you have any kind of queries regarding wherever along with how you can work with [http://prometeu.net clash Of clans hack deutsch], it is possible to contact us on our webpage. Their eye can choose to be stuck towards the monitor for hours as these kinds of products play their preferred pastimes. If you want aid regulating your baby's clash of clans Hack time, the pursuing article has a few recommendations for you.<br><br>Workstation games offer entertaining - everybody, and they unquestionably are surely more complicated than Frogger was! And get all you also can out of game titles, use the advice put out here. You are going to find one exciting new world into gaming, and you would wonder how you previously got by without the company!<br><br>Waiting for game of these season editions of significant titles. These come out per august or higher after all the initial headline, but take into account a lot of down-loadable and extra posts which was released in steps once the primary headline. These pastime [http://www.twitpic.com/tag/titles+supply titles supply] a huge amount more bang for your current buck.<br><br>Preserve some money on your company's games, think about following into a assistance in which you can rent payments games from. The charge of these lease accords for the year is regarded as normally under the amount of two video party games. You can preserve the match titles until you hit them and simply pass out them back a lot more and purchase another one.<br><br>It's a nice technique. Breaking the appraisement bottomward into chunks of all their time that accomplish college to be able so that you can bodies (hour/day/week) causes this particular to be accessible to visualize. Everybody comprehends what it appears the same as to accept to hold off on a day. It happens to be additionally actual accessible for tune. If you change your current apperception after and adjudge just that one day should group more, all you accusation to complete is amend 1 value.

	~~:<math>H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a~~,~~b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.</math>~~

	~~It can~~ be ~~confirmed that ''H'' is closed under quaternion multiplication and addition, which makes it~~ a ~~[[subring]]~~ of the ~~[[ring (mathematics)\|ring]] of all quaternions '''H'''~~.

	A ~~'''Lipschitz quaternion''' (or '''Lipschitz integer''') is a quaternion whose components are all~~ [~~[integer~~]]s. ~~The set of all Lipschitz quaternions~~

	:<~~math~~>~~L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}~~<~~/math~~>

	~~forms~~ a ~~subring of~~ the ~~Hurwitz quaternions ''H''~~.

	As a ~~[[group (mathematics)\|group]]~~, ~~''H'' is [[free abelian group\|free abelian]] with generators {(1 + ''i'' + ''j'' + ''k'')/2, ''i'', ''j'', ''k''}. It therefore forms~~ a ~~[[lattice (group)\|lattice]] in '''R'''<sup>4</sup>~~. ~~This lattice is known as~~ the ~~[[F4 lattice\|''F'~~'<~~sub~~>4<~~/sub~~> ~~lattice]] since~~ it is the ~~[[root lattice]]~~ of ~~the~~ [~~[semisimple Lie algebra]] [[F4 (mathematics)\|''F''<sub>4<~~/~~sub>]]~~. ~~The Lipschitz quaternions ''L'' form an index 2 sublattice of ''H''.~~

	~~The [[group of units]] in ''L'' is the order 8 [[quaternion group]~~] ~~''Q'' = {±1~~, ~~±''i'', ±''j'', ±''k''}. The [[group of units]] in ''H''~~ is ~~a nonabelian group of order 24 known as the [[binary tetrahedral group]]~~. ~~The elements of this group include~~ the ~~8 elements~~ of ~~''Q'' along with the 16 quaternions {(±1 ± ''i'' ± ''j'' ± ''k'')/2} where signs may be taken in any combination~~. ~~The quaternion group is a [[normal subgroup]] of the binary tetrahedral group~~ '~~'U''(''H''). The elements~~ of ~~''U''(''H''), which all have norm 1~~, ~~form the vertices of the [[24-cell]] inscribed in the [[3-sphere]].~~

	~~The Hurwitz quaternions form an [[order (ring theory)\|order]] (in the sense of [[ring theory]]) in~~ the ~~[[division ring]] of quaternions with [[rational number\|rational]] components. It is in fact~~ a ~~[[maximal order]]; this accounts~~ for ~~its importance~~. ~~The Lipschitz quaternions~~, ~~which~~ are ~~the~~ more ~~obvious candidate for the idea~~ of ~~an ''integral quaternion''~~, ~~also form an order~~. ~~However, this latter order is not a maximal~~ one, and ~~therefore (as it turns out) less suitable~~ for ~~developing a theory~~ of ~~[[left ideal]]s comparable to that~~ of ~~[[algebraic number theory]]~~. ~~What [[Adolf Hurwitz]] realised~~, ~~therefore, was that this definition~~ of ~~Hurwitz integral quaternion is the better one to operate with. This~~ was ~~one major step~~ in the ~~theory of maximal orders, the other being the remark that they will not, for a non-commutative ring such as '''H''', be unique~~. ~~One therefore needs to fix a maximal order to work with, in carrying over the concept of an~~ [~~[algebraic integer]]~~.

	~~The [[field norm\|(arithmetic, or field) norm~~]~~] of~~ a ~~Hurwitz quaternion, given by~~ <~~math~~>~~a^2+b^2+c^2+d^2~~<~~/math~~>, ~~is always an integer. By~~ a ~~[[Lagrange's four-square theorem\|theorem of Lagrange]] every nonnegative integer~~ can ~~be written as a sum~~ of ~~at most four [[square (algebra)\|squares]]. Thus, every nonnegative integer~~ is the ~~norm~~ of ~~some Lipschitz (or Hurwitz) quaternion~~. ~~A Hurwitz integer is~~ a ~~[[prime element]] if~~ and ~~only if its norm is a [[prime number]]~~.

	~~==See also==~~
	* [[Gaussian integer]]
	* [[Eisenstein integer]]
	* The Lie group [[F4 (mathematics)\|F<~~sub~~>4<~~/sub~~>]]
	* The [[E8 lattice\|E<sub>8</sub> lattice]]

	~~==References==~~
	*John Horton Conway, Derek Alan Smith (~~2003), [http:~~//~~books~~.~~google~~.co.~~uk/books?id=E_HCwwxMbfMC On quaternions~~ and ~~octonions: their geometry~~, ~~arithmetic, and symmetry], A K Peters Ltd~~.~~, ISBN 978-1-56881-134-5~~

	~~[[Category:Quaternions]]~~

117.211.86.74: /* History */

2014-01-07T10:50:45Z

History

← Older revision		Revision as of 12:50, 7 January 2014
Line 1:		Line 1:
			In [[mathematics]], a '''Hurwitz quaternion''' (or '''Hurwitz integer''') is a [[quaternion]] whose components are ''either'' all [[integer]]s ''or'' all [[half-integer]]s (halves of an odd integer; a mixture of integers and half-integers is not allowed). The set of all Hurwitz quaternions is

			:<math>H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.</math>

	~~Individual who wrote~~ the [~~http~~://~~www~~.~~sharkbayte~~.~~com~~/~~keyword~~/~~article article~~] is ~~called Eusebio~~. ~~His friends say it~~'~~s no real for him but what exactly he loves doing~~ is ~~acting and he~~'~~s just lately been doing it~~ for ~~many years~~. ~~Filing has been his profession~~ for a ~~short time. Massachusetts has always been his everyday living place and his wife~~ and ~~kids loves~~ it. Go to ~~his website~~ to ~~discover~~ a ~~out more: http://circuspartypanama~~.~~com~~<br><br>~~Take~~ a ~~look~~ at ~~my blog;~~ [http://~~circuspartypanama~~.~~com clash of clans hack download free~~]		It can be confirmed that ''H'' is closed under quaternion multiplication and addition, which makes it a [[subring]] of the [[ring (mathematics)\|ring]] of all quaternions '''H'''.

			A '''Lipschitz quaternion''' (or '''Lipschitz integer''') is a quaternion whose components are all [[integer]]s. The set of all Lipschitz quaternions

			:<math>L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}</math>

			forms a subring of the Hurwitz quaternions ''H''.

			As a [[group (mathematics)\|group]], ''H'' is [[free abelian group\|free abelian]] with generators {(1 + ''i'' + ''j'' + ''k'')/2, ''i'', ''j'', ''k''}. It therefore forms a [[lattice (group)\|lattice]] in '''R'''<sup>4</sup>. This lattice is known as the [[F4 lattice\|''F''<sub>4</sub> lattice]] since it is the [[root lattice]] of the [[semisimple Lie algebra]] [[F4 (mathematics)\|''F''<sub>4</sub>]]. The Lipschitz quaternions ''L'' form an index 2 sublattice of ''H''.

			The [[group of units]] in ''L'' is the order 8 [[quaternion group]] ''Q'' = {±1, ±''i'', ±''j'', ±''k''}. The [[group of units]] in ''H'' is a nonabelian group of order 24 known as the [[binary tetrahedral group]]. The elements of this group include the 8 elements of ''Q'' along with the 16 quaternions {(±1 ± ''i'' ± ''j'' ± ''k'')/2} where signs may be taken in any combination. The quaternion group is a [[normal subgroup]] of the binary tetrahedral group ''U''(''H''). The elements of ''U''(''H''), which all have norm 1, form the vertices of the [[24-cell]] inscribed in the [[3-sphere]].

			The Hurwitz quaternions form an [[order (ring theory)\|order]] (in the sense of [[ring theory]]) in the [[division ring]] of quaternions with [[rational number\|rational]] components. It is in fact a [[maximal order]]; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an ''integral quaternion'', also form an order. However, this latter order is not a maximal one, and therefore (as it turns out) less suitable for developing a theory of [[left ideal]]s comparable to that of [[algebraic number theory]]. What [[Adolf Hurwitz]] realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. This was one major step in the theory of maximal orders, the other being the remark that they will not, for a non-commutative ring such as '''H''', be unique. One therefore needs to fix a maximal order to work with, in carrying over the concept of an [[algebraic integer]].

			The [[field norm\|(arithmetic, or field) norm]] of a Hurwitz quaternion, given by <math>a^2+b^2+c^2+d^2</math>, is always an integer. By a [[Lagrange's four-square theorem\|theorem of Lagrange]] every nonnegative integer can be written as a sum of at most four [[square (algebra)\|squares]]. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. A Hurwitz integer is a [[prime element]] if and only if its norm is a [[prime number]].

			==See also==
			* [[Gaussian integer]]
			* [[Eisenstein integer]]
			* The Lie group [[F4 (mathematics)\|F<sub>4</sub>]]
			* The [[E8 lattice\|E<sub>8</sub> lattice]]

			==References==
			*John Horton Conway, Derek Alan Smith (2003), [http://books.google.co.uk/books?id=E_HCwwxMbfMC On quaternions and octonions: their geometry, arithmetic, and symmetry], A K Peters Ltd., ISBN 978-1-56881-134-5

			[[Category:Quaternions]]

en>Stevenj: /* Algorithm */ restore sourced information about crossover point that was deleted without discussion

2012-08-18T21:02:29Z

Algorithm: restore sourced information about crossover point that was deleted without discussion

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