Enumerative combinatorics - Revision history

en>Myasuda: /* References */ added missing diacritic

2014-11-28T15:20:07Z

References: added missing diacritic

← Older revision		Revision as of 17:20, 28 November 2014
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	~~Nice~~ to ~~meet you,~~ my ~~title is Refugia~~. To play baseball is the ~~hobby~~ he will ~~by no means~~ quit doing~~. Managing people has been his working day occupation for a whilst. South Dakota is where I've usually been residing~~.<br><br>~~Feel free to surf to~~ my web ~~page;~~ [http://~~exchange~~.~~craftemergency.org~~/~~thankyous/can-men-be-sexually-harassed-ladies exchange.craftemergency~~.~~org~~]		I am Oscar and I totally dig that name. She is a librarian but she's usually needed her own business. Years in the past we moved to Puerto Rico and my family loves it. To play baseball is the pastime he will never quit doing.<br><br>my web site [http://kard.dk/?p=17207 kard.dk]

en>Joel B. Lewis: Undid revision 596444868 by Olenyash (talk) combinatorics does not belong to analysis

2014-02-21T14:55:40Z

Undid revision 596444868 by Olenyash (talk) combinatorics does not belong to analysis

← Older revision		Revision as of 16:55, 21 February 2014
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	~~In [[mathematics]]~~, ~~especially the field of [[computational group theory]], a '''Schreier vector'''~~ is ~~a tool for reducing the time and space complexity required to calculate [[orbit (group theory)\|orbits]] of a [[permutation group]]~~.		Nice to meet you, my title is Refugia. To play baseball is the hobby he will by no means quit doing. Managing people has been his working day occupation for a whilst. South Dakota is where I've usually been residing.<br><br>Feel free to surf to my web page; [http://exchange.craftemergency.org/thankyous/can-men-be-sexually-harassed-ladies exchange.craftemergency.org]

	~~==Overview==~~

	~~Suppose G~~ is ~~a finite [[Group (mathematics)\|group]] with generating sequence <math>X = \{x_1,x_2,...,x_r\}</math> which [[Group action\|acts]] on~~ the ~~finite set <math>\Omega = \{1,2,~~..~~.,n\}</math>. A common task in computational group theory~~ is ~~to compute the [[Orbit (group theory)\|orbit]] of some element <math>\omega \in \Omega</math> under G~~. ~~At the same time, one can record a Schreier vector for~~ <~~math~~>~~\omega~~<~~/math~~>~~. This vector can then be used~~ to ~~find an element <math>g \in G</math> satisfying <math>\omega^g = \alpha</math>, for any <math>\alpha \in \omega^G</math>. Use of Schreier vectors~~ to ~~perform this requires less storage space and time complexity than storing these g explicitly.~~

	~~==Formal definition==~~

	~~All variables used here are defined in the overview.~~

	~~A Schreier vector for <math>\omega \in \Omega</math> is a vector <math>\mathbf{v} = (v[1],v[2],...,v~~[~~n])</math> such that~~:

	~~# <math>v[\omega] = -1<~~/~~math>~~
	~~# For <math>\alpha \in \omega^G \setminus \{ {\omega} \} , v[\alpha] \in \{1,...,r\}</math> (the manner in which the <math>v[\alpha]<~~/~~math> are chosen will be made clear in the next section)~~
	~~# <math>v[\alpha] = 0</math> for <math>\alpha \notin \omega^G</math>~~

	~~==Use in algorithms==~~

	~~Here we illustrate, using [[pseudocode]], the use of Schreier vectors in two algorithms~~

	* Algorithm to compute the orbit of ''ω'' under ''G'' and the corresponding Schreier vector

	~~:Input: ''ω'' in ''Ω'', <math>X = \{x_1,x_2,~~..~~.,x_r\}<~~/~~math>~~

	~~:for ''i'' in { 0, 1, …, ''n'' }:~~
	~~::set ''v''[''i''] = 0~~

	~~:set ''orbit'' = { ''ω'' }, ''v''[''ω''] = −1~~

	~~:for ''α'' in ''orbit'' and ''i'' in { 1, 2, …, ''r'' }:~~
	~~::if <math>\alpha^{x_i}<~~/~~math> is not in ''orbit'':~~
	~~:::append <math>\alpha^{x_i}</math> to ''orbit''~~
	~~:::set <math>v[\alpha^{x_i}] = i</math>~~

	~~:return ''orbit'', ''v''~~

	* Algorithm to find a ''g'' in ''G'' such that ''ω''<sup>''g''</sup> = ''α'' for some ''α'' in ''Ω'', using the ''v'' from the first algorithm

	~~:Input: ''v'', ''α'', ''X''~~

	~~:if ''v''[''α''] = 0:~~
	~~::return false~~

	~~:set ''g'' = ''e'', and ''k'' = ''v''[''α''] (where ''e'' is the identity element of ''G'')~~

	~~:while ''k'' ≠ −1:~~
	~~::set <math>g = {x_k}g, \alpha = \alpha^{x_k^{~~-~~1}}, k = v[\alpha]</math>~~

	~~:return ''g''~~

	~~==References==~~
	~~{{refimprove\|date=February 2008}}~~
	*{{Citation \| last1=Butler \| first1=G. \| title=Fundamental algorithms for permutation groups \| publisher=[[Springer-~~Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Computer Science \| isbn=978~~-3-~~540~~-~~54955-0 \| id={{MathSciNet \| id = 1225579}} \| year=1991 \| volume=559}}~~
	*{{Citation \| last1=Holt \| first1=Derek F. ~~\| title=A Handbook of Computational Group Theory \| publisher=[[CRC Press]] \| location=London \| isbn=978-1-58488-372-2 \| year=2005}}~~
	*{{Citation \| last1=Seress \| first1=Ákos \| title=Permutation group algorithms \| publisher=[[Cambridge University Press]] \| series=Cambridge Tracts in Mathematics \| isbn=978-0-521-66103-4 \| id={{MathSciNet \| id = 1970241}} \| year=2003 \| volume=152}}



	~~[[Category:Computational group theory]]~~
	~~[[Category:Permutation groups]~~]

en>Michael Hardy: links and copy-editing

2013-10-16T18:23:01Z

links and copy-editing

← Older revision		Revision as of 20:23, 16 October 2013
Line 1:		Line 1:
	~~The author is called Irwin Wunder but it~~'~~s not the most masucline title out there. For many years home std test I~~'~~ve been operating as~~ a ~~payroll clerk. One of the very very best things in~~ the ~~world for me is to do aerobics~~ and ~~now I'm attempting~~ to ~~make cash with std testing at home it~~. ~~For~~ a ~~while~~ [~~http:~~//~~vip~~.~~akwacity~~.~~com~~/~~oxwall~~/~~blogs~~/~~post/10278 at home std testing] std testing at home she's been~~ in ~~[http:~~/~~/Www~~.~~genitalherpestreatment~~.co/~~what-does-genital-herpes-look-like South Dakota~~].<br><br>[~~http:~~/~~/Blog~~.~~Catherinecoaches~~.~~com~~/~~2011~~/11/the~~-hollywood-herpes-tree-celebrities-who-have-herpes-allegedly.html Feel free]~~ to ~~visit my webpage~~ ... ~~over the~~ [~~http~~://~~www.associazioneitalianafotografi.it~~/~~community~~/~~gruppi~~/~~points~~-to-~~know~~-~~when~~-~~confronted~~-~~with-candidiasis/ http://www~~.~~associazioneitalianafotografi.it/community/gruppi/points~~-to-~~know~~-~~when~~-~~confronted~~-~~with~~-~~candidiasis~~] ~~counter std test,~~ [~~http~~:~~//moodle.kspu.karelia.ru/user/view.php?id=23302&course=1 click through the up coming web site~~],		In [[mathematics]], especially the field of [[computational group theory]], a '''Schreier vector''' is a tool for reducing the time and space complexity required to calculate [[orbit (group theory)\|orbits]] of a [[permutation group]].

			==Overview==

			Suppose G is a finite [[Group (mathematics)\|group]] with generating sequence <math>X = \{x_1,x_2,...,x_r\}</math> which [[Group action\|acts]] on the finite set <math>\Omega = \{1,2,...,n\}</math>. A common task in computational group theory is to compute the [[Orbit (group theory)\|orbit]] of some element <math>\omega \in \Omega</math> under G. At the same time, one can record a Schreier vector for <math>\omega</math>. This vector can then be used to find an element <math>g \in G</math> satisfying <math>\omega^g = \alpha</math>, for any <math>\alpha \in \omega^G</math>. Use of Schreier vectors to perform this requires less storage space and time complexity than storing these g explicitly.

			==Formal definition==

			All variables used here are defined in the overview.

			A Schreier vector for <math>\omega \in \Omega</math> is a vector <math>\mathbf{v} = (v[1],v[2],...,v[n])</math> such that:

			# <math>v[\omega] = -1</math>
			# For <math>\alpha \in \omega^G \setminus \{ {\omega} \} , v[\alpha] \in \{1,...,r\}</math> (the manner in which the <math>v[\alpha]</math> are chosen will be made clear in the next section)
			# <math>v[\alpha] = 0</math> for <math>\alpha \notin \omega^G</math>

			==Use in algorithms==

			Here we illustrate, using [[pseudocode]], the use of Schreier vectors in two algorithms

			* Algorithm to compute the orbit of ''ω'' under ''G'' and the corresponding Schreier vector

			:Input: ''ω'' in ''Ω'', <math>X = \{x_1,x_2,...,x_r\}</math>

			:for ''i'' in { 0, 1, …, ''n'' }:
			::set ''v''[''i''] = 0

			:set ''orbit'' = { ''ω'' }, ''v''[''ω''] = −1

			:for ''α'' in ''orbit'' and ''i'' in { 1, 2, …, ''r'' }:
			::if <math>\alpha^{x_i}</math> is not in ''orbit'':
			:::append <math>\alpha^{x_i}</math> to ''orbit''
			:::set <math>v[\alpha^{x_i}] = i</math>

			:return ''orbit'', ''v''

			* Algorithm to find a ''g'' in ''G'' such that ''ω''<sup>''g''</sup> = ''α'' for some ''α'' in ''Ω'', using the ''v'' from the first algorithm

			:Input: ''v'', ''α'', ''X''

			:if ''v''[''α''] = 0:
			::return false

			:set ''g'' = ''e'', and ''k'' = ''v''[''α''] (where ''e'' is the identity element of ''G'')

			:while ''k'' ≠ −1:
			::set <math>g = {x_k}g, \alpha = \alpha^{x_k^{-1}}, k = v[\alpha]</math>

			:return ''g''

			==References==
			{{refimprove\|date=February 2008}}
			*{{Citation \| last1=Butler \| first1=G. \| title=Fundamental algorithms for permutation groups \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Computer Science \| isbn=978-3-540-54955-0 \| id={{MathSciNet \| id = 1225579}} \| year=1991 \| volume=559}}
			*{{Citation \| last1=Holt \| first1=Derek F. \| title=A Handbook of Computational Group Theory \| publisher=[[CRC Press]] \| location=London \| isbn=978-1-58488-372-2 \| year=2005}}
			*{{Citation \| last1=Seress \| first1=Ákos \| title=Permutation group algorithms \| publisher=[[Cambridge University Press]] \| series=Cambridge Tracts in Mathematics \| isbn=978-0-521-66103-4 \| id={{MathSciNet \| id = 1970241}} \| year=2003 \| volume=152}}



			[[Category:Computational group theory]]
			[[Category:Permutation groups]]

en>David Eppstein: definitely not a stub

2012-08-24T06:25:14Z

definitely not a stub

New page

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