Enstrophy - Revision history

87.10.61.143 at 11:02, 8 January 2015

2015-01-08T11:02:46Z

← Older revision		Revision as of 13:02, 8 January 2015
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	~~{{about\|\|the cyclic decomposition of [[Graph theory\|graphs]]\|Cycle decomposition (graph theory)\|[[cycling]] terminology\|glossary of bicycling}}~~		The author's title is Christy. For years she's been residing in Kentucky but her spouse wants them to move. What me and my family adore is bungee leaping but I've been taking on new things recently. For years she's been working as a travel agent.<br><br>My blog; [http://appin.co.kr/board_Zqtv22/688025 psychics online]

	~~In [[combinatorics\|combinatorial]] [[mathematics]], the '''cycle notation''~~' is ~~a useful convention for writing down a [[permutation]] in terms of its constituent [[cycle (mathematics)\|cycle]]s~~.~~<ref>Fraleigh 2002:89; Hungerford 1997:230</ref> This is also called~~ '~~''circular notation''' and the permutation called a '''cyclic''' or '''circular''' permutation~~.~~<ref>Dehn 1930:19</ref>~~

	~~==Definition==~~
	~~Let <math>S</math> be the set <math>\{1,\dots,n\}, n \in \mathbb{N}</math>,~~ and
	~~:<math> a_1,\ldots,a_k,\quad 1 \leq k \leq n</math>~~

	~~be distinct elements of <math>S</math>. The expression~~
	~~:<math>(a_1\ \ldots\ a_k)</math>~~

	~~denotes the cycle σ whose [[Group action\|action]]~~ is
	~~:<math> a_1\mapsto a_2\mapsto a_3\mapsto \ldots \mapsto a_k \mapsto a_1~~.~~</math>~~
	For ~~each index~~ '~~'i'',~~
	~~:<math>\sigma (a_i) = a_{i+1},</math>~~
	~~where <math>a_{k+1}</math> is taken to mean <math>a_1</math>.~~

	~~There are <math>k</math> different expressions for the same cycle; the following all represent the same cycle:~~

	~~:<math> (a_1\ a_2\ a_3\ \ldots\ a_k) = (a_2\ a_3\ \ldots\ a_k\ a_1) = \cdots = (a_k\ a_1\ a_2\ \ldots\ a_{k-1}).\, </math>~~

	A 1-element cycle such as (3) is the [[identity function\|identity]] permutation.<ref>Hungerford 1997:231</ref> The identity permutation can also be written as an empty cycle, "()".<ref>Johnson 2003:691</ref>

	~~==Permutation as product of cycles==~~
	~~Let <math>\pi</math> be a permutation of <math>S</math>, and let~~
	~~:<math> S_1,\ldots, S_k\subset S,\quad k\in\mathbb{N}</math>~~

	~~be the [[Orbit (group theory)#Orbits and stabilizers\|orbit]]~~s of <math>\pi</math> with more than 1 element. Consider an element <math>S_j</math>, <math>j=1,\ldots,k</math>, let <math>n_j</math> denote the cardinality of <math>S_j</math>,<math>\|S_j\|</math> =<math>n_j</math>. Also, choose an <math>a_{1,j}\in S_j</math>, and define
	~~:<math> a_{i+1,j} = \pi(a_{i,j}),\quad\text{for } 1\leq i<n_j;\quad\text{then also }\pi(a_{n_j,j})=a_{1,j}.\,</math>~~

	~~We can now express <math>\pi</math>~~ as a ~~product of disjoint cycles, namely~~
	~~:<math> \pi = (a_{1,1}\ \ldots a_{n_1,1}) (a_{1,2}\ \ldots\ a_{n_2,2}) \ldots (a_{1,k}\ \ldots\ a_{n_k,k}).\,</math>~~

	Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for [[function composition]]), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.

	~~==Example==~~
	~~Here are the 24 elements of the [[symmetric group]] on <math>\{1,2,3,4\}</math> expressed using the cycle notation, and grouped according to their [[conjugacy class]]es:~~

	::<~~math~~> ~~( )\,~~<~~/math~~>
	~~::<math> (1 2), \;(1 3),\; (1 4),\~~; ~~(2 3),\; (2 4),\; (3 4)</math> (~~[~~[Transposition (mathematics)\|transpositions]])~~
	:~~:<math> (1 2 3),\; (1 3 2),\; (1 2 4),\; (1 4 2),\; (1 3 4),\; (1 4 3),\; (2 3 4),\; (2 4 3)<~~/~~math>~~
	~~::<math> (1 2)(3 4),\;(1 3)(2 4),\; (1 4)(2 3)<~~/~~math>~~
	~~::<math> (1 2 3 4),\; (1 2 4 3),\; (1 3 2 4),\; (1 3 4 2),\; (1 4 2 3),\; (1 4 3 2)</math>~~

	~~==See also==~~
	*[[Cyclic permutation]]
	*[[Cycles and fixed points]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{citation\|title=Algebraic Equations \|first=Edgar \|last=Dehn\|publisher=Dover \|year=1960 \|origyear=1930}}.
	*{{citation\|first=John\|last=Fraleigh\|authorlink=\|title=A first course in abstract algebra\|edition=7th\|year=2003\|publisher=Addison Wesley\|isbn=978-0-201-76390-4\|page=88–90}}.
	*{{citation\|first=Thomas W.\|last=Hungerford\|title=Abstract Algebra: An Introduction\|year=1997\|publisher=Brooks/~~Cole\|isbn=978-0-03-010559-3}}.~~
	*{{citation\|first=James L.\|last=Johnson\|title=Probability and Statistics for Computer Science\|year=2003\|publisher=Wiley Interscience\|isbn=978-0-471-32672-4}}.

	~~{{PlanetMath attribution\|id=2808\|title=cycle notation}}~~

	~~[[Category:Permutations]~~]

en>Michael H 34: Fix link

2013-03-20T12:10:54Z

Fix link

← Older revision		Revision as of 14:10, 20 March 2013
Line 1:		Line 1:
	~~The individual who wrote~~ the ~~article~~ is called ~~Jayson Hirano~~ and ~~he totally digs that title~~. ~~To perform lacross~~ is ~~some thing I really appreciate doing~~. ~~Office supervising~~ is ~~exactly where my main earnings arrives from but I've always needed my own company~~. ~~Alaska~~ is ~~where he's always been residing~~.<br><br>~~Take~~ a ~~look at my web blog~~ [~~http~~://~~netwk~~.~~hannam~~.ac.kr/xe/~~data_2~~/~~85669 best psychic readings~~]		{{about\|\|the cyclic decomposition of [[Graph theory\|graphs]]\|Cycle decomposition (graph theory)\|[[cycling]] terminology\|glossary of bicycling}}

			In [[combinatorics\|combinatorial]] [[mathematics]], the '''cycle notation''' is a useful convention for writing down a [[permutation]] in terms of its constituent [[cycle (mathematics)\|cycle]]s.<ref>Fraleigh 2002:89; Hungerford 1997:230</ref> This is also called '''circular notation''' and the permutation called a '''cyclic''' or '''circular''' permutation.<ref>Dehn 1930:19</ref>

			==Definition==
			Let <math>S</math> be the set <math>\{1,\dots,n\}, n \in \mathbb{N}</math>, and
			:<math> a_1,\ldots,a_k,\quad 1 \leq k \leq n</math>

			be distinct elements of <math>S</math>. The expression
			:<math>(a_1\ \ldots\ a_k)</math>

			denotes the cycle σ whose [[Group action\|action]] is
			:<math> a_1\mapsto a_2\mapsto a_3\mapsto \ldots \mapsto a_k \mapsto a_1.</math>
			For each index ''i'',
			:<math>\sigma (a_i) = a_{i+1},</math>
			where <math>a_{k+1}</math> is taken to mean <math>a_1</math>.

			There are <math>k</math> different expressions for the same cycle; the following all represent the same cycle:

			:<math> (a_1\ a_2\ a_3\ \ldots\ a_k) = (a_2\ a_3\ \ldots\ a_k\ a_1) = \cdots = (a_k\ a_1\ a_2\ \ldots\ a_{k-1}).\, </math>

			A 1-element cycle such as (3) is the [[identity function\|identity]] permutation.<ref>Hungerford 1997:231</ref> The identity permutation can also be written as an empty cycle, "()".<ref>Johnson 2003:691</ref>

			==Permutation as product of cycles==
			Let <math>\pi</math> be a permutation of <math>S</math>, and let
			:<math> S_1,\ldots, S_k\subset S,\quad k\in\mathbb{N}</math>

			be the [[Orbit (group theory)#Orbits and stabilizers\|orbit]]s of <math>\pi</math> with more than 1 element. Consider an element <math>S_j</math>, <math>j=1,\ldots,k</math>, let <math>n_j</math> denote the cardinality of <math>S_j</math>,<math>\|S_j\|</math> =<math>n_j</math>. Also, choose an <math>a_{1,j}\in S_j</math>, and define
			:<math> a_{i+1,j} = \pi(a_{i,j}),\quad\text{for } 1\leq i<n_j;\quad\text{then also }\pi(a_{n_j,j})=a_{1,j}.\,</math>

			We can now express <math>\pi</math> as a product of disjoint cycles, namely
			:<math> \pi = (a_{1,1}\ \ldots a_{n_1,1}) (a_{1,2}\ \ldots\ a_{n_2,2}) \ldots (a_{1,k}\ \ldots\ a_{n_k,k}).\,</math>

			Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for [[function composition]]), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.

			==Example==
			Here are the 24 elements of the [[symmetric group]] on <math>\{1,2,3,4\}</math> expressed using the cycle notation, and grouped according to their [[conjugacy class]]es:

			::<math> ( )\,</math>
			::<math> (1 2), \;(1 3),\; (1 4),\; (2 3),\; (2 4),\; (3 4)</math> ([[Transposition (mathematics)\|transpositions]])
			::<math> (1 2 3),\; (1 3 2),\; (1 2 4),\; (1 4 2),\; (1 3 4),\; (1 4 3),\; (2 3 4),\; (2 4 3)</math>
			::<math> (1 2)(3 4),\;(1 3)(2 4),\; (1 4)(2 3)</math>
			::<math> (1 2 3 4),\; (1 2 4 3),\; (1 3 2 4),\; (1 3 4 2),\; (1 4 2 3),\; (1 4 3 2)</math>

			==See also==
			*[[Cyclic permutation]]
			*[[Cycles and fixed points]]

			==Notes==
			{{reflist}}

			==References==
			*{{citation\|title=Algebraic Equations \|first=Edgar \|last=Dehn\|publisher=Dover \|year=1960 \|origyear=1930}}.
			*{{citation\|first=John\|last=Fraleigh\|authorlink=\|title=A first course in abstract algebra\|edition=7th\|year=2003\|publisher=Addison Wesley\|isbn=978-0-201-76390-4\|page=88–90}}.
			*{{citation\|first=Thomas W.\|last=Hungerford\|title=Abstract Algebra: An Introduction\|year=1997\|publisher=Brooks/Cole\|isbn=978-0-03-010559-3}}.
			*{{citation\|first=James L.\|last=Johnson\|title=Probability and Statistics for Computer Science\|year=2003\|publisher=Wiley Interscience\|isbn=978-0-471-32672-4}}.

			{{PlanetMath attribution\|id=2808\|title=cycle notation}}

			[[Category:Permutations]]

en>Gabbe: changed hyphen (-) to en dash (–) per WP:ENDASH; removed odd "Stub" section

2012-08-26T21:48:08Z

changed hyphen (-) to en dash (–) per WP:ENDASH; removed odd "Stub" section

New page

The individual who wrote the article is called Jayson Hirano and he totally digs that title. To perform lacross is some thing I really appreciate doing. Office supervising is exactly where my main earnings arrives from but I've always needed my own company. Alaska is where he's always been residing.<br><br>Take a look at my web blog [http://netwk.hannam.ac.kr/xe/data_2/85669 best psychic readings]