Volume fraction - Revision history

84.226.178.205 at 21:15, 19 October 2014

2014-10-19T21:15:17Z

← Older revision		Revision as of 23:15, 19 October 2014
Line 1:		Line 1:
	~~Hello and welcome~~. ~~My name~~ is ~~Figures Wunder~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my spouse reside~~. ~~He utilized to be unemployed but now he~~ is a meter reader. ~~One of the extremely very best issues~~ in ~~the globe for him is to gather badges~~ but ~~he is struggling~~ to ~~find time for it~~.<br><br>~~Look at~~ my ~~page ...~~ [http://~~3bbc~~.~~com~~/~~index.php?do=~~/~~profile~~-~~548128/info~~/ ~~at home std testing~~]		Nice to satisfy [http://support.kaponline.com/entries/46228144-Endocrine-Illnesses-Canines-Part-1 at home std test] you, my title at home std test is Numbers Held though [http://www.yellowpages.com/hialeah-fl/free-std-testing-centers I don't] really like becoming called like that. His wife doesn't like it the way he does but what he really likes performing over the counter std test is to do aerobics and he's been performing it for quite a whilst. [http://www.ubi-cation.com/ubication/node/6056 www.ubi-cation.com] My working day job is a meter reader. Her husband and her reside in Puerto Rico but she will have to move one working day or home std test kit an [http://www.careresource.org/hivaids/testing-hours-locations/ additional].<br><br>Feel free to surf to my homepage - [http://mcb-law.net/candidiasis-tips-you-have-to-remember/ http://mcb-law.net/]

en>RolfSander: for consistency, "i" is now always used inside the sum, and "j" outside

2014-03-04T23:27:47Z

for consistency, "i" is now always used inside the sum, and "j" outside

← Older revision		Revision as of 01:27, 5 March 2014
Line 1:		Line 1:
	~~In [[mathematics]], the '''Herzog–Schönheim conjecture''' is a combinatorial problem in the area of [[group theory]], posed by Marcel Herzog~~ and ~~Jochanan Schönheim in 1974~~.~~<ref>{{citation~~		Hello and welcome. My name is Figures Wunder. South Dakota is exactly where me and my spouse reside. He utilized to be unemployed but now he is a meter reader. One of the extremely very best issues in the globe for him is to gather badges but he is struggling to find time for it.<br><br>Look at my page ... [http://3bbc.com/index.php?do=/profile-548128/info/ at home std testing]
	~~\| last1 = Herzog \| first1 = M~~.
	~~\| last2 = Schönheim \| first2 = J.~~
	~~\| journal = Canadian Mathematical Bulletin~~
	~~\| page = 150~~
	~~\| title = Research problem No. 9~~
	~~\| volume = 17~~
	~~\| year = 1974}}. As cited by {{harvtxt\|Sun\|2004}}.</ref>~~

	~~Let <math>G</math> be a [[group (mathematics)\|group]],~~ and ~~let~~

	~~:<math>A=\{a_1G_1,\ \ldots,\ a_kG_k\}</math>~~

	~~be a finite system of left [[coset]]s of [[subgroup]]s~~
	~~<math>G_1,\ldots,G_k</math> of <math>G</math>~~.

	~~Herzog and Schönheim conjectured~~
	~~that if <math>A</math> forms a [[Partition of a set\|partition]] of <math>G</math>~~
	~~with <math>k>1</math>,~~
	~~then the (finite) indices <math>[G:G_1],\ldots,[G:G_k]</math> cannot~~ be ~~distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if <math>H</math>~~ is ~~any subgroup of <math>G</math>~~
	~~with [[Index of~~ a ~~subgroup\|index]] <math>k=[G:H]<\infty</math> then <math>G</math> can be partitioned into <math>k</math> left cosets of <math>H</math>~~.

	~~==Subnormal subgroups==~~
	~~In 2004 [[Zhi-Wei Sun]] proved an extended version~~
	of the ~~Herzog–Schönheim conjecture~~ in the ~~case where~~ <~~math~~>~~G_1,\ldots,G_k~~<~~/math> are [[subnormal subgroup\|subnormal]] in <math>G</math~~>.~~<ref>{{citation~~
	~~\| last = Sun \| first = Zhi-Wei \| authorlink = Sun Zhiwei~~
	~~\| arxiv = math/0306099~~
	~~\| doi = 10~~.~~1016/S0021-8693(03)00526-X~~
	~~\| issue = 1~~
	~~\| journal = Journal of Algebra~~
	~~\| mr = 2032455~~
	~~\| pages = 153–175~~
	~~\| title = On the Herzog-Schönheim conjecture for uniform covers of groups~~
	~~\| volume = 273~~
	~~\| year = 2004}}~~.~~</ref> A basic lemma in Sun's proof states that if <math>G_1,\ldots,G_k</math> are subnormal and of finite index in <math>G</math>, then~~

	~~:<math>\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg\|\ \prod_{i=1}^k[G:G_i]</math>~~

	~~and hence~~

	~~:<math>P\bigg(\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg)~~
	~~=\bigcup_{i=1}^kP(~~[G:~~G_i]),<~~/~~math>~~

	~~where <math>P(n)<~~/~~math> denotes the set of [[Prime number\|prime]]~~
	~~[[divisor]]s of <math>n<~~/~~math>~~.

	=~~=Mirsky–Newman theorem==~~
	~~When <math>G<~~/~~math> is the additive group <math>\Z<~~/~~math> of integers, the cosets of <math>G<~~/~~math> are the [[arithmetic progression]]s.~~
	In this case, the Herzog–Schönheim conjecture states that every [[covering system]], a family of arithmetic progressions that together cover all the integers, must either cover some integers more than once or include at least one pair of progressions that have the same difference as each other. This result was conjectured in 1950 by [[Paul Erdős]] and proved soon thereafter by [[Leon Mirsky]] and [[Donald J. Newman]]. However, Mirsky and Newman never published their proof. The same proof was also found independently by [[Harold Davenport]] and [[Richard Rado]].<ref name="soifer">{{citation
	~~\| last = Soifer \| first = Alexander \| author-link = Alexander Soifer~~
	~~\| isbn = 978-0-387-74640-1~~
	~~\| location = New York~~
	~~\| publisher = Springer~~
	~~\| title = The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators~~
	~~\| year = 2008~~
	~~\| contribution = Chapter 1. A story of colored polygons and arithmetic progressions~~
	~~\| pages = 1–9}}.</ref>~~

	In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a [[regular polygon]] are colored in such a way that every color class itself forms the vertices of a regular polygon. Then, there exist two color classes that form congruent polygons.<ref name="soifer"/>

	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Herzog-Schonheim conjecture}}~~
	~~[[Category:Group theory]]~~
	~~[[Category:Conjectures]~~]

en>TAB: changed SI unit

2014-01-27T18:07:03Z

changed SI unit

← Older revision		Revision as of 20:07, 27 January 2014
Line 1:		Line 1:
	~~Hello~~. ~~Let me introduce the author~~. ~~Her name is Refugia Shryock~~. ~~North Dakota is her beginning location but she~~ ~~at home std testing will have to~~ [~~http~~://~~www~~.~~care2~~.~~com~~/~~greenliving~~/~~5-myths-about-herpes.html transfer~~] ~~one~~ [~~http~~://~~www.hotporn123.com~~/~~user~~/~~RWhiteman hotporn123~~.~~com~~] ~~day or another. He is truly fond~~ of ~~doing ceramics but he is having difficulties to discover time for it~~. He ~~[http:~~//~~www~~.~~streaming.iwarrior.net~~/~~blog~~/~~258268 std home test~~] ~~used to be unemployed over the counter std test but now he is a meter reader.~~<br><br>~~Look home std test into my website~~ [~~http~~://~~www~~.~~zavodpm.ru~~/~~blogs~~/~~ashelykeplerlkws~~/~~9653~~-~~valuable~~-~~guidance~~-~~successfully~~-~~treating~~-~~candida~~-~~albicans home std test kit~~]		In [[mathematics]], the '''Herzog–Schönheim conjecture''' is a combinatorial problem in the area of [[group theory]], posed by Marcel Herzog and Jochanan Schönheim in 1974.<ref>{{citation
			\| last1 = Herzog \| first1 = M.
			\| last2 = Schönheim \| first2 = J.
			\| journal = Canadian Mathematical Bulletin
			\| page = 150
			\| title = Research problem No. 9
			\| volume = 17
			\| year = 1974}}. As cited by {{harvtxt\|Sun\|2004}}.</ref>

			Let <math>G</math> be a [[group (mathematics)\|group]], and let

			:<math>A=\{a_1G_1,\ \ldots,\ a_kG_k\}</math>

			be a finite system of left [[coset]]s of [[subgroup]]s
			<math>G_1,\ldots,G_k</math> of <math>G</math>.

			Herzog and Schönheim conjectured
			that if <math>A</math> forms a [[Partition of a set\|partition]] of <math>G</math>
			with <math>k>1</math>,
			then the (finite) indices <math>[G:G_1],\ldots,[G:G_k]</math> cannot be distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if <math>H</math> is any subgroup of <math>G</math>
			with [[Index of a subgroup\|index]] <math>k=[G:H]<\infty</math> then <math>G</math> can be partitioned into <math>k</math> left cosets of <math>H</math>.

			==Subnormal subgroups==
			In 2004 [[Zhi-Wei Sun]] proved an extended version
			of the Herzog–Schönheim conjecture in the case where <math>G_1,\ldots,G_k</math> are [[subnormal subgroup\|subnormal]] in <math>G</math>.<ref>{{citation
			\| last = Sun \| first = Zhi-Wei \| authorlink = Sun Zhiwei
			\| arxiv = math/0306099
			\| doi = 10.1016/S0021-8693(03)00526-X
			\| issue = 1
			\| journal = Journal of Algebra
			\| mr = 2032455
			\| pages = 153–175
			\| title = On the Herzog-Schönheim conjecture for uniform covers of groups
			\| volume = 273
			\| year = 2004}}.</ref> A basic lemma in Sun's proof states that if <math>G_1,\ldots,G_k</math> are subnormal and of finite index in <math>G</math>, then

			:<math>\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg\|\ \prod_{i=1}^k[G:G_i]</math>

			and hence

			:<math>P\bigg(\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg)
			=\bigcup_{i=1}^kP([G:G_i]),</math>

			where <math>P(n)</math> denotes the set of [[Prime number\|prime]]
			[[divisor]]s of <math>n</math>.

			==Mirsky–Newman theorem==
			When <math>G</math> is the additive group <math>\Z</math> of integers, the cosets of <math>G</math> are the [[arithmetic progression]]s.
			In this case, the Herzog–Schönheim conjecture states that every [[covering system]], a family of arithmetic progressions that together cover all the integers, must either cover some integers more than once or include at least one pair of progressions that have the same difference as each other. This result was conjectured in 1950 by [[Paul Erdős]] and proved soon thereafter by [[Leon Mirsky]] and [[Donald J. Newman]]. However, Mirsky and Newman never published their proof. The same proof was also found independently by [[Harold Davenport]] and [[Richard Rado]].<ref name="soifer">{{citation
			\| last = Soifer \| first = Alexander \| author-link = Alexander Soifer
			\| isbn = 978-0-387-74640-1
			\| location = New York
			\| publisher = Springer
			\| title = The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators
			\| year = 2008
			\| contribution = Chapter 1. A story of colored polygons and arithmetic progressions
			\| pages = 1–9}}.</ref>

			In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a [[regular polygon]] are colored in such a way that every color class itself forms the vertices of a regular polygon. Then, there exist two color classes that form congruent polygons.<ref name="soifer"/>

			==References==
			{{reflist}}

			{{DEFAULTSORT:Herzog-Schonheim conjecture}}
			[[Category:Group theory]]
			[[Category:Conjectures]]

en>ZéroBot: r2.7.1) (Robot: Adding zh:体积分数

2012-08-01T22:48:48Z

r2.7.1) (Robot: Adding zh:体积分数

New page

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← Older revision		Revision as of 01:27, 5 March 2014
Line 1:		Line 1:
	~~In [[mathematics]], the '''Herzog–Schönheim conjecture''' is a combinatorial problem in the area of [[group theory]], posed by Marcel Herzog~~ and ~~Jochanan Schönheim in 1974~~.~~<ref>{{citation~~		Hello and welcome. My name is Figures Wunder. South Dakota is exactly where me and my spouse reside. He utilized to be unemployed but now he is a meter reader. One of the extremely very best issues in the globe for him is to gather badges but he is struggling to find time for it.<br><br>Look at my page ... [http://3bbc.com/index.php?do=/profile-548128/info/ at home std testing]
	~~\| last1 = Herzog \| first1 = M~~.
	~~\| last2 = Schönheim \| first2 = J.~~
	~~\| journal = Canadian Mathematical Bulletin~~
	~~\| page = 150~~
	~~\| title = Research problem No. 9~~
	~~\| volume = 17~~
	~~\| year = 1974}}. As cited by {{harvtxt\|Sun\|2004}}.</ref>~~

	~~Let <math>G</math> be a [[group (mathematics)\|group]],~~ and ~~let~~

	~~:<math>A=\{a_1G_1,\ \ldots,\ a_kG_k\}</math>~~

	~~be a finite system of left [[coset]]s of [[subgroup]]s~~
	~~<math>G_1,\ldots,G_k</math> of <math>G</math>~~.

	~~Herzog and Schönheim conjectured~~
	~~that if <math>A</math> forms a [[Partition of a set\|partition]] of <math>G</math>~~
	~~with <math>k>1</math>,~~
	~~then the (finite) indices <math>[G:G_1],\ldots,[G:G_k]</math> cannot~~ be ~~distinct. In contrast, if repeated indices are allowed, then partitioning a group into cosets is easy: if <math>H</math>~~ is ~~any subgroup of <math>G</math>~~
	~~with [[Index of~~ a ~~subgroup\|index]] <math>k=[G:H]<\infty</math> then <math>G</math> can be partitioned into <math>k</math> left cosets of <math>H</math>~~.

	~~==Subnormal subgroups==~~
	~~In 2004 [[Zhi-Wei Sun]] proved an extended version~~
	of the ~~Herzog–Schönheim conjecture~~ in the ~~case where~~ <~~math~~>~~G_1,\ldots,G_k~~<~~/math> are [[subnormal subgroup\|subnormal]] in <math>G</math~~>.~~<ref>{{citation~~
	~~\| last = Sun \| first = Zhi-Wei \| authorlink = Sun Zhiwei~~
	~~\| arxiv = math/0306099~~
	~~\| doi = 10~~.~~1016/S0021-8693(03)00526-X~~
	~~\| issue = 1~~
	~~\| journal = Journal of Algebra~~
	~~\| mr = 2032455~~
	~~\| pages = 153–175~~
	~~\| title = On the Herzog-Schönheim conjecture for uniform covers of groups~~
	~~\| volume = 273~~
	~~\| year = 2004}}~~.~~</ref> A basic lemma in Sun's proof states that if <math>G_1,\ldots,G_k</math> are subnormal and of finite index in <math>G</math>, then~~

	~~:<math>\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg\|\ \prod_{i=1}^k[G:G_i]</math>~~

	~~and hence~~

	~~:<math>P\bigg(\bigg[G:\bigcap_{i=1}^kG_i\bigg]\ \bigg)~~
	~~=\bigcup_{i=1}^kP(~~[G:~~G_i]),<~~/~~math>~~

	~~where <math>P(n)<~~/~~math> denotes the set of [[Prime number\|prime]]~~
	~~[[divisor]]s of <math>n<~~/~~math>~~.

	=~~=Mirsky–Newman theorem==~~
	~~When <math>G<~~/~~math> is the additive group <math>\Z<~~/~~math> of integers, the cosets of <math>G<~~/~~math> are the [[arithmetic progression]]s.~~
	In this case, the Herzog–Schönheim conjecture states that every [[covering system]], a family of arithmetic progressions that together cover all the integers, must either cover some integers more than once or include at least one pair of progressions that have the same difference as each other. This result was conjectured in 1950 by [[Paul Erdős]] and proved soon thereafter by [[Leon Mirsky]] and [[Donald J. Newman]]. However, Mirsky and Newman never published their proof. The same proof was also found independently by [[Harold Davenport]] and [[Richard Rado]].<ref name="soifer">{{citation
	~~\| last = Soifer \| first = Alexander \| author-link = Alexander Soifer~~
	~~\| isbn = 978-0-387-74640-1~~
	~~\| location = New York~~
	~~\| publisher = Springer~~
	~~\| title = The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators~~
	~~\| year = 2008~~
	~~\| contribution = Chapter 1. A story of colored polygons and arithmetic progressions~~
	~~\| pages = 1–9}}.</ref>~~

	In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a [[regular polygon]] are colored in such a way that every color class itself forms the vertices of a regular polygon. Then, there exist two color classes that form congruent polygons.<ref name="soifer"/>

	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Herzog-Schonheim conjecture}}~~
	~~[[Category:Group theory]]~~
	~~[[Category:Conjectures]~~]