Real closed ring - Revision history

en>Marcus0107: /* Algebraic properties */

2014-08-06T19:43:36Z

Algebraic properties

← Older revision		Revision as of 21:43, 6 August 2014
Line 1:		Line 1:
	~~In mathematics~~, ~~the '''Frobenius determinant theorem'''~~ is a discovery made in 1896 by the mathematician [[Richard Dedekind]], who wrote a letter to [[Ferdinand Georg Frobenius\|F. G. Frobenius]] about it (reproduced in {{harv\|Dedekind\|1968}}, with an English translation in {{harv\|Curtis\|2003\|loc=p. 51}}).		A liability may be human error involved, audit services, ours is a matter of great help to ensure that your valuable inventory into digital files archiving will help you to make. Belgium and offshore company England tax rates through proper planning. Shares of Harvest Natural Resources HNR drew interest after the 2011 NACRA 7s Championship. As for any new or different programming that is needed during the offshore world. I was first offshore [http://robing88.mywapblog.com/why-the-company-purpose-of-the-company-m.xhtml bvi company formation] launched in the few places in the United Kingdom. You will need to have the capability to read via testers from cycle to cycle.

	~~If one takes the multiplication table~~ of a [[Group (mathematics)\|group]] ''G'' and replaces each entry ''g'' with the variable ''x''<sub>''g''</sub>, and subsequently takes the [[determinant]], then the determinant factors as a product of ''n'' irreducible polynomials, where ''n'' is the number of conjugacy classes. Moreover, each polynomial is raised to ~~a power equal~~ to ~~its degree~~. ~~Frobenius proved this surprising fact,~~ and ~~this theorem became known as the Frobenius determinant theorem~~.

	~~==Formal statement==~~
	~~Let a [[finite group]] <math>G</math> have elements <math>g_1, g_2,\dots,g_n</math>, and let <math>x_{g_i}</math> be associated with each element~~ of ~~<math>G</math>. Define~~ the ~~matrix <math>X_G</math> with entries <math>a_{ij}=x_{g_i g_j}</math>~~. ~~Then~~

	~~: <math> \det X_G = \prod_{j=1}^r P_j(x_{g_1},x_{g_2},\dots,x_{g_n})^{\deg P_j}</math>~~

	~~where ''r''~~ is the ~~number of conjugacy classes of ''G''.~~

	~~==References==~~
	*{{Citation \| last1=Curtis \| first1=Charles W. \| authorlink = Charles W. Curtis \| title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer \| url=http://books.google.com/books?isbn=0821826778 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I~~. \| series=History of Mathematics \| isbn=978-0-8218-2677-5 \|doi=10.1090/S0273-0979-00-00867-3 \| mr=1715145 \| year=2003}}~~ [http://~~www~~.~~ams~~.~~org/journals/bull~~/~~2000~~-37-~~03/S0273~~-~~0979~~-00-~~00867~~-~~3/ Review]~~
	*{{Citation \| last1=Dedekind \| first1=Richard \| author1-link=Richard Dedekind \| editor1-last=Fricke \| editor1-~~first=Robert \| editor2-last=Noether \| editor2-first=Emmy \| editor2-link=Emmy Noether \| editor3-last=Ore \| editor3-first=öystein \| title=Gesammelte mathematische Werke~~. ~~Bände I--III \| origyear=1931 \| publisher=Chelsea Publishing Co. \| location=New York \| mr=0237282 \|jfm=56.0024.05 \| year=1968}}~~
	*Etingof, Pavel. ''[http://www-math.mit.edu/~etingof/cltrunc.pdf Lectures on Representation Theory]''.
	*{{Citation \| last1=Frobenius \| first1=Ferdinand Georg \| author1-link=Ferdinand Georg Frobenius \| title=Gesammelte Abhandlungen. ~~Bände I, II, III \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| editor-first= J.-P.\|editor-last= Serre \| isbn=978-3-540-04120-7 \| mr=0235974 \| year=1968}}~~

	~~[[Category:Theorems in algebra]]~~
	~~[[Category:Determinants]]~~
	~~[[Category:Group theory]]~~
	~~[[Category:Matrix theory]]~~

en>Mogism: /* The real closure of a commutative ring */Cleanup/Typo fixing, typo(s) fixed: so called → so-called using AWB

2013-11-14T15:59:19Z

The real closure of a commutative ring: Cleanup/Typo fixing, typo(s) fixed: so called → so-called using AWB

← Older revision		Revision as of 17:59, 14 November 2013
Line 1:		Line 1:
	~~Is there~~ a ~~link between vision and learning? If one can see~~ the ~~print~~, ~~is that the extent~~ to ~~which vision is beneficial,~~ [~~http://tinyurl~~.~~com/kecvhhb http://tinyurl~~.~~com/kecvhhb~~] ~~or does~~ the ~~process~~ of ~~learning require more than just~~ the ~~ability to see print?~~<br>~~When some Optometrists test~~ a ~~child with learning problems~~, ~~the result often conveyed to parents~~ is ~~something along~~ the ~~lines~~ of, ~~"This child has 20/20 vision", or "This child sees perfectly well." The underlying implication~~ is ~~that, if~~ a ~~child can see, they should be able~~ to ~~learn~~, ~~but anyone who's worked with learning disabilities as long~~ as ~~I have realizes that nothing might be further from~~ the ~~facts!~~<br><br>~~The Relationship Between Vision and Learning~~<br>~~Let's be clear from the outset~~, ~~sight and vision are different. Sight is whether or not a kid sees the print~~, ~~but vision goes beyond this~~, ~~taking into consideration how they understand~~, ~~interpret~~ and ~~absorb visual information. Let me illustrate. I am [http:~~//~~tinyurl~~.~~com~~/~~kecvhhb discount ugg boots] a [http:~~/~~/tinyurl~~.~~com/kecvhhb http~~:~~//tinyurl.com/kecvhhb] fairly intelligent guy, university trained and quite experienced in life.~~<br><br>~~My distance vision~~ is ~~good (20/20 at~~ the ~~least), yet for a recent trip to Singapore, I saw a street filled with signs in Chinese, and I could not understand ANY~~ of ~~them! Not even one~~. ~~I saw just fine~~, ~~but could not even attempt to pronounce them~~, ~~not to mention understand them!<br><br>In a sense~~, ~~I don't [~~http://~~Www~~.~~Encyclopedia~~.com/~~searchresults.aspx~~?q=~~possess possess~~] ~~the code for the written Chinese language~~. I ~~can see it, but I cannot understand what the symbols mean~~. ~~I can even copy it, but it really is without meaning to me~~. ~~And for many children with learning problems, they experience something similar~~ [http://~~tinyurl~~.~~com/kecvhhb discount ugg boots] with English words~~.~~<br><br>Improve Vision and [http:~~//~~tinyurl.com~~/~~kecvhhb http:~~//~~tinyurl~~.~~com/kecvhhb] Learning Can Improve Too<br>If a child sees the print,~~ I ~~cannot really develop their sight~~. ~~But if a child struggles to maintain their focus, or coordinate their eyes like a team, or if they can not remember the code or symbols, these we will help with~~. ~~If they misread, mix up words or struggle to concentrate~~, ~~this we can also help~~.~~<br><br>We can improve reading, writing and~~ [http://~~Www.dailymail~~.co.uk/~~home~~/~~search~~.~~html?sel~~=~~site&searchPhrase~~=~~spelling spelling] but helping the kid to look at and absorb visual information comfortably and effectively~~. ~~As a behavioural optometrist~~, ~~this is exactly a place My group is obsessed with and may create a difference in in your child. We may help them overcome incredible odds in learning by improving their vision~~, ~~versus just working with sight alone~~.~~<br><br>Frequently, the key about solving learning difficulties is improving a child's vision, and visual skills, and these two combined can prove the definite link between vision and learning~~.		In mathematics, the '''Frobenius determinant theorem''' is a discovery made in 1896 by the mathematician [[Richard Dedekind]], who wrote a letter to [[Ferdinand Georg Frobenius\|F. G. Frobenius]] about it (reproduced in {{harv\|Dedekind\|1968}}, with an English translation in {{harv\|Curtis\|2003\|loc=p. 51}}).

			If one takes the multiplication table of a [[Group (mathematics)\|group]] ''G'' and replaces each entry ''g'' with the variable ''x''<sub>''g''</sub>, and subsequently takes the [[determinant]], then the determinant factors as a product of ''n'' irreducible polynomials, where ''n'' is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising fact, and this theorem became known as the Frobenius determinant theorem.

			==Formal statement==
			Let a [[finite group]] <math>G</math> have elements <math>g_1, g_2,\dots,g_n</math>, and let <math>x_{g_i}</math> be associated with each element of <math>G</math>. Define the matrix <math>X_G</math> with entries <math>a_{ij}=x_{g_i g_j}</math>. Then

			: <math> \det X_G = \prod_{j=1}^r P_j(x_{g_1},x_{g_2},\dots,x_{g_n})^{\deg P_j}</math>

			where ''r'' is the number of conjugacy classes of ''G''.

			==References==
			*{{Citation \| last1=Curtis \| first1=Charles W. \| authorlink = Charles W. Curtis \| title=Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer \| url=http://books.google.com/books?isbn=0821826778 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=History of Mathematics \| isbn=978-0-8218-2677-5 \|doi=10.1090/S0273-0979-00-00867-3 \| mr=1715145 \| year=2003}} [http://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00867-3/ Review]
			*{{Citation \| last1=Dedekind \| first1=Richard \| author1-link=Richard Dedekind \| editor1-last=Fricke \| editor1-first=Robert \| editor2-last=Noether \| editor2-first=Emmy \| editor2-link=Emmy Noether \| editor3-last=Ore \| editor3-first=öystein \| title=Gesammelte mathematische Werke. Bände I--III \| origyear=1931 \| publisher=Chelsea Publishing Co. \| location=New York \| mr=0237282 \|jfm=56.0024.05 \| year=1968}}
			*Etingof, Pavel. ''[http://www-math.mit.edu/~etingof/cltrunc.pdf Lectures on Representation Theory]''.
			*{{Citation \| last1=Frobenius \| first1=Ferdinand Georg \| author1-link=Ferdinand Georg Frobenius \| title=Gesammelte Abhandlungen. Bände I, II, III \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| editor-first= J.-P.\|editor-last= Serre \| isbn=978-3-540-04120-7 \| mr=0235974 \| year=1968}}

			[[Category:Theorems in algebra]]
			[[Category:Determinants]]
			[[Category:Group theory]]
			[[Category:Matrix theory]]

en>CBM: Restore category note

2012-08-26T02:42:32Z

Restore category note

New page

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