Chen model - Revision history

en>Niceguyedc: WPCleaner v1.34 - Repaired 1 link to disambiguation page - (You can help) - Lin Chen

2014-12-31T10:17:14Z

WPCleaner v1.34 - Repaired 1 link to disambiguation page - (You can help) - Lin Chen

← Older revision		Revision as of 12:17, 31 December 2014
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	~~In [[mathematics]], the '''Brauer–Siegel theorem''', named after [[Richard Brauer]] and [[Carl Ludwig Siegel]],~~ is an asymptotic result on the behaviour of [[algebraic number field]]s, obtained by [[Richard Brauer]] and [[Carl Ludwig Siegel]]. It attempts to generalise the results known on the [[class number (number theory)\|class number]]s of [[imaginary quadratic field]]s, to a more general sequence of number fields		The writer is called Araceli Gulledge. Camping is something that I've done for many years. Interviewing is what I do for a residing but I plan on altering it. I currently reside in Arizona but now I'm contemplating other options.<br><br>Feel free to visit my web site [http://Www.gauhaticommercecollege.in/content/dont-allow-auto-repair-best-you extended car warranty]

	~~:<math>K_1, K_2, \ldots.\ </math>~~

	In all cases other than the rational field '''Q''' and imaginary quadratic fields, the [[regulator of a number field\|regulator]] ''R''<sub>''i''</sub> of ''K''<sub>''i''</sub> must be taken into account, because ''K''<sub>i</sub> then has units of infinite order by [[Dirichlet's unit theorem]]. ~~The quantitative hypothesis of the standard Brauer–Siegel theorem~~ is that if '~~'D''<sub>''i''</sub>~~ is ~~the [[discriminant of an algebraic number field\|discriminant]] of ''K'~~'<~~sub>''i''</sub~~>~~, then~~

	: <~~math~~> ~~\frac{[K_i : Q]}{\log\|D_i\|} \~~to ~~0\text{ as }i \to\infty. </math>~~

	~~Assuming that, and the algebraic hypothesis that ''K''<sub>''i''</sub> is a~~ [~~[Galois extension]] of '''Q''', the conclusion is that~~

	: ~~<math> \frac{ \log(h_i R_i) }{ \log\sqrt{\|D_i\|} } \to 1\text{ as }i \to\infty </math>~~

	~~where ''h''<sub>''i''<~~/~~sub> is the class number of ''K''<sub>''i''<~~/~~sub>~~.

	~~This result is [[effective results in number theory\|ineffective]], as indeed was the result on quadratic fields on which it built~~. ~~Effective results~~ in ~~the same direction were initiated in work of [[Harold Stark]] from the early 1970s.~~

	~~==References==~~
	* [[Richard Brauer]], ''On the Zeta-~~Function of Algebraic Number Fields'', ''[[American Journal of Mathematics]]'' 69 (1947), 243–250.~~

	~~{{DEFAULTSORT:Brauer~~-~~Siegel theorem}}~~
	~~[[Category:Analytic number theory]]~~
	~~[[Category:Theorems in algebraic number theory]~~]

en>Quebec99: fixed references

2013-11-25T16:06:44Z

fixed references

← Older revision		Revision as of 18:06, 25 November 2013
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	~~Nestor~~ is the ~~title~~ [~~http~~:~~//www~~.~~frag5.com~~/~~index.php?mod=users&action=view&id=13770 auto warranty] my mothers~~ and ~~fathers gave me but~~ [~~https:~~//~~autoclubsouth.aaa.com~~/~~Automotive/auto_warranty.aspx?nvbar=Automotive:ExtendedWarranty I don~~'t] ~~like when individuals use my complete name~~. ~~Arizona~~ is ~~her beginning location and she will by no means move. The favorite pastime for my~~ [~~http~~:~~//www~~.~~Carwarranty.com~~/ ~~children]~~ and [~~http~~://81.30.~~156~~.~~125/index.php?mod~~=~~users&action~~=~~view&id~~=~~14403 car warranty~~] ~~me is taking part in crochet and now I~~'~~m trying to make money with it. Meter auto warranty~~ [~~http://www.toyotacertified.com/warranty.html reading~~] ~~is exactly where my main income arrives from but soon I~~'~~ll be on my own~~.~~<br><br>My web page~~ :: [~~http~~:~~//Exitus.bplaced.net/index.php?mod=users&action=view&id=139 car warranty companies~~]		In [[mathematics]], the '''Brauer–Siegel theorem''', named after [[Richard Brauer]] and [[Carl Ludwig Siegel]], is an asymptotic result on the behaviour of [[algebraic number field]]s, obtained by [[Richard Brauer]] and [[Carl Ludwig Siegel]]. It attempts to generalise the results known on the [[class number (number theory)\|class number]]s of [[imaginary quadratic field]]s, to a more general sequence of number fields

			:<math>K_1, K_2, \ldots.\ </math>

			In all cases other than the rational field '''Q''' and imaginary quadratic fields, the [[regulator of a number field\|regulator]] ''R''<sub>''i''</sub> of ''K''<sub>''i''</sub> must be taken into account, because ''K''<sub>i</sub> then has units of infinite order by [[Dirichlet's unit theorem]]. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if ''D''<sub>''i''</sub> is the [[discriminant of an algebraic number field\|discriminant]] of ''K''<sub>''i''</sub>, then

			: <math> \frac{[K_i : Q]}{\log\|D_i\|} \to 0\text{ as }i \to\infty. </math>

			Assuming that, and the algebraic hypothesis that ''K''<sub>''i''</sub> is a [[Galois extension]] of '''Q''', the conclusion is that

			: <math> \frac{ \log(h_i R_i) }{ \log\sqrt{\|D_i\|} } \to 1\text{ as }i \to\infty </math>

			where ''h''<sub>''i''</sub> is the class number of ''K''<sub>''i''</sub>.

			This result is [[effective results in number theory\|ineffective]], as indeed was the result on quadratic fields on which it built. Effective results in the same direction were initiated in work of [[Harold Stark]] from the early 1970s.

			==References==
			* [[Richard Brauer]], ''On the Zeta-Function of Algebraic Number Fields'', ''[[American Journal of Mathematics]]'' 69 (1947), 243–250.

			{{DEFAULTSORT:Brauer-Siegel theorem}}
			[[Category:Analytic number theory]]
			[[Category:Theorems in algebraic number theory]]

en>Cydebot: Robot - Speedily moving category Short-rate model to :Category:Short-rate models per CFDS.

2012-05-23T22:20:27Z

Robot - Speedily moving category Short-rate model to Category:Short-rate models per CFDS.

New page

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