Experimental uncertainty analysis - Revision history

138.51.145.1: /* Example expansion: p = 2 */

2014-11-28T23:47:25Z

Example expansion: p = 2

← Older revision		Revision as of 01:47, 29 November 2014
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	~~In [[mathematics]], more specifically in the [[Representation theory of Lie groups\|representation theory]] of [[reductive Lie group]]s~~, a <~~math~~>~~(\mathfrak{g},K~~)<~~/math~~>~~'''-module''' is an algebraic object, first introduced by [[Harish-Chandra]],~~<~~ref~~>~~Page 73 of {{harvnb\|Wallach\|1988}}<~~/~~ref> used to deal with continuous infinite-dimensional representations using algebraic techniques~~. ~~Harish-Chandra showed that the study of~~ [~~[Unitary representation\|irreducible unitary representations]] of a real reductive Lie group, ''G'', could be reduced to the study of irreducible <math>(\mathfrak{g},K)<~~/~~math>~~-~~modules, where <math>\mathfrak{g}<~~/~~math> is the~~ [~~[Lie algebra~~]~~] of ''G'' and ''K'' is a~~ [~~[maximal compact subgroup]] of ''G''~~.<~~ref~~>~~Page 12 of {{harvnb\|Doran\|Varadarajan\|2000}}~~<~~/ref~~>		Hi, everybody! <br>I'm Italian male ;=). <br>I really like 2 Broke Girls!<br>xunjie 5日ZOL：2013年8月26日13時46分19秒のいずれか1製品は本土日本のオリジナルのライセンスを取得することができますが、
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	~~==Definition==~~		My website; [http://www.mazz.ch/mall/fashionbag.html シャネル財布]
	Let ''G'' be a real Lie group. Let <math>\mathfrak{g}</math> be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra <math>\mathfrak{k}</math>. A <math>(\mathfrak{g},K)</math>-module is defined as follows:<ref>This is James Lepowsky's more general definition, as given in section 3.3.1 of {{harvnb\|Wallach\|1988}}</ref> it is a [[vector space]] ''V'' that is both a [[Lie algebra representation]] of <math>\mathfrak{g}</math> and a [[group representation]] of ''K'' (without regard to the [[topological space\|topology]] of ''K'') satisfying the following three conditions
	~~:1. for any ''v'' ∈ ''V'', ''k'' ∈ ''K'', and ''X'' ∈ <math>\mathfrak{g}</math>~~
	~~::<math>k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v)</math>~~
	~~:2. for any ''v'' ∈ ''V'', ''Kv'' spans a ''finite-dimensional'' subspace of ''V'' on which the action of ''K'' is continuous~~
	~~:3. for any ''v'' ∈ ''V'' and ''Y'' ∈ <math>\mathfrak{k}</math>~~
	~~::<math>\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right\|_{t=0}=Y\cdot v.</math>~~
	~~In the above, the dot, <math>\cdot</math>, denotes both the action of <math>\mathfrak{g}</math> on ''V'' and that of ''K''. The notation Ad(''k'') denotes the~~ [~~[adjoint representation of a Lie group\|adjoint action]] of ''G'' on <math>\mathfrak{g}</math>, and ''Kv'' is the set of vectors <math>k\cdot v</math> as ''k'' varies over all of ''K''.~~

	~~The first condition can be understood as follows~~: ~~if ''G'' is the [[general linear group]] GL(''n'', '''R'''), then <math>\mathfrak{g}<~~/~~math> is the algebra of all ''n'' by ''n'' matrices, and the adjoint action of ''k'' on ''X'' is ''kXk''<sup>−1<~~/~~sup>; condition 1 can then be read as~~
	~~:<math>kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv~~.~~</math>~~
	~~In other words, it is a compatibility requirement among the actions of ''K'' on ''V'', <math>\mathfrak{g}</math> on ''V'', and ''K'' on <math>\mathfrak{g}</math>~~. ~~The third condition is also a compatibility condition, this time between the action of <math>\mathfrak{k}</math> on ''V'' viewed as a sub-Lie algebra of <math>\mathfrak{g}<~~/~~math> and its action viewed as the differential of the action of ''K'' on ''V''.~~

	~~==Notes==~~
	~~<references~~ />

	~~==References==~~
	*{{Citation
	~~\| editor1-last=Doran~~
	~~\| editor1-first=Robert S.~~
	~~\| editor2-last=Varadarajan~~
	~~\| editor2-first=V~~. S.
	~~\| title=The mathematical legacy of Harish-Chandra~~
	~~\| publisher=[[American Mathematical Society\|AMS]]~~
	~~\| series=Proceedings of Symposia in Pure Mathematics~~
	~~\| volume=68~~
	~~\| year=2000~~
	~~\| mr=1767886~~
	~~\| isbn=978-0-8218-1197-9~~
	}}
	*{{Citation
	~~\| last=Wallach~~
	~~\| first=Nolan R.~~
	~~\| title=Real reductive groups I~~
	~~\| year=1988~~
	~~\| publisher=Academic Press~~
	~~\| series=Pure and Applied Mathematics~~
	~~\| volume=132~~
	~~\| mr=0929683~~
	~~\| isbn=978-0-12-732960-4~~
	}}

	~~{{DEFAULTSORT:(G,K)-Module}}~~
	~~[[Category:Representation theory of Lie groups]~~]

72.27.7.8: /* Sensitivity analysis */

2013-11-19T22:55:37Z

Sensitivity analysis

← Older revision		Revision as of 00:55, 20 November 2013
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	~~CMS provides~~ the ~~best platform to create websites that fulfill all the specifications~~ of ~~SEO. The next step~~ is to ~~visit your Word~~ - ~~Press blog dashboard~~. ~~I thought about what would happen by placing~~ a ~~text widget in~~ the ~~sidebar beneath my banner ad~~, ~~and so it went. Word~~ - ~~Press also provides protection against spamming~~, ~~as security~~ is ~~a measure issue. All this is very simple, and~~ the ~~best thing is that it is totally free,~~ and ~~you don~~'~~t need~~ a ~~domain name or web hosting~~. <br><br>		In [[mathematics]], more specifically in the [[Representation theory of Lie groups\|representation theory]] of [[reductive Lie group]]s, a <math>(\mathfrak{g},K)</math>'''-module''' is an algebraic object, first introduced by [[Harish-Chandra]],<ref>Page 73 of {{harvnb\|Wallach\|1988}}</ref> used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of [[Unitary representation\|irreducible unitary representations]] of a real reductive Lie group, ''G'', could be reduced to the study of irreducible <math>(\mathfrak{g},K)</math>-modules, where <math>\mathfrak{g}</math> is the [[Lie algebra]] of ''G'' and ''K'' is a [[maximal compact subgroup]] of ''G''.<ref>Page 12 of {{harvnb\|Doran\|Varadarajan\|2000}}</ref>

	~~Choosing what kind of links you~~'ll be ~~using is~~ a ~~ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines~~. ~~But as expected the level of support you get with them can~~ be ~~hit or miss based on the developer~~'~~s free time and desire. With the free Word - Press blog, you have the liberty to come up~~ with ~~your own personalized domain name~~. ~~From my very own experiences~~, ~~I will let you know why you should choose WPZOOM Live journal templates. By using Word - Press, you can develop very rich, user-friendly and full-functional website.~~ <br><br>~~Saying that~~, ~~despite the launch of Wordpress Express many months ago, there has still been no sign~~ of a ~~Wordpress video tutorial on offer UNTIL NOW. After sending these details, your Word - Press blog will be setup within~~ a ~~few days. Use this section to change many formatting elements. Newer programs allow website owners~~ and ~~internet marketers~~ to ~~automatically and dynamically change words in their content to match~~ the ~~keywords entered by their web visitors in their search queries~~'~~a feat that they cannot easily achieve with older software~~. ~~Search engine optimization pleasant picture~~ and ~~solution links suggest you will have a much better adjust at gaining considerable natural site visitors.~~ <br><br>~~Whether your Word~~ - ~~Press themes~~ is ~~premium or not, but nowadays every theme is designed with widget-ready. High Quality Services~~: ~~These companies help you in creating high quality Word - Press websites~~. ~~When you have almost~~ any ~~issues regarding wherever~~ and ~~also tips on how to use [http~~://~~blbuh~~.ru/~~wordpress_dropbox_backup_115257 backup plugin]~~, it is ~~possible to e mail us on our own web page~~. ~~Some examples of its additional features include; code inserter~~ (~~for use with adding Google Analytics~~, ~~Adsense section targeting etc~~) ~~Webmaster verification assistant~~, ~~Link Mask Generator~~, ~~Robots. Fast Content Update~~ - ~~It's easy to edit or add posts with free Wordpress websites~~. ~~Fortunately~~, ~~Word - Press Customization Service~~ is ~~available these days~~, ~~right from custom theme design~~, ~~to plugin customization~~ and ~~modifying your website, you can take any bespoke service for your Word - Press development project.~~ <br><br>~~You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products and services for your customers~~. ~~In fact portfolio Word - Press themes~~ is a ~~smooth and attractive but considerably flawed Word - Press theme in creating simpler to~~ the ~~photographers or designers to develop~~ a ~~specific internet site showcasing their most current perform since it appear modern~~-~~day and has fantastic typography and large photographs which would develop an attractive wanting portfolio internet site. It can be concluded that white label SEO comprise~~ of ~~a third party who resells a contract involving IT expert or consultant, SEO professional~~ and ~~end user. Web developers and newbies alike will have~~ the ~~ability to extend your web site and fit other incredible functions with out having to spend more. Your topic is going to be~~ the ~~basis~~ of ~~your site~~'~~s name~~.		==Definition==
			Let ''G'' be a real Lie group. Let <math>\mathfrak{g}</math> be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra <math>\mathfrak{k}</math>. A <math>(\mathfrak{g},K)</math>-module is defined as follows:<ref>This is James Lepowsky's more general definition, as given in section 3.3.1 of {{harvnb\|Wallach\|1988}}</ref> it is a [[vector space]] ''V'' that is both a [[Lie algebra representation]] of <math>\mathfrak{g}</math> and a [[group representation]] of ''K'' (without regard to the [[topological space\|topology]] of ''K'') satisfying the following three conditions
			:1. for any ''v'' ∈ ''V'', ''k'' ∈ ''K'', and ''X'' ∈ <math>\mathfrak{g}</math>
			::<math>k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v)</math>
			:2. for any ''v'' ∈ ''V'', ''Kv'' spans a ''finite-dimensional'' subspace of ''V'' on which the action of ''K'' is continuous
			:3. for any ''v'' ∈ ''V'' and ''Y'' ∈ <math>\mathfrak{k}</math>
			::<math>\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right\|_{t=0}=Y\cdot v.</math>
			In the above, the dot, <math>\cdot</math>, denotes both the action of <math>\mathfrak{g}</math> on ''V'' and that of ''K''. The notation Ad(''k'') denotes the [[adjoint representation of a Lie group\|adjoint action]] of ''G'' on <math>\mathfrak{g}</math>, and ''Kv'' is the set of vectors <math>k\cdot v</math> as ''k'' varies over all of ''K''.

			The first condition can be understood as follows: if ''G'' is the [[general linear group]] GL(''n'', '''R'''), then <math>\mathfrak{g}</math> is the algebra of all ''n'' by ''n'' matrices, and the adjoint action of ''k'' on ''X'' is ''kXk''<sup>−1</sup>; condition 1 can then be read as
			:<math>kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.</math>
			In other words, it is a compatibility requirement among the actions of ''K'' on ''V'', <math>\mathfrak{g}</math> on ''V'', and ''K'' on <math>\mathfrak{g}</math>. The third condition is also a compatibility condition, this time between the action of <math>\mathfrak{k}</math> on ''V'' viewed as a sub-Lie algebra of <math>\mathfrak{g}</math> and its action viewed as the differential of the action of ''K'' on ''V''.

			==Notes==
			<references />

			==References==
			*{{Citation
			\| editor1-last=Doran
			\| editor1-first=Robert S.
			\| editor2-last=Varadarajan
			\| editor2-first=V. S.
			\| title=The mathematical legacy of Harish-Chandra
			\| publisher=[[American Mathematical Society\|AMS]]
			\| series=Proceedings of Symposia in Pure Mathematics
			\| volume=68
			\| year=2000
			\| mr=1767886
			\| isbn=978-0-8218-1197-9
			}}
			*{{Citation
			\| last=Wallach
			\| first=Nolan R.
			\| title=Real reductive groups I
			\| year=1988
			\| publisher=Academic Press
			\| series=Pure and Applied Mathematics
			\| volume=132
			\| mr=0929683
			\| isbn=978-0-12-732960-4
			}}

			{{DEFAULTSORT:(G,K)-Module}}
			[[Category:Representation theory of Lie groups]]

206.174.203.250: This article makes no sense without the equations provided in the introduction. Undid revision 474926540 by 66.176.73.89 (talk)

2012-04-20T16:53:11Z

This article makes no sense without the equations provided in the introduction. Undid revision 474926540 by 66.176.73.89 (talk)

New page

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