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en>Wa3frp: Undid revision 636319226 by 217.179.194.98 (talk)RV vandalism

2014-12-02T14:49:28Z

Undid revision 636319226 by 217.179.194.98 (talk)RV vandalism

← Older revision		Revision as of 16:49, 2 December 2014
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en>WadeSimMiser: Reverted edits by 116.251.187.6 (talk) to last revision by ClueBot NG (HG)

2014-02-24T22:46:26Z

Reverted edits by 116.251.187.6 (talk) to last revision by ClueBot NG (HG)

← Older revision		Revision as of 00:46, 25 February 2014
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	In [[representation theory]], a branch of mathematics, the '''Whittaker model''' is a realization of a [[Group representation\|representation]] of a [[reductive algebraic group]] such as ''GL''<sub>2</sub> over a [[finite field\|finite]] or [[local field\|local]] or [[global field]] on a space of functions on the group. It is named after [[E. T. Whittaker]] even though he never worked in this area, because {{harvs\|last=Jacquet\|year1=1966\|year2=1967}} pointed out that for the group SL<sub>2</sub>('''R''') some of the functions involved in the representation are [[Whittaker function]]s.

	[[Irreducible representation]]s without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation [[θ10\|θ<sub>10</sub>]] of the [[symplectic group]] Sp<sub>4</sub> is the simplest example of a degenerate representation.

	~~==Whittaker models for GL<sub>2</sub>==~~		By investing in a premium Word - Press theme, you're investing in the future of your website. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <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. If you beloved this write-up and you would like to obtain extra info regarding [http://shortener.us/backup_plugin_9225338 wordpress backup] kindly go to our page. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. Use this section to change many formatting elements. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. After that the developer adds the unordered list for navigations. <br><br>The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. php file in the Word - Press root folder and look for this line (line 73 in our example):. To do this, you should link your posts to other relevant posts that you've created. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Word - Press is the most popular personal publishing platform which was launched in 2003. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account.

	~~If ''G'' is the [[algebraic group]] ''GL''<sub>2</sub> and '''F''' is~~ a ~~local field~~,
	~~and τ is a fixed non-trivial character of the additive group of~~ '~~''F''' and π is an irreducible representation of ''G''('''F'''), then~~ the ~~Whittaker model for π is a representation π on a space~~ of ~~functions ''f'' on ''G''('''F''') satisfying~~

	~~:<math>f\left(\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix}g\right) = \tau(b)f(g)~~. ~~</math>~~

	~~{{harvtxt\|Jacquet\|Langlands\|1970}} used Whittaker models to assign L~~-~~functions~~ to ~~[[admissible representation]]s of ''GL''<sub>2</sub>~~.

	~~==Whittaker models~~ for ~~GL<sub>''n''</sub>==~~

	~~Let <math>G</math> be~~ the ~~[[general linear~~ group~~]] <math>{\rm GL}_n</math>~~, ~~<math>\psi</math>~~ a ~~smooth complex valued non~~-~~trivial additive character~~ of ~~<math>F</math>~~ and ~~<math>U</math>~~ the ~~subgroup~~ of <~~math~~>~~{\rm GL}_n~~<~~/math~~> ~~consisting~~ of ~~unipotent upper triangular matrices~~. ~~A non-degenerate character~~ on ~~<math>U</math>~~ is of the ~~form~~

	~~:<math>\chi(u)=\psi(\alpha_1x_{12}+\alpha_2x_{23}+\cdots+\alpha_{n-1}x_{n-1n}),</math>~~

	~~for <math>u=(x_{ij})</math> ∈ <math>U</math>~~ and ~~non~~-~~zero~~ <~~math~~>~~\alpha_1~~<~~/math~~>, .~~.., <math>\alpha_{n~~-~~1}<~~/~~math> ∈ <math>F<~~/~~math>~~. ~~If <math>(\pi,V)<~~/~~math> is~~ a ~~smooth representation of <math>G(F)</math>,~~ a ~~'''Whittaker functional'''~~ <~~math~~>~~\lambda~~<~~/math~~> is a ~~continuous linear functional on <math>V</math> such that <math>\lambda~~(~~\pi(u~~)~~v)=\chi(u)\lambda(v)</math> for all <math>u</math> ∈ <math>U</math>~~, ~~<math>v</math> ∈ <math>V</math>~~. '~~''Multiplicity~~ one~~''' states~~ that, ~~for~~ <~~math~~>~~\pi~~<~~/math~~> ~~unitary irreducible,~~ the ~~space~~ of ~~Whittaker functionals has dimension at most equal~~ to ~~one~~.

	~~==Whittaker models for reductive groups==~~

	If '~~'G'' is~~ a ~~split reductive group and ''U'' is~~ the ~~unipotent radical of a Borel subgroup ''B''~~, ~~then a Whittaker model for a representation is an embedding of~~ it ~~into the induced ([[Gelfand–Graev representation\|Gelfand–Graev]]) representation Ind{{su\|b=~~''~~U''\|p=''G''}}(χ)~~, ~~where χ is~~ a ~~non-degenerate character of~~ ''~~U''.~~

	~~==See also==~~

	*[[Gelfand–Graev representation]], roughly the sum of Whittaker models over a ~~finite field~~.
	*[[Kirillov model]]

	~~==References==~~

	*{{Citation \| last1=Jacquet \| first1=Hervé \| title=Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples \| mr=0200390 \| year=1966 \| journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B \| issn=0151-0509 \| volume=262 \| pages=A943––A945}}
	*{{Citation \| last1=Jacquet \| first1=Hervé \| title=Fonctions de Whittaker associées aux groupes de Chevalley \| url=http://www.numdam.org/item?id=BSMF_1967__95__243_0 \| mr=0271275 \| year=1967 \| journal=Bulletin de la Société Mathématique de France \| issn=0037-9484 \| volume=95 \| pages=243–309}}
	*{{Citation \| author2-link=Robert Langlands \| last1=Jacquet \| first1=H. \| last2=Langlands \| first2=Robert P. \| title=Automorphic forms on GL(2) \| url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in ~~Mathematics, Vol. 114 \| doi=10~~.~~1007/BFb0058988 \| mr=0401654 \| year=1970 \| volume=114 \| isbn=978-3-540-04903-6}}~~

	*J. A. Shalika, '~~'The multiplicity one theorem~~ for ~~<math>GL_n</math>'', The Annals of Mathematics, 2nd. Ser~~.~~, Vol. 100, No. 2 (1974), 171-193.~~

	~~[[Category:Representation theory]]~~
	~~[[Category:Automorphic forms]]~~

en>Conquerist: /* Broadband */ add link. wording.

2014-02-02T00:13:46Z

Broadband: add link. wording.

← Older revision		Revision as of 02:13, 2 February 2014
Line 1:		Line 1:
	~~If you are looking for~~ a ~~specific plugin~~, ~~then you can just search for~~ the ~~name~~ of the ~~plugin~~. ~~Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company~~. ~~PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up~~ in ~~browser. Dead links are listed~~ out ~~simply because it will negatively have an influence on~~ the ~~website~~'s ~~search engine rating~~. ~~In the most current edition you can customize your retailer layout~~ and ~~display hues and fonts similar to your site or blog~~. <br><br>~~Word - Press~~ is ~~known as~~ the ~~most popular blogging platform all over the web and is used by millions~~ of ~~blog enthusiasts worldwide~~. ~~Some of~~ the ~~Wordpress development services offered by us are:. Some plugins ask users to match pictures or add numbers,~~ and ~~although effective~~, ~~they appear unprofessional~~ and ~~unnecessary. So, if you are looking for some option to build~~ a ~~giant e~~-~~commerce website, then e-shopping preferable CMS tools will be helpful for you. As soon as you start developing your Word - Press MLM website you~~'~~ll see how straightforward~~ and ~~simple it~~ is ~~to create~~ an ~~online presence~~ for ~~you and the products and services you offer.~~ <br><br>~~You can down load it here at this link: and utilize your FTP software program to upload it~~ to ~~your Word - Press Plugin folder. It was also the very first year that the category of Martial Arts was included in the Parents~~ - ~~Connect nationwide online poll, allowing parents~~ to ~~vote for their favorite San Antonio Martial Arts Academy. For those who have any concerns about where by in addition to the best way to utilize~~ [~~http:~~//~~u.9col.com~~/~~wordpress_backup_plugin_579995 wordpress dropbox backup~~], ~~you'll be able to e mail us from our web~~-~~page. You can now search through~~ the ~~thousands~~ of ~~available plugins to add all kinds~~ of ~~functionality to your Word - Press site~~. ~~To turn the Word~~ - ~~Press Plugin~~ on~~, click Activate on the far right side~~ of the ~~list~~. So, ~~if you are looking online to hire dedicated Wordpress developers~~, ~~India PHP Expert can give~~ a ~~hand you in each and every best possible way~~. <br><br>~~There~~ has ~~been~~ a ~~huge increase in~~ the ~~number~~ of ~~developers releasing free premium Word~~ - ~~Press themes~~ over ~~the years~~. ~~High Quality Services~~: ~~These companies help you in creating high quality Word~~ - ~~Press websites~~. ~~Enterprise, when they plan to hire Word~~ - ~~Press developer resources still PHP~~, ~~My - SQL and watch with great expertise~~ in ~~codebase~~. ~~Fast Content Update~~ - ~~It's easy to edit or add posts with free Wordpress websites~~. ~~Make sure you have the latest versions of all your plugins are updated~~. <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~~. ~~It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area~~. ~~Word - Press can also be quickly extended however improvement API is not as potent as Joomla's~~. ~~Word~~ - ~~Press is the most popular personal publishing platform which was launched in 2003. Your topic is going to be the basis of your site's name~~.		In [[representation theory]], a branch of mathematics, the '''Whittaker model''' is a realization of a [[Group representation\|representation]] of a [[reductive algebraic group]] such as ''GL''<sub>2</sub> over a [[finite field\|finite]] or [[local field\|local]] or [[global field]] on a space of functions on the group. It is named after [[E. T. Whittaker]] even though he never worked in this area, because {{harvs\|last=Jacquet\|year1=1966\|year2=1967}} pointed out that for the group SL<sub>2</sub>('''R''') some of the functions involved in the representation are [[Whittaker function]]s.

			[[Irreducible representation]]s without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation [[θ10\|θ<sub>10</sub>]] of the [[symplectic group]] Sp<sub>4</sub> is the simplest example of a degenerate representation.

			==Whittaker models for GL<sub>2</sub>==

			If ''G'' is the [[algebraic group]] ''GL''<sub>2</sub> and '''F''' is a local field,
			and τ is a fixed non-trivial character of the additive group of '''F''' and π is an irreducible representation of ''G''('''F'''), then the Whittaker model for π is a representation π on a space of functions ''f'' on ''G''('''F''') satisfying

			:<math>f\left(\begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix}g\right) = \tau(b)f(g). </math>

			{{harvtxt\|Jacquet\|Langlands\|1970}} used Whittaker models to assign L-functions to [[admissible representation]]s of ''GL''<sub>2</sub>.

			==Whittaker models for GL<sub>''n''</sub>==

			Let <math>G</math> be the [[general linear group]] <math>{\rm GL}_n</math>, <math>\psi</math> a smooth complex valued non-trivial additive character of <math>F</math> and <math>U</math> the subgroup of <math>{\rm GL}_n</math> consisting of unipotent upper triangular matrices. A non-degenerate character on <math>U</math> is of the form

			:<math>\chi(u)=\psi(\alpha_1x_{12}+\alpha_2x_{23}+\cdots+\alpha_{n-1}x_{n-1n}),</math>

			for <math>u=(x_{ij})</math> ∈ <math>U</math> and non-zero <math>\alpha_1</math>, ..., <math>\alpha_{n-1}</math> ∈ <math>F</math>. If <math>(\pi,V)</math> is a smooth representation of <math>G(F)</math>, a '''Whittaker functional''' <math>\lambda</math> is a continuous linear functional on <math>V</math> such that <math>\lambda(\pi(u)v)=\chi(u)\lambda(v)</math> for all <math>u</math> ∈ <math>U</math>, <math>v</math> ∈ <math>V</math>. '''Multiplicity one''' states that, for <math>\pi</math> unitary irreducible, the space of Whittaker functionals has dimension at most equal to one.

			==Whittaker models for reductive groups==

			If ''G'' is a split reductive group and ''U'' is the unipotent radical of a Borel subgroup ''B'', then a Whittaker model for a representation is an embedding of it into the induced ([[Gelfand–Graev representation\|Gelfand–Graev]]) representation Ind{{su\|b=''U''\|p=''G''}}(χ), where χ is a non-degenerate character of ''U''.

			==See also==

			*[[Gelfand–Graev representation]], roughly the sum of Whittaker models over a finite field.
			*[[Kirillov model]]

			==References==

			*{{Citation \| last1=Jacquet \| first1=Hervé \| title=Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples \| mr=0200390 \| year=1966 \| journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B \| issn=0151-0509 \| volume=262 \| pages=A943––A945}}
			*{{Citation \| last1=Jacquet \| first1=Hervé \| title=Fonctions de Whittaker associées aux groupes de Chevalley \| url=http://www.numdam.org/item?id=BSMF_1967__95__243_0 \| mr=0271275 \| year=1967 \| journal=Bulletin de la Société Mathématique de France \| issn=0037-9484 \| volume=95 \| pages=243–309}}
			*{{Citation \| author2-link=Robert Langlands \| last1=Jacquet \| first1=H. \| last2=Langlands \| first2=Robert P. \| title=Automorphic forms on GL(2) \| url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/JL.html#book \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics, Vol. 114 \| doi=10.1007/BFb0058988 \| mr=0401654 \| year=1970 \| volume=114 \| isbn=978-3-540-04903-6}}

			*J. A. Shalika, ''The multiplicity one theorem for <math>GL_n</math>'', The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.

			[[Category:Representation theory]]
			[[Category:Automorphic forms]]

en>RussBot: Robot: fix links to disambiguation page Hardware

2012-08-28T15:41:24Z

Robot: fix links to disambiguation page Hardware

New page

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