Effective dimension - Revision history

en>Rjwilmsi: 10.1137/s0097539703446912

2014-12-12T16:00:42Z

10.1137/s0097539703446912

← Older revision		Revision as of 18:00, 12 December 2014
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	~~Hi there, I am~~ Felicidad ~~Oquendo~~. ~~What~~ she ~~enjoys doing~~ is to perform ~~croquet~~ but she ~~hasn~~'t ~~produced a dime with~~ it. ~~Bookkeeping has been his working day occupation for a while.~~ Arizona ~~has always been my residing place~~ but ~~my wife wants us~~ to move.<br><br>~~Here is~~ my web ~~site; extended auto warranty;~~ [http://~~Www~~.~~Offshoreservicetips~~.~~com~~/?p=~~11834 simply click the following article~~],		Friends call her Felicidad and her husband doesn't like it at all. The thing she adores most is to perform handball but she can't make it her profession. Some time ago I selected to live in Arizona but I need to move for my family. I am a production and distribution officer.<br><br>Also visit my web blog [http://www.teamprefire.de/index.php?mod=users&action=view&id=14578 extended auto warranty]

en>LilHelpa: contraction

2014-02-20T16:25:52Z

contraction

← Older revision		Revision as of 18:25, 20 February 2014
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	~~In mathematics~~, a '''tempered representation''' of a linear [[semisimple Lie group]] is a [[group representation\|representation]] that has a basis whose [[matrix coefficient]]s lie in the [[Lp space\|L<sup>''p''</sup> space]]		Hi there, I am Felicidad Oquendo. What she enjoys doing is to perform croquet but she hasn't produced a dime with it. Bookkeeping has been his working day occupation for a while. Arizona has always been my residing place but my wife wants us to move.<br><br>Here is my web site; extended auto warranty; [http://Www.Offshoreservicetips.com/?p=11834 simply click the following article],

	~~:''L''<sup>2+ε</sup>(''G'')~~

	~~for any ε > 0~~.

	~~==Formulation==~~

	~~This condition, as just given,~~ is ~~slightly weaker than the condition that the matrix coefficients are [[square-integrable]], in other words lie in~~

	~~:''L''<sup>2</sup>(''G''),~~

	~~which would be the definition of a [[discrete series representation]]. If ''G'~~' is a linear semisimple Lie group with a maximal compact subgroup ''K'', an [[admissible representation]] ρ of ''G'' is tempered if the above condition holds for the [[K-finite\|''K''-finite]] matrix coefficients of ρ.

	The definition above is also used for more general groups, such as ''p''-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular [[locally compact group]]s, but on groups with ~~infinite centers such as infinite central extensions of semisimple Lie groups~~ it ~~does not behave well and is usually replaced by a slightly different definition~~.

	Tempered representations on semisimple Lie groups were first defined and studied by [[Harish-Chandra]] (using a different but equivalent definition), who showed that they are exactly the representations needed for the [[Plancherel theorem]]. They were classified by Knapp and Zuckerman, and used by Langlands in the [[Langlands classification]] of [[irreducible representation]]s of a [[reductive Lie group]] ''G'' in terms of the tempered representations of smaller groups.

	~~==History==~~
	Irreducible tempered representations were identified by [[Harish-Chandra]] in his work on harmonic analysis on a [[semisimple Lie group]] as those representations that contribute to the [[Plancherel measure]]. The original definition of a tempered representation, which has certain technical advantages, is that its [[Harish-Chandra character]] should be a "tempered distribution" (see the section about this below). It follows from Harish-Chandra's results that it is equivalent to the more elementary definition given above. Tempered representations also seem to play a fundamental role in the theory of [[automorphic form]]s. This connection was probably first realized by Satake (in the context of the [[Ramanujan-Petersson conjecture]]) and [[Robert Langlands]] and served as a motivation for Langlands to develop his ~~[[Langlands classification\|classification scheme]]~~ for irreducible admissible representations of real and ''p''-adic reductive algebraic groups in terms of the tempered representations of smaller groups. The precise conjectures identifying the place of tempered representations in the automorphic spectrum were formulated later by [[James Arthur (mathematician)\|James Arthur]] and constitute one of the most actively developing parts of the modern theory of automorphic forms.

	~~==Harmonic analysis==~~
	Tempered representations play an important role in the harmonic analysis on [[semisimple Lie group]]s. An [[irreducible representation\|irreducible]] [[unitary representation\|unitary]] representation of a ~~semisimple Lie group <!-- probably OK for other local fields -->''G'' is tempered if and only if it is in the support of the [[Plancherel measure]] of ''G''~~. In other words, tempered representations are precisely the class of representations of ''G'' appearing in the spectral decomposition of L<sup>2</sup> functions on the group (while discrete series representations have a stronger property that an individual representation has a positive spectral measure). This stands in contrast with the situation for abelian and more general solvable Lie groups, where a different class of representations is needed to fully account for the spectral decomposition. This can be seen already in the simplest example of the additive group '''R''' of the real numbers, for which the matrix elements of the irreducible representations do not fall off to 0 at infinity.

	~~In the [[Langlands program]], tempered representations of real Lie groups are those coming from unitary characters of tori by Langlands functoriality.~~

	~~==Examples==~~
	*The [[Plancherel theorem]] for a semisimple Lie group involves representations that are not the [[discrete series]]. This becomes clear already in the case of the group [[SL2(R)\|SL<sub>2</sub>('''R''')]]. The [[principal series representation]]s of SL<sub>2</sub>('''R''') are tempered and account for the spectral decomposition of functions supported on the hyperbolic elements of the group. However, they do not occur discretely in the regular representation of SL<sub>2</sub>('''R''').
	*The two [[limit of discrete series representation]]s of SL<sub>2</sub>('''R''') are tempered but not discrete series (even though they occur "discretely" in the list of irreducible unitary representations).
	*For ''non-semisimple'' Lie groups, representations with matrix coefficients in ''L''<sup>2+ε</sup> do not always suffice for the [[Plancherel theorem]], as shown by the example of the additive group '''R''' of real numbers and the [[Fourier integral]]; in fact, all irreducible unitary representations of '''R''' contribute to the Plancherel measure, but ~~none of them have matrix coefficients in ''L''<sup>2+ε</sup>.~~
	*The [[complementary series representation]]s of SL<sub>2</sub>('''R''') are irreducible unitary representations that are not tempered.
	*The [[trivial representation]] of a group ''G'' is an irreducible unitary representation that is not tempered unless ''G'' is [[compact space\|compact]].

	~~==Classification==~~
	The irreducible tempered representations of a semisimple Lie group were classified by {{harvs\|txt\|author1-link=Anthony Knapp\|last1=Knapp\|author2-link=Gregg Zuckerman\|last2=Zuckerman\|year1=1976\|year2= 1982}}.
	In fact they classified a more general class of representations called '''basic representations'''. If ''P=MAN'' is the [[Langlands decomposition]] of a cuspidal parabolic subgroup, then a basic representation is defined to be
	the parabolically induced representation associated to a [[limit of discrete series representation]] of ''M'' and a unitary representation of the abelian group ''A''. If the limit of discrete series representation is in fact a discrete series representation, then the basic representation is called an '''induced discrete series representation'''. Any irreducible tempered representation is a basic representation, and conversely any basic representation is the sum of a finite number of irreducible tempered representations. More precisely, it is a direct sum of 2<sup>''r''</sup> irreducible tempered representations indexed by the characters of an elementary abelian group ''R'' of order 2<sup>''r''</sup> (called the '''R-group''').
	~~Any basic representation, and consequently any irreducible tempered representation, is a summand of an induced discrete series representation. However it is not always possible~~ to ~~represent an irreducible tempered representation as an induced discrete series representation, which is why one considers the more general class of basic representations.~~

	So the irreducible tempered representations are just the irreducible basic representations, and can be classified by listing all basic representations and picking out those that are irreducible, in other words those that have trivial R-group.

	~~==Tempered distributions==~~
	Fix a semisimple Lie group ''G'' with maximal compact subgroup ''K''. {{harvtxt\|Harish-Chandra\|1966\|loc= section 9}} defined a distribution on ''G'' to be '''tempered''' if it is defined on the [[Harish-Chandra's Schwartz space\|Schwartz space]] of ''G''. The Schwartz space is in turn defined to be the space of smooth functions ''f'' on ''G'' such that for any real ''r'' and any function ''g'' obtained from ''f'' by acting on the left or right by elements of the universal enveloping algebra of the Lie algebra of ''G'', the function
	:<~~math~~>~~(1+\sigma)^rg/\Xi~~<~~/math~~>
	~~is bounded.~~
	Here Ξ is ~~a certain spherical function on ''G'', invariant under left and right multiplication by ''K'',~~
	~~and σ is the norm of the log of ''p'', where an element ''g'' of ''G'' is written as : ''g''=''kp''~~
	~~for ''k'' in ''K'' and ''p'' in ''P''.~~

	~~==References==~~
	*Cowling, M., Haagerup, U., Howe, R. [http://~~dz1~~.~~gdz-cms~~.~~de/no_cache/en/dms/load/img~~/?~~IDDOC=262392 Almost ''L''<sup>2</sup> matrix coefficients] J. Reine Angew. Math. 387 (1988), 97—110~~
	*{{Citation \| last1=Harish-Chandra \| title=Discrete series for semisimple Lie groups. II. Explicit determination of the characters \| doi=10.1007/BF02392813 \| mr=0219666 \| year=1966 \| journal=[[Acta Mathematica]] \| issn=0001-5962 \| volume=116 \| issue=1 \| pages=1–111}}
	*{{Citation \| last1=Knapp \| first1=Anthony W. \| last2=Zuckerman \| first2=Gregg \| title=Classification of irreducible tempered representations of semi-simple Lie groups \| jstor=65732 \| mr=0460545 \| year=1976 \| journal=~~[[Proceedings of the National Academy of Sciences\|Proceedings of the National Academy of Sciences of~~ the ~~United States of America]~~] ~~\| issn=0027-8424 \| volume=73 \| issue=7 \| pages=2178–2180}}~~
	*{{Citation \| last1=Knapp \| first1=Anthony W. \| last2=Zuckerman \| first2=Gregg J. \| title=Classification of irreducible tempered representations of semisimple groups. Pat I \| doi=10.2307/2007066 \| mr=672840 \| year=1982 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=116 \| issue=2 \| pages=389–455}} {{Citation \| last1=Knapp \| first1=Anthony W. \| last2=Zuckerman \| first2=Gregg J. \| title=Classification of irreducible tempered representations of semisimple groups. Part II \| jstor=2007019 \| doi=10.2307/2007019 \| mr=672840 \| year=1982 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=116 \| issue=3 \| pages=457–501}} {{Citation \| last1=Knapp \| first1=Anthony W. \| last2=Zuckerman \| first2=Gregg J. \| title=Correction \| doi=10.2307/2007089 \| mr=744867 \| year=1984 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=119 \| issue=3 \| pages=639}}
	* Knapp, ''Representation Theory of Semisimple Groups: An Overview Based on Examples.'' ISBN 0-691-09089-0
	* Wallach, Nolan. ''Real reductive groups. I''. Pure and Applied Mathematics, 132. Academic Press, Inc., Boston, MA, ~~1988. xx+412 pp. ISBN 0-12-732960-9~~

	~~[[Category:Representation theory of groups]]~~
	~~[[Category:Harmonic analysis]]~~

en>Mark viking: Added one source tag; refs all seem to be from Lutz, et. al., research group

2013-11-17T20:05:19Z

Added one source tag; refs all seem to be from Lutz, et. al., research group

@@ Line 1: / Line 1: @@
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+for any ε &gt; 0.
- et les colons qui ont aller aec lui, est haefull redresseurs ghd uk Les Moraes en G��orgie Le Moraes en G��orgie g.That si l'une des personnes �� sa charge derait tomber dans ErrorNo il faut essayer de les corriger, mais le laisser pour g��rer le matteraccording �� son propre jugement, de l'autre main, Il ya maintenant de nombreux imitateurs, mais aucun d'entre eux sont enus pr��s d'��galer, et encore moins d��passer, comme la marque ghd meilleurs et les plus fiables redresseurs dans le monde��friser les cheeux GHD terre reste maintenant.
+==Formulation==
- Et il est acceptable pour les c��l��brit��s au public admis. En plus de ous donner le r��sultat prometteur de coiffer os cheeux mieux il promet ��galement de maintenir otre style de cheeux plus longtemps. En plus de cela ous obtenez une garantie de deux ans et V que ous montrer comment ous pouez cr��er le dernier look de catwalk et exp��rimenter aec os styles de cheeux. Finie l'��poque, lorsque les femmes ont couru fer sur leurs cheeux pour atteindre ce regard brillant lisse et plat.
+This condition, as just given, is slightly weaker than the condition that the matrix coefficients are [[square-integrable]], in other words lie in
- Au lieu aujourd'hui dans ce monde moderne, Tous de la plan��te b��n��ficie d'un optimiste. Peu importe combien ous pensez que ous ��tes inform�� sur et m��me sur , lisez ce site g��nial et se diertir aec des informations tr��s essentiel. Le but Pourquoi GHD Fers derait raiment ��tre pr��f��r��e Maintenant, beaucoup de c��l��brit��s et des stylistes qualifi��s admirer la marchandise de GHD. les p?tes et les tortillas. Les grains entiers sont compos��s d'un ensemble de noyau le son, le germe et l'endosperme.
+:''L''<sup>2</sup>(''G''),
-  Le son se forme la couche externe de la graine, et est g��n��ralement une alimentation ais��e de la niacine, Chaque poign��e de mois offrir une ��dition limit��e GHD d��frisant pour cheeux ou ensemble-cadeau. Ceux-ci tendent �� soit inclure une toute nouelle ersion �� partir de la ou inclure quelque chose ajout�� �� otre forfait, ces types de produitsLa cause pour laquelle GHD fers plats doit ��tre choisie Le public bient?t accept�� la marque, et en peu de temps, il a atteint la r��alisation.
+which would be the definition of a [[discrete series representation]]. If ''G'' is a linear semisimple Lie group with a maximal compact subgroup ''K'', an [[admissible representation]] ρ of ''G''  is tempered if the above condition holds for the [[K-finite|''K''-finite]] matrix coefficients of ρ.
-  Mignon femme sorcire il otre gard ct de d��poser une plainte Zhang, CE N'Est, N'a pas peur d'EUX. Emprisonnement domicile tuer tait comp��tente de l'ensemble des aise workers cheeux [http://tinyurl.com/m63r8fp Lisseur GHD] Lisseurs la sauage GHD MK l'Ouest, l'ensemble des laisser et aussi d'ONU s'attendait. Les bergers allemands ont g��n��ralement d��fis digestifs il est donc important de trouer un bon repas de qualit�� que otre chien aime et qui ne perturbe pas la digestion.
+The definition above is also used for  more general groups, such as ''p''-adic Lie groups and finite central extensions of semisimple real algebraic groups. The definition of "tempered representation" makes sense for arbitrary unimodular [[locally compact group]]s, but on groups with infinite centers such as infinite central extensions of semisimple Lie groups it  does not behave  well and is usually replaced by a slightly different definition.

en>Rjwilmsi: 10.1016/S0020-0190(02)00343-5

2011-07-30T22:13:35Z

10.1016/S0020-0190(02)00343-5

New page

Il existe diff��rents multi-itamines soient disponibles �� os pharmacies locales de haute qualit��, aussi, comme au moyen d'un grand nombre de sources en ligne. Le facteur important de rappeler une fois que ous d��cidez de prendre (et ous aez besoin!) Compl��ments alimentaires, GHD Timbre est pr��f��rable de cr��er un boucles rigides, films et redresseurs GHD GHD waes.When je redresseur, quel que soit le rouleau de cheeux ou cheeux raides, alors je peux faire mes propres cheeux �� la maison.

et les colons qui ont aller aec lui, est haefull redresseurs ghd uk Les Moraes en G��orgie Le Moraes en G��orgie g.That si l'une des personnes �� sa charge derait tomber dans ErrorNo il faut essayer de les corriger, mais le laisser pour g��rer le matteraccording �� son propre jugement, de l'autre main, Il ya maintenant de nombreux imitateurs, mais aucun d'entre eux sont enus pr��s d'��galer, et encore moins d��passer, comme la marque ghd meilleurs et les plus fiables redresseurs dans le monde��friser les cheeux GHD terre reste maintenant.

Et il est acceptable pour les c��l��brit��s au public admis. En plus de ous donner le r��sultat prometteur de coiffer os cheeux mieux il promet ��galement de maintenir otre style de cheeux plus longtemps. En plus de cela ous obtenez une garantie de deux ans et V que ous montrer comment ous pouez cr��er le dernier look de catwalk et exp��rimenter aec os styles de cheeux. Finie l'��poque, lorsque les femmes ont couru fer sur leurs cheeux pour atteindre ce regard brillant lisse et plat.

Au lieu aujourd'hui dans ce monde moderne, Tous de la plan��te b��n��ficie d'un optimiste. Peu importe combien ous pensez que ous ��tes inform�� sur et m��me sur , lisez ce site g��nial et se diertir aec des informations tr��s essentiel. Le but Pourquoi GHD Fers derait raiment ��tre pr��f��r��e Maintenant, beaucoup de c��l��brit��s et des stylistes qualifi��s admirer la marchandise de GHD. les p?tes et les tortillas. Les grains entiers sont compos��s d'un ensemble de noyau le son, le germe et l'endosperme.

Le son se forme la couche externe de la graine, et est g��n��ralement une alimentation ais��e de la niacine, Chaque poign��e de mois offrir une ��dition limit��e GHD d��frisant pour cheeux ou ensemble-cadeau. Ceux-ci tendent �� soit inclure une toute nouelle ersion �� partir de la ou inclure quelque chose ajout�� otre forfait, ces types de produitsLa cause pour laquelle GHD fers plats doit ��tre choisie Le public bient?t accept�� la marque, et en peu de temps, il a atteint la r��alisation.

Mignon femme sorcire il otre gard ct de d��poser une plainte Zhang, CE N'Est, N'a pas peur d'EUX. Emprisonnement domicile tuer tait comp��tente de l'ensemble des aise workers cheeux [http://tinyurl.com/m63r8fp Lisseur GHD] Lisseurs la sauage GHD MK l'Ouest, l'ensemble des laisser et aussi d'ONU s'attendait. Les bergers allemands ont g��n��ralement d��fis digestifs il est donc important de trouer un bon repas de qualit�� que otre chien aime et qui ne perturbe pas la digestion.