Uncertain inference - Revision history

en>Kku: /* Definitions */ lf

2014-04-14T12:36:36Z

Definitions: lf

← Older revision		Revision as of 14:36, 14 April 2014
Line 1:		Line 1:
	In the mathematical theory of [[automorphic representation]]s, a '''multiplicity-one theorem''' is a result about the [[representation theory]] of an [[adelic algebraic group\|adelic]] [[reductive algebraic group]]. The multiplicity in question is the number of times a given abstract [[group representation]] is realised in a certain space, of [[square integrable function]]s, given in a concrete way.

	~~==Definition==~~
	Let ''G'' be a reductive algebraic group over a [[number field]] ''K'' and let '''A''' denote the [[adele ring\|adele]]s of ''K''. Let ''Z'' denote the [[center of a group\|centre]] of ''G'' and let ω be a [[continuous (mathematics)\|continuous]] [[character (mathematics)\|unitary character]] from ''Z''(''K'')\Z('''A''')<sup>×</sup> to '''C'''<sup>×</sup>. Let ''L''<sup>2</sup><sub>0</sub>(''G''(''K'')/''G''('''A'''), ω) denote the [[cuspidal representation\|space of cusp forms with central character ω]] on ''G''('''A'''). This space decomposes into a [[direct sum of Hilbert spaces]]
	~~:<math>L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi</math>~~
	~~where the sum is over [[irreducible representation\|irreducible]] [[subrepresentation]]s and ''m''<sub>π</sup> are non-negative [[integer]]s.~~

	~~The group of adelic points of ''G'~~', ~~''G''('''A''')~~, is ~~said to satisfy the '''multiplicity~~-~~one property'''~~ if ~~any [[smooth representation\|smooth]] irreducible [[admissible representation]] of~~ '~~'G''('''A''') occurs~~ with ~~multiplicity at most one in the space of~~ [~~[cusp form]]s of central character ω, i~~.e. ~~''m''<sub>π</sub> is 0 or 1 for all such π~~.		I've seen worse, fallen from worse, but there is a nice ride-around if you don't want to start the ride off with wounded bike or body parts. Should you cherished this information as well as you want to obtain details with regards to [http://snakzstock.com/dressup4ever/profile/954401/MoJVL.html Size guaide mountain bike sizing.] kindly check out our own web site. For more serious cyclists, next to bicycles themselves, proper shoes are the most important piece of equipment. If you go on a regular mountain biking, you will develop a healthier body and disposition. There should not be a strain and similarly you shouldn't find yourself catching your breath. However, due to its high price tag and light weight, it is seldom used for jumping and downhill races. <br><br>The prices offered by these online stores are very competitive and realistic according to the market. One way to increase exercise fun is to find a friend to run with you. Fitting is logical, the instructions are clear few tools are required. - Get a bike that is lightweight and made out of carbon fiber materials. The micro-brewery and pub sits right along the White Pine Trail, a paved rail-trail that runs from Grand Rapids to Cadillac. <br><br>They have great suspension that fits the need of both amateur and professional bikers. That's a level of quality you just don't get from many brands these days. With mountain biking being a very popular sport, there are many bikes to choose from. Hence, having the bicycle that suits you perfectly is much easier now. You might want to pitch a tent or you may find that there are many other options for lodging as you move along the trails on your mountain biking trips. <br><br>It works great unless you have some physical issue. As a form of transport in itself bike riding is great but because of the design of these bikes you can travel over harsher terrain which will lead to you tossing in your old bike and traveling to work on your newly discovered fun machine. But older children may have more adventurous tastes, and want for toys that are - well, still miniatures, but are flashier, more expensive, and more functional. Mountain bikes are equipped with knobby tires, large number of gears and handle bars that are wider; they also include shocks and a much stronger suspension. But that wasn't the bad part 'cause, see, those two almost invisible grey hairs were only a distraction from the real culprit. <br><br>If you can afford it, a full suspension mountain bike is always worth the purchase. Why pollute the air when you can experience an electrical folding bike. The price may generally differ depending on the condition of the bike and on how new the model really is. A twenty-inch wheeled bike are good for longer traveling time. It's about the journey and all that it entails, learning about yourself, making friends, being helped, and of course, helping others along the way".

	~~==Results==~~
	~~The fact that the [[general linear group]~~], ~~''GL''(''n'')~~, ~~has~~ the ~~multiplicity-one property was proved by {{harvtxt\|Jacquet\|Langlands\|1970}} for ''n'' = 2~~ and ~~independently by {{harvtxt\|Piatetski-Shapiro\|1979}}~~ and ~~{{harvs\|txt\|authorlink=Joseph Shalika\|last=Shalika\|year=1974}} for ''n~~'~~' > 2 using the uniqueness of the [[Whittaker model]]~~. ~~Multiplicity-one also holds for [[Special linear group\|''SL''(2)]]~~, ~~but not~~ for ~~''SL''(''n'') for ''n'' ~~>~~ 2 {{harv\|Blasius\|1994}}.~~

	~~==Strong multiplicity one theorem==~~

	The ~~strong multiplicity one theorem of {{harvtxt\|Piatetski-Shapiro\|1979}}~~ and ~~{{harvtxt\|Jacquet\|Shalika\|1981}} states that two cuspidal automorphic representations of~~ the ~~general linear group~~ are ~~isomorphic if their local components~~ are ~~isomorphic for all but~~ a ~~finite number~~ of ~~places~~.

	~~==References==~~

	*{{Citation \| last1=Blasius \| first1=Don \| title=On multiplicities for SL(n) \| doi=10.1007/BF02937513 \| mr=1303497 \| year=1994 \| journal=Israel Journal of Mathematics \| issn=0021-~~2172 \| volume=88 \| issue=1 \| pages=237–251}}~~
	*{{Citation \| last1=Cogdell \| first1=James W. \| editor1-~~last=Cogdell \| editor1-first=James W~~. ~~\| editor2-last=Kim \| editor2-first=Henry H~~. ~~\| editor3-last=Murty \| editor3-first=Maruti Ram \| title=Lectures on automorphic L-functions \| url=http://books~~.~~google~~.~~com/books?id=jb3ZCp0-MQsC \| publisher=[[American Mathematical Society]] \| location=Providence~~, R.~~I. \| series=Fields Inst. Monogr~~. ~~\| isbn=978-0-8218-3516-6 \| mr=2071506 \| year=2004 \| volume=20 \| chapter=Lectures on L-functions, converse theorems, and functoriality for GL~~<~~sub~~>n<~~/sub~~> ~~\| chapterurl=http://www~~.~~math.osu.edu/~cogdell/ \| pages=1–96}}~~
	*{{Citation
	~~\| last=Jacquet~~
	~~\| first=Hervé~~
	~~\| last2=Langlands~~
	~~\| first2=Robert~~
	~~\| title=Automorphic forms on GL(2)~~
	~~\| series=Lecture Notes~~ in ~~Mathematics~~
	~~\| publisher=Springer-Verlag~~
	~~\| volume=114~~
	~~\| year=1970~~
	}}
	*{{Citation \| last1=Jacquet \| first1=H. \| last2=Shalika \| first2=J. A. \| title=On Euler products and the ~~classification~~ of automorphic representations. I \| doi=10.2307/2374103 \| mr=618323 \| year=1981 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=103 \| issue=3 \| pages=499–558}} {{Citation \| last1=Jacquet \| first1=H. \| last2=Shalika \| first2=J. A. \| title=On Euler products and ~~the classification of automorphic representations~~. ~~II \| jstor=2374050 \| mr=618323 \| year=1981 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=103 \| issue=4 \| pages=777–815}}~~
	*{{Citation \| last1=Piatetski-Shapiro \| first1=I. I. \| editor1-last=Borel \| editor1-first=Armand \| editor1-link=Armand Borel \| editor2-last=Casselman. \| editor2-first=W. \| title=Automorphic forms, ~~representations~~ and L-~~functions (Proc. Sympos. Pure Math.~~, ~~Oregon State Univ.~~, ~~Corvallis~~, ~~Ore~~., ~~1977), Part 1 \| url=http://www~~.~~ams.org/publications/online-books/pspum331-index \| publisher=[[American Mathematical Society]] \| location=Providence~~, ~~R.I. \| series=Proc. Sympos. Pure Math.~~, ~~XXXIII \| isbn=978-0-8218-1435-2 \| mr=546599 \| year=1979 \| chapter=Multiplicity one theorems \| pages=209–212}}~~
	*{{Citation \| last1=Shalika \| first1=J. A. ~~\| title=The multiplicity one theorem for GL~~<~~sub~~>n<~~/sub~~> ~~\| jstor=1971071 \| mr=0348047 \| year=1974 \| journal=[[Annals~~ of ~~Mathematics\|Annals of Mathematics~~. ~~Second Series]] \| issn=0003~~-~~486X \| volume=100 \| pages=171–193}}~~

	~~[[Category:Representation theory~~ of ~~groups]]~~
	~~[[Category:Automorphic forms]]~~
	~~[[Category:Theorems in number theory]]~~
	~~[[Category:Theorems in representation theory]]~~

173.56.70.112: /* Definitions */

2012-12-25T06:49:13Z

Definitions

← Older revision		Revision as of 08:49, 25 December 2012
Line 1:		Line 1:
	~~With~~ the ~~backdrop~~ of ~~the Superstition Mountains~~, ~~you have~~ a ~~scenic mountain bike trail network that can embody every quality of off~~-~~road riding in~~ the ~~Southwest~~. ~~Each hole~~ is ~~made up differently and your driver is not always~~ the ~~best way to start~~ a ~~hole. For maximum safety, it~~ is ~~important to have a bike-repair multi-tool,~~ a ~~patch kit for flat tires~~, ~~and tire levers. Generally speaking~~, a ~~good, durable mountain bike starts at about $600 and goes up from there. Why would I want to change, even for a day, the most important and shaping event in my life~~. ~~<br><br>~~		In the mathematical theory of [[automorphic representation]]s, a '''multiplicity-one theorem''' is a result about the [[representation theory]] of an [[adelic algebraic group\|adelic]] [[reductive algebraic group]]. The multiplicity in question is the number of times a given abstract [[group representation]] is realised in a certain space, of [[square integrable function]]s, given in a concrete way.

	~~Therefore all sides~~ of the ~~square are~~ of ~~equal length~~. ~~General Use~~ (~~Commuting, Mountain Bike, Road Bike~~, ~~BMX, and Universal Fit~~) ~~General use helmets are usually about. So how exactly would you go about making this dream a reality. Now please continue~~ on ~~additionally so you can get added info for this topic~~. ~~It is~~ a ~~choice you make that determines the level~~ of ~~commitment you have.~~ <br><br>~~If you are looking for the best safety mountain bike accessories then~~ the ~~first thing you need to look into~~ is ~~a helmet~~. The ~~big three has a major part to play on this matter and so they should step up to be strong and mature leaders. Apply a generous amount~~ of ~~mountain bike lube~~ to ~~your chain as you move~~ the ~~pedals around backward. Mounting biking is a fun sport for professionals and beginners alike.~~ -~~Del Plomo Hot Springs: Ideal for bikes~~, ~~low traffic~~. <br><br>The ~~inaugural Big Bear Gran Fondo takes place on July 26th in Big Bear~~, ~~with distance options of 20K~~, ~~30K, 70K~~ and ~~100K. There are also tires with detailed treads which are very prominent~~ and ~~these are very useful~~ for the ~~biker because~~ of the ~~grip they provide~~. ~~Just remember your heart is a muscle~~ and ~~can be developed~~ the ~~more you practice~~. ~~Here is more information regarding [~~http://~~ddm~~.~~nicoschlitter~~.~~de/DistributedDataMining~~/~~view_profile.php~~?~~userid~~=~~1161386 Womens mountain bike sizing~~.~~] have a look at the web-page~~. ~~With a Shimano 105 groupset and FSA Gossamer chain set, it's a great choice for under~~. ~~With~~ ~~Adventure Kerala ( you will enjoy adventure sports like mountaineering~~, ~~rapelling, rock climbing, river crossing~~, and ~~a whole lot of adventure activities.~~ <br><br>~~When the helmet is~~ in the ~~correct position on the head, the straps can be adjusted so that they are snug across the side~~ of ~~the head~~ and ~~under~~ the ~~chin~~. ~~The correct quill stems are sized down with the inside diameter~~ of ~~the fork's steer tube~~. ~~Usually~~, ~~alloy stems tend to be the more cost~~-~~effective of the two~~, ~~however the high end of the level~~, ~~aluminum stems which have been produced from high~~-~~grade alloys could be more expensive than carbon counterparts because they are lighter plus more responsive~~. ~~It is also easier to hold the handlebars if they are covered with rubber than exposed handlebars~~, ~~especially if it is too hot because steel absorbs heat and makes it difficult~~ for ~~you to hold. Some riders advocate the additional technical the trail the greater a hardtail mountain bikeis sought after~~.		==Definition==
			Let ''G'' be a reductive algebraic group over a [[number field]] ''K'' and let '''A''' denote the [[adele ring\|adele]]s of ''K''. Let ''Z'' denote the [[center of a group\|centre]] of ''G'' and let ω be a [[continuous (mathematics)\|continuous]] [[character (mathematics)\|unitary character]] from ''Z''(''K'')\Z('''A''')<sup>×</sup> to '''C'''<sup>×</sup>. Let ''L''<sup>2</sup><sub>0</sub>(''G''(''K'')/''G''('''A'''), ω) denote the [[cuspidal representation\|space of cusp forms with central character ω]] on ''G''('''A'''). This space decomposes into a [[direct sum of Hilbert spaces]]
			:<math>L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi</math>
			where the sum is over [[irreducible representation\|irreducible]] [[subrepresentation]]s and ''m''<sub>π</sup> are non-negative [[integer]]s.

			The group of adelic points of ''G'', ''G''('''A'''), is said to satisfy the '''multiplicity-one property''' if any [[smooth representation\|smooth]] irreducible [[admissible representation]] of ''G''('''A''') occurs with multiplicity at most one in the space of [[cusp form]]s of central character ω, i.e. ''m''<sub>π</sub> is 0 or 1 for all such π.

			==Results==
			The fact that the [[general linear group]], ''GL''(''n''), has the multiplicity-one property was proved by {{harvtxt\|Jacquet\|Langlands\|1970}} for ''n'' = 2 and independently by {{harvtxt\|Piatetski-Shapiro\|1979}} and {{harvs\|txt\|authorlink=Joseph Shalika\|last=Shalika\|year=1974}} for ''n'' > 2 using the uniqueness of the [[Whittaker model]]. Multiplicity-one also holds for [[Special linear group\|''SL''(2)]], but not for ''SL''(''n'') for ''n'' > 2 {{harv\|Blasius\|1994}}.

			==Strong multiplicity one theorem==

			The strong multiplicity one theorem of {{harvtxt\|Piatetski-Shapiro\|1979}} and {{harvtxt\|Jacquet\|Shalika\|1981}} states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places.

			==References==

			*{{Citation \| last1=Blasius \| first1=Don \| title=On multiplicities for SL(n) \| doi=10.1007/BF02937513 \| mr=1303497 \| year=1994 \| journal=Israel Journal of Mathematics \| issn=0021-2172 \| volume=88 \| issue=1 \| pages=237–251}}
			*{{Citation \| last1=Cogdell \| first1=James W. \| editor1-last=Cogdell \| editor1-first=James W. \| editor2-last=Kim \| editor2-first=Henry H. \| editor3-last=Murty \| editor3-first=Maruti Ram \| title=Lectures on automorphic L-functions \| url=http://books.google.com/books?id=jb3ZCp0-MQsC \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Fields Inst. Monogr. \| isbn=978-0-8218-3516-6 \| mr=2071506 \| year=2004 \| volume=20 \| chapter=Lectures on L-functions, converse theorems, and functoriality for GL<sub>n</sub> \| chapterurl=http://www.math.osu.edu/~cogdell/ \| pages=1–96}}
			*{{Citation
			\| last=Jacquet
			\| first=Hervé
			\| last2=Langlands
			\| first2=Robert
			\| title=Automorphic forms on GL(2)
			\| series=Lecture Notes in Mathematics
			\| publisher=Springer-Verlag
			\| volume=114
			\| year=1970
			}}
			*{{Citation \| last1=Jacquet \| first1=H. \| last2=Shalika \| first2=J. A. \| title=On Euler products and the classification of automorphic representations. I \| doi=10.2307/2374103 \| mr=618323 \| year=1981 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=103 \| issue=3 \| pages=499–558}} {{Citation \| last1=Jacquet \| first1=H. \| last2=Shalika \| first2=J. A. \| title=On Euler products and the classification of automorphic representations. II \| jstor=2374050 \| mr=618323 \| year=1981 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=103 \| issue=4 \| pages=777–815}}
			*{{Citation \| last1=Piatetski-Shapiro \| first1=I. I. \| editor1-last=Borel \| editor1-first=Armand \| editor1-link=Armand Borel \| editor2-last=Casselman. \| editor2-first=W. \| title=Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1 \| url=http://www.ams.org/publications/online-books/pspum331-index \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Proc. Sympos. Pure Math., XXXIII \| isbn=978-0-8218-1435-2 \| mr=546599 \| year=1979 \| chapter=Multiplicity one theorems \| pages=209–212}}
			*{{Citation \| last1=Shalika \| first1=J. A. \| title=The multiplicity one theorem for GL<sub>n</sub> \| jstor=1971071 \| mr=0348047 \| year=1974 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=100 \| pages=171–193}}

			[[Category:Representation theory of groups]]
			[[Category:Automorphic forms]]
			[[Category:Theorems in number theory]]
			[[Category:Theorems in representation theory]]

en>Linas: /* Definitions */ autoepistemic logic

2011-06-10T18:54:54Z

Definitions: autoepistemic logic

New page

With the backdrop of the Superstition Mountains, you have a scenic mountain bike trail network that can embody every quality of off-road riding in the Southwest. Each hole is made up differently and your driver is not always the best way to start a hole. For maximum safety, it is important to have a bike-repair multi-tool, a patch kit for flat tires, and tire levers. Generally speaking, a good, durable mountain bike starts at about $600 and goes up from there. Why would I want to change, even for a day, the most important and shaping event in my life. 

Therefore all sides of the square are of equal length. General Use (Commuting, Mountain Bike, Road Bike, BMX, and Universal Fit) General use helmets are usually about. So how exactly would you go about making this dream a reality. Now please continue on additionally so you can get added info for this topic. It is a choice you make that determines the level of commitment you have. If you are looking for the best safety mountain bike accessories then the first thing you need to look into is a helmet. The big three has a major part to play on this matter and so they should step up to be strong and mature leaders. Apply a generous amount of mountain bike lube to your chain as you move the pedals around backward. Mounting biking is a fun sport for professionals and beginners alike. -Del Plomo Hot Springs: Ideal for bikes, low traffic. The inaugural Big Bear Gran Fondo takes place on July 26th in Big Bear, with distance options of 20K, 30K, 70K and 100K. There are also tires with detailed treads which are very prominent and these are very useful for the biker because of the grip they provide. Just remember your heart is a muscle and can be developed the more you practice. Here is more information regarding [http://ddm.nicoschlitter.de/DistributedDataMining/view_profile.php?userid=1161386 Womens mountain bike sizing.] have a look at the web-page. With a Shimano 105 groupset and FSA Gossamer chain set, it's a great choice for under. With Adventure Kerala ( you will enjoy adventure sports like mountaineering, rapelling, rock climbing, river crossing, and a whole lot of adventure activities. When the helmet is in the correct position on the head, the straps can be adjusted so that they are snug across the side of the head and under the chin. The correct quill stems are sized down with the inside diameter of the fork's steer tube. Usually, alloy stems tend to be the more cost-effective of the two, however the high end of the level, aluminum stems which have been produced from high-grade alloys could be more expensive than carbon counterparts because they are lighter plus more responsive. It is also easier to hold the handlebars if they are covered with rubber than exposed handlebars, especially if it is too hot because steel absorbs heat and makes it difficult for you to hold. Some riders advocate the additional technical the trail the greater a hardtail mountain bikeis sought after.

← Older revision		Revision as of 14:36, 14 April 2014
Line 1:		Line 1:
	In the mathematical theory of [[automorphic representation]]s, a '''multiplicity-one theorem''' is a result about the [[representation theory]] of an [[adelic algebraic group\|adelic]] [[reductive algebraic group]]. The multiplicity in question is the number of times a given abstract [[group representation]] is realised in a certain space, of [[square integrable function]]s, given in a concrete way.

	~~==Definition==~~
	Let ''G'' be a reductive algebraic group over a [[number field]] ''K'' and let '''A''' denote the [[adele ring\|adele]]s of ''K''. Let ''Z'' denote the [[center of a group\|centre]] of ''G'' and let ω be a [[continuous (mathematics)\|continuous]] [[character (mathematics)\|unitary character]] from ''Z''(''K'')\Z('''A''')<sup>×</sup> to '''C'''<sup>×</sup>. Let ''L''<sup>2</sup><sub>0</sub>(''G''(''K'')/''G''('''A'''), ω) denote the [[cuspidal representation\|space of cusp forms with central character ω]] on ''G''('''A'''). This space decomposes into a [[direct sum of Hilbert spaces]]
	~~:<math>L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi</math>~~
	~~where the sum is over [[irreducible representation\|irreducible]] [[subrepresentation]]s and ''m''<sub>π</sup> are non-negative [[integer]]s.~~

	~~The group of adelic points of ''G'~~', ~~''G''('''A''')~~, is ~~said to satisfy the '''multiplicity~~-~~one property'''~~ if ~~any [[smooth representation\|smooth]] irreducible [[admissible representation]] of~~ '~~'G''('''A''') occurs~~ with ~~multiplicity at most one in the space of~~ [~~[cusp form]]s of central character ω, i~~.e. ~~''m''<sub>π</sub> is 0 or 1 for all such π~~.		I've seen worse, fallen from worse, but there is a nice ride-around if you don't want to start the ride off with wounded bike or body parts. Should you cherished this information as well as you want to obtain details with regards to [http://snakzstock.com/dressup4ever/profile/954401/MoJVL.html Size guaide mountain bike sizing.] kindly check out our own web site. For more serious cyclists, next to bicycles themselves, proper shoes are the most important piece of equipment. If you go on a regular mountain biking, you will develop a healthier body and disposition. There should not be a strain and similarly you shouldn't find yourself catching your breath. However, due to its high price tag and light weight, it is seldom used for jumping and downhill races. <br><br>The prices offered by these online stores are very competitive and realistic according to the market. One way to increase exercise fun is to find a friend to run with you. Fitting is logical, the instructions are clear few tools are required. - Get a bike that is lightweight and made out of carbon fiber materials. The micro-brewery and pub sits right along the White Pine Trail, a paved rail-trail that runs from Grand Rapids to Cadillac. <br><br>They have great suspension that fits the need of both amateur and professional bikers. That's a level of quality you just don't get from many brands these days. With mountain biking being a very popular sport, there are many bikes to choose from. Hence, having the bicycle that suits you perfectly is much easier now. You might want to pitch a tent or you may find that there are many other options for lodging as you move along the trails on your mountain biking trips. <br><br>It works great unless you have some physical issue. As a form of transport in itself bike riding is great but because of the design of these bikes you can travel over harsher terrain which will lead to you tossing in your old bike and traveling to work on your newly discovered fun machine. But older children may have more adventurous tastes, and want for toys that are - well, still miniatures, but are flashier, more expensive, and more functional. Mountain bikes are equipped with knobby tires, large number of gears and handle bars that are wider; they also include shocks and a much stronger suspension. But that wasn't the bad part 'cause, see, those two almost invisible grey hairs were only a distraction from the real culprit. <br><br>If you can afford it, a full suspension mountain bike is always worth the purchase. Why pollute the air when you can experience an electrical folding bike. The price may generally differ depending on the condition of the bike and on how new the model really is. A twenty-inch wheeled bike are good for longer traveling time. It's about the journey and all that it entails, learning about yourself, making friends, being helped, and of course, helping others along the way".

	~~==Results==~~
	~~The fact that the [[general linear group]~~], ~~''GL''(''n'')~~, ~~has~~ the ~~multiplicity-one property was proved by {{harvtxt\|Jacquet\|Langlands\|1970}} for ''n'' = 2~~ and ~~independently by {{harvtxt\|Piatetski-Shapiro\|1979}}~~ and ~~{{harvs\|txt\|authorlink=Joseph Shalika\|last=Shalika\|year=1974}} for ''n~~'~~' > 2 using the uniqueness of the [[Whittaker model]]~~. ~~Multiplicity-one also holds for [[Special linear group\|''SL''(2)]]~~, ~~but not~~ for ~~''SL''(''n'') for ''n'' ~~>~~ 2 {{harv\|Blasius\|1994}}.~~

	~~==Strong multiplicity one theorem==~~

	The ~~strong multiplicity one theorem of {{harvtxt\|Piatetski-Shapiro\|1979}}~~ and ~~{{harvtxt\|Jacquet\|Shalika\|1981}} states that two cuspidal automorphic representations of~~ the ~~general linear group~~ are ~~isomorphic if their local components~~ are ~~isomorphic for all but~~ a ~~finite number~~ of ~~places~~.

	~~==References==~~

	*{{Citation \| last1=Blasius \| first1=Don \| title=On multiplicities for SL(n) \| doi=10.1007/BF02937513 \| mr=1303497 \| year=1994 \| journal=Israel Journal of Mathematics \| issn=0021-~~2172 \| volume=88 \| issue=1 \| pages=237–251}}~~
	*{{Citation \| last1=Cogdell \| first1=James W. \| editor1-~~last=Cogdell \| editor1-first=James W~~. ~~\| editor2-last=Kim \| editor2-first=Henry H~~. ~~\| editor3-last=Murty \| editor3-first=Maruti Ram \| title=Lectures on automorphic L-functions \| url=http://books~~.~~google~~.~~com/books?id=jb3ZCp0-MQsC \| publisher=[[American Mathematical Society]] \| location=Providence~~, R.~~I. \| series=Fields Inst. Monogr~~. ~~\| isbn=978-0-8218-3516-6 \| mr=2071506 \| year=2004 \| volume=20 \| chapter=Lectures on L-functions, converse theorems, and functoriality for GL~~<~~sub~~>n<~~/sub~~> ~~\| chapterurl=http://www~~.~~math.osu.edu/~cogdell/ \| pages=1–96}}~~
	*{{Citation
	~~\| last=Jacquet~~
	~~\| first=Hervé~~
	~~\| last2=Langlands~~
	~~\| first2=Robert~~
	~~\| title=Automorphic forms on GL(2)~~
	~~\| series=Lecture Notes~~ in ~~Mathematics~~
	~~\| publisher=Springer-Verlag~~
	~~\| volume=114~~
	~~\| year=1970~~
	}}
	*{{Citation \| last1=Jacquet \| first1=H. \| last2=Shalika \| first2=J. A. \| title=On Euler products and the ~~classification~~ of automorphic representations. I \| doi=10.2307/2374103 \| mr=618323 \| year=1981 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=103 \| issue=3 \| pages=499–558}} {{Citation \| last1=Jacquet \| first1=H. \| last2=Shalika \| first2=J. A. \| title=On Euler products and ~~the classification of automorphic representations~~. ~~II \| jstor=2374050 \| mr=618323 \| year=1981 \| journal=[[American Journal of Mathematics]] \| issn=0002-9327 \| volume=103 \| issue=4 \| pages=777–815}}~~
	*{{Citation \| last1=Piatetski-Shapiro \| first1=I. I. \| editor1-last=Borel \| editor1-first=Armand \| editor1-link=Armand Borel \| editor2-last=Casselman. \| editor2-first=W. \| title=Automorphic forms, ~~representations~~ and L-~~functions (Proc. Sympos. Pure Math.~~, ~~Oregon State Univ.~~, ~~Corvallis~~, ~~Ore~~., ~~1977), Part 1 \| url=http://www~~.~~ams.org/publications/online-books/pspum331-index \| publisher=[[American Mathematical Society]] \| location=Providence~~, ~~R.I. \| series=Proc. Sympos. Pure Math.~~, ~~XXXIII \| isbn=978-0-8218-1435-2 \| mr=546599 \| year=1979 \| chapter=Multiplicity one theorems \| pages=209–212}}~~
	*{{Citation \| last1=Shalika \| first1=J. A. ~~\| title=The multiplicity one theorem for GL~~<~~sub~~>n<~~/sub~~> ~~\| jstor=1971071 \| mr=0348047 \| year=1974 \| journal=[[Annals~~ of ~~Mathematics\|Annals of Mathematics~~. ~~Second Series]] \| issn=0003~~-~~486X \| volume=100 \| pages=171–193}}~~

	~~[[Category:Representation theory~~ of ~~groups]]~~
	~~[[Category:Automorphic forms]]~~
	~~[[Category:Theorems in number theory]]~~
	~~[[Category:Theorems in representation theory]]~~