Cationic polymerization - Revision history

92.224.253.221: /* Lewis acids/Friedel-Crafts catalysts */

2014-12-09T21:44:07Z

Lewis acids/Friedel-Crafts catalysts

← Older revision		Revision as of 23:44, 9 December 2014
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	~~{{Orphan\|date=September 2011}}~~		There are lots of distinctive sorts of interior LED develop light , which includes many extra, in addition to metal halide, ruthless sodium bulbs fluorescent lights.Use MHs for 18-24 hours everyday to market growth. Indoor plants may possibly not receive sufficient all-natural light and will need extra artificial lighting for most effective development. Coverage: two.52.5ft( Veg), 22ft(Flowering) New 2014 300W LED Develop Light for Indoor Plants Growth and Flowering. Coverage: 3.52ft( Veg), 31.5ft(Flowering) Package Include:1 piece 300w led develop light, energy cord and steel hanging kit. Reflector design 100% LED lumen output to plants, improve yields! 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	~~In [[mathematics]], the concept~~ of ~~'''groupoid algebra''' generalizes the notion~~ of ~~[[group algebra]]~~.~~<ref>Khalkhali~~ (~~2009~~), ~~{{Google books quote\|id=UInc5AyTAikC\|page=48\|text=groupoid algebra\|p~~. ~~48}}</ref>~~

	~~== Definition ==~~
	~~Given~~ a [~~[groupoid]~~] <~~math~~>~~(G, \cdot)~~<~~/math~~> ~~and a [[field (mathematics)\|field]]~~ <~~math~~>K<~~/math~~>, it is ~~possible~~ to ~~define~~ the ~~groupoid algebra~~ <~~math~~>KG<~~/math~~> as the ~~[[Algebra over a field\|algebra]]~~ over <~~math~~>K<~~/math~~> ~~formed by~~ the ~~[[vector space]] having~~ the ~~elements~~ of ~~<math>G</math> as [[generator (mathematics)\|generator]]s~~ and ~~having~~ the ~~[[multiplication]]~~ of ~~these elements defined by~~ <~~math~~>~~g * h = g \cdot h~~<~~/math~~>, ~~whenever~~ this ~~product~~ is ~~defined~~, and <~~math~~>~~g * h = 0~~<~~/math~~> ~~otherwise~~. ~~The product~~ is ~~then extended by [[linearity]]~~.<~~ref~~>~~Dokuchaev~~, ~~Exel & Piccione (2000)~~, p. ~~7</ref>~~

	~~== Examples ==~~
	~~Some examples of groupoid algebras~~ are the ~~following:~~<~~ref~~>~~da Silva & Weinstein (1999), {{Google books quote\|id=2fcC1EGKz08C\|page=97\|text=groupoid algebras\|p. 97}}~~<~~/ref~~>
	* [[Group algebra]]s
	* [[Matrix ring\|Matrix algebra]]s
	* [[Algebra of ~~functions\|Algebras of functions]]~~

	~~== Properties ==~~
	* When a groupoid has a [~~[Wikt~~:~~finite\|finite~~]~~] number of [[Object (category theory~~)~~\|objects]] and a finite number~~ of ~~[[morphism (category theory)\|morphisms]],~~ the ~~groupoid algebra~~ is a ~~[[direct sum]] of [[tensor product]]s of group algebras and matrix algebras~~.<~~ref~~>~~Khalkhali & Marcolli (2008), {{Google books quote\|id=HsTkPOj0iusC\|page=210\|text=Groupoid algebra of a finite groupoid\|p. 210}}~~</~~ref>~~

	~~== See also ==~~
	* [[Hopf algebra]]
	* [[Partial group algebra]]

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

	~~== References ==~~
	* {{cite book \|last1=Khalkhali \|first1=Masoud \|authorlink1= \|last2= \|first2= \|authorlink2= \|title=Basic Noncommutative Geometry \|url= \|edition= \|series=EMS Series of ~~Lectures~~ in ~~Mathematics \|volume= \|year=2009 \|publisher=European Mathematical Society \|location= \|isbn=978-3-03719-061-6 }}~~
	* {{cite book \|last1=da Silva \|first1=Ana Cannas \|authorlink1= \|last2=Weinstein \|first2=Alan \|authorlink2= \|title=Geometric models for noncommutative algebras \|url= \|edition=2 \|series=Berkeley mathematics lecture notes \|volume=10 \|year=1999 \|publisher=AMS Bookstore \|location= \|isbn=978-0-8218-0952-5 }}
	* {{cite journal \|last1=Dokuchaev \|first1=M. ~~\|last2=Exel \|first2=R~~. ~~\|last3=Piccione \|first3=P. \|year=2000 \|title=Partial Representations and Partial Group Algebras \|journal=Journal~~ of ~~Algebra \|volume=226 \|issue= \|pages=505–532 \|publisher=Elsevier \|issn=0021-8693 \|doi= 10~~.~~1006/jabr.1999~~.~~8204\|url=\|arxiv= math/9903129~~
	}}
	* {{cite book \|last1=Khalkhali \|first1=Masoud \|authorlink1= \|last2=Marcolli \|first2=Matilde \|authorlink2= \|title=An invitation to noncommutative geometry \|url= \|edition= \|series= \|volume= \|year=2008 \|publisher=World Scientific \|location= \|isbn=978-981-270-616-4 }}

	~~[[Category:Algebra]]~~


	~~{{algebra-stub}}~~

en>Jamesx12345: Cleaned up using AutoEd | Assisted by Citation bot r432

2013-07-15T13:59:13Z

Cleaned up using AutoEd | Assisted by Citation bot r432

← Older revision		Revision as of 15:59, 15 July 2013
Line 1:		Line 1:
			{{Orphan\|date=September 2011}}

			In [[mathematics]], the concept of '''groupoid algebra''' generalizes the notion of [[group algebra]].<ref>Khalkhali (2009), {{Google books quote\|id=UInc5AyTAikC\|page=48\|text=groupoid algebra\|p. 48}}</ref>

	It is definitely much common that prefer to have household doctor. Coverage it comes to oral health, most of this people won't be careful enough. They are going for the consultation belonging to the specialized doctor i.e. dentist when they get some dental or oral irritation. Most of the people don't prefer to have family dentist. But that is not a ~~right thing. Will be very much necessary have an understanding of the incredible importance of having family dentists. Together with having a truck driver dentist~~, ~~its very much necessary to go to him on regular good reason. This will help to keep the dental problems~~ and ~~also to cya of any dental problems in first stages itself.~~<br><br>~~Worn out front teeth: Is that your imagination or are your front teeth looking shorter than they used to? Actually~~, it~~'s true. A of depreciation can make teeth millimeters shorter compared~~ to ~~what they were several decades formerly. This can be fixed by treatments such~~ as ~~bonding or veneers~~ the ~~cosmetic dentist. Bonding can be~~ a product with this increasing applied towards surface in a mere the right shape. Via the internet that dentist have staying adept at sculpting to graduate from dental high school? It's true. Veneers are thin coverings that are put over ~~the tooth to hide flaws or create the more even appearance.~~<br><br>~~Contact insurance companies to get a list of acceptable St Paul family dentist ry professionals to choose from. This will narrow down your choices and a person find one close~~ by ~~that is accepting new men and women. Once you obtain that list many start calling around observe when are able to get everyone in regarding any check up. This is mindful yourself . way for~~ the ~~dentist to obtain a from everyone's teeth and for your family to obtain used into~~ the ~~dentist, especially the children.~~<br><br>~~Saliva plays an important part maintaining your mouth clean. In case your mouth always feel dry, you will have to hydrate more frequently dentist~~ and ~~use antibacterial rinses to make up for~~ the ~~associated with saliva. If you are taking any medication, look up a listing~~ of ~~side effects to the ads your medication could be causing mouth area to be dry.~~<br><br>~~If your teeth are heavily stained~~, ~~you needs to know that it may take multiple treatments to be able to obtain that bright smile that you want dentist . Teeth in which stained~~ and ~~discolored, just a few shades off white, much more to treat, no matter which cures you choose~~. ~~Badly discolored teeth are likely to require several treatments before improvement~~ is ~~observed~~.<br><br>~~If you~~ are ~~you looking for more info regarding [https~~:~~//www~~.~~youtube.com~~/~~watch?v~~=~~4To0cBm6IsM tampa Bay Health center~~] ~~check out the web site. Use~~ a ~~toothbrush~~ of ~~medium hardness~~ (~~assuming that experience healthy teeth and gums~~)~~. If during brushing your gums bleeding~~, ~~then use softer toothbrush. Soft are less traumatic for~~ the ~~affected gums~~ and ~~not damaging gums during clearing off~~.<br><br>~~The tooth is not strong enough~~ for ~~the to relish his favorite food following the medicine~~. ~~A cosmetic dentist in New York recommends the position~~ of ~~a crown together with the tooth to prevent damage whilst healing~~. ~~Following days, the infection at the pulp chamber of the tooth will disappear and the fundamental will be saved~~.		== Definition ==
			Given a [[groupoid]] <math>(G, \cdot)</math> and a [[field (mathematics)\|field]] <math>K</math>, it is possible to define the groupoid algebra <math>KG</math> as the [[Algebra over a field\|algebra]] over <math>K</math> formed by the [[vector space]] having the elements of <math>G</math> as [[generator (mathematics)\|generator]]s and having the [[multiplication]] of these elements defined by <math>g * h = g \cdot h</math>, whenever this product is defined, and <math>g * h = 0</math> otherwise. The product is then extended by [[linearity]].<ref>Dokuchaev, Exel & Piccione (2000), p. 7</ref>

			== Examples ==
			Some examples of groupoid algebras are the following:<ref>da Silva & Weinstein (1999), {{Google books quote\|id=2fcC1EGKz08C\|page=97\|text=groupoid algebras\|p. 97}}</ref>
			* [[Group algebra]]s
			* [[Matrix ring\|Matrix algebra]]s
			* [[Algebra of functions\|Algebras of functions]]

			== Properties ==
			* When a groupoid has a [[Wikt:finite\|finite]] number of [[Object (category theory)\|objects]] and a finite number of [[morphism (category theory)\|morphisms]], the groupoid algebra is a [[direct sum]] of [[tensor product]]s of group algebras and matrix algebras.<ref>Khalkhali & Marcolli (2008), {{Google books quote\|id=HsTkPOj0iusC\|page=210\|text=Groupoid algebra of a finite groupoid\|p. 210}}</ref>

			== See also ==
			* [[Hopf algebra]]
			* [[Partial group algebra]]

			== Notes ==
			<references />

			== References ==
			* {{cite book \|last1=Khalkhali \|first1=Masoud \|authorlink1= \|last2= \|first2= \|authorlink2= \|title=Basic Noncommutative Geometry \|url= \|edition= \|series=EMS Series of Lectures in Mathematics \|volume= \|year=2009 \|publisher=European Mathematical Society \|location= \|isbn=978-3-03719-061-6 }}
			* {{cite book \|last1=da Silva \|first1=Ana Cannas \|authorlink1= \|last2=Weinstein \|first2=Alan \|authorlink2= \|title=Geometric models for noncommutative algebras \|url= \|edition=2 \|series=Berkeley mathematics lecture notes \|volume=10 \|year=1999 \|publisher=AMS Bookstore \|location= \|isbn=978-0-8218-0952-5 }}
			* {{cite journal \|last1=Dokuchaev \|first1=M. \|last2=Exel \|first2=R. \|last3=Piccione \|first3=P. \|year=2000 \|title=Partial Representations and Partial Group Algebras \|journal=Journal of Algebra \|volume=226 \|issue= \|pages=505–532 \|publisher=Elsevier \|issn=0021-8693 \|doi= 10.1006/jabr.1999.8204\|url=\|arxiv= math/9903129
			}}
			* {{cite book \|last1=Khalkhali \|first1=Masoud \|authorlink1= \|last2=Marcolli \|first2=Matilde \|authorlink2= \|title=An invitation to noncommutative geometry \|url= \|edition= \|series= \|volume= \|year=2008 \|publisher=World Scientific \|location= \|isbn=978-981-270-616-4 }}

			[[Category:Algebra]]


			{{algebra-stub}}

en>Fraggle81: Reverted edit(s) by 216.67.22.196 identified as test/vandalism using STiki

2012-07-03T21:50:26Z

Reverted edit(s) by 216.67.22.196 identified as test/vandalism using STiki

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It is definitely much common that prefer to have household doctor. Coverage it comes to oral health, most of this people won't be careful enough. They are going for the consultation belonging to the specialized doctor i.e. dentist when they get some dental or oral irritation. Most of the people don't prefer to have family dentist. But that is not a right thing. Will be very much necessary have an understanding of the incredible importance of having family dentists. Together with having a truck driver dentist, its very much necessary to go to him on regular good reason. This will help to keep the dental problems and also to cya of any dental problems in first stages itself. Worn out front teeth: Is that your imagination or are your front teeth looking shorter than they used to? Actually, it's true. A of depreciation can make teeth millimeters shorter compared to what they were several decades formerly. This can be fixed by treatments such as bonding or veneers the cosmetic dentist. Bonding can be a product with this increasing applied towards surface in a mere the right shape. Via the internet that dentist have staying adept at sculpting to graduate from dental high school? It's true. Veneers are thin coverings that are put over the tooth to hide flaws or create the more even appearance. Contact insurance companies to get a list of acceptable St Paul family dentist ry professionals to choose from. This will narrow down your choices and a person find one close by that is accepting new men and women. Once you obtain that list many start calling around observe when are able to get everyone in regarding any check up. This is mindful yourself . way for the dentist to obtain a from everyone's teeth and for your family to obtain used into the dentist, especially the children. Saliva plays an important part maintaining your mouth clean. In case your mouth always feel dry, you will have to hydrate more frequently dentist and use antibacterial rinses to make up for the associated with saliva. If you are taking any medication, look up a listing of side effects to the ads your medication could be causing mouth area to be dry. If your teeth are heavily stained, you needs to know that it may take multiple treatments to be able to obtain that bright smile that you want dentist . Teeth in which stained and discolored, just a few shades off white, much more to treat, no matter which cures you choose. Badly discolored teeth are likely to require several treatments before improvement is observed. If you are you looking for more info regarding [https://www.youtube.com/watch?v=4To0cBm6IsM tampa Bay Health center] check out the web site. Use a toothbrush of medium hardness (assuming that experience healthy teeth and gums). If during brushing your gums bleeding, then use softer toothbrush. Soft are less traumatic for the affected gums and not damaging gums during clearing off. The tooth is not strong enough for the to relish his favorite food following the medicine. A cosmetic dentist in New York recommends the position of a crown together with the tooth to prevent damage whilst healing. Following days, the infection at the pulp chamber of the tooth will disappear and the fundamental will be saved.