Hooke's atom - Revision history

en>Monkbot: /* Further reading */Task 3: Fix CS1 deprecated coauthor parameter errors

2014-05-12T03:10:47Z

Further reading: Task 3: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 05:10, 12 May 2014
Line 1:		Line 1:
	In [[mathematics]], especially in the areas of [[abstract algebra]] dealing with [[group cohomology]] or relative homological algebra, '''Shapiro's lemma''', also known as the '''Eckmann–Shapiro lemma''', relates extensions of modules over one ring to extensions over another, especially the [[group ring]] of a [[group (mathematics)\|group]] and of a [[subgroup]]. It thus relates the [[group cohomology]] with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold Shapiro, who proved it in 1961;<ref>{{citation\|title=Differential algebra and algebraic groups\|volume=54\|series=Pure and applied mathematics\|first=Ellis Robert\|last=Kolchin\|publisher=Academic Press\|year=1973\|isbn=978-0-12-417650-8\|page=53\|url=http://books.google.com/books?id=yDCfhIjka-8C&pg=PA53}}.</ref> however, [[Beno Eckmann]] had discovered it earlier, in 1953.<ref>{{citation\|series=Lectures Notes in Mathematics\|volume=1758\|year=2001\|title=Continuous Bounded Cohomology of Locally Compact Groups\|first=Nicolas\|last=Monod\|publisher=Springer-Verlag\|contribution=Cohomological techniques\|pages=129–168\|doi=10.1007/3-540-44962-0_5\|isbn=978-3-540-42054-5}}.</ref>

	~~==Statement for rings==~~
	Let ''R'' → ''S'' be a [[ring homomorphism]], so that ''S'' becomes a left and right ''R''-module. Let ''M'' be a left ''S''-module and ''N'' a left ''R''-module. By restriction of scalars, ''M'' is also a left ''R''-module.
	* If ''S'' is projective as a right ''R''-module, then:
	~~:<math>\operatorname{Ext}^n_R(N, {}_R M) \cong \operatorname{Ext}^n_S(S \otimes_R N, M)</math>~~
	* If ''S'' is projective as a left ''R''-module, then:
	~~:<math>\operatorname{Ext}^n_R({}_R M,N) \cong \operatorname{Ext}^n_S(M,\operatorname{Hom}_R(S,N))</math>~~

	~~See {{harv\|Benson\|1991\|p=47}}. The projectivity conditions~~ can ~~be weakened into conditions on~~ the ~~vanishing~~ of ~~certain Tor~~- ~~or Ext-groups: see {{harv\|Cartan\|Eilenberg\|1956\|p=118\|loc=VI~~.~~§5}}~~.		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. This CMS has great flexibility to adapt various extensions and add-ons. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. <br><br>You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. This plugin is a must have for anyone who is serious about using Word - Press. They provide many such popular products which you can buy for your baby. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business. <br><br>Minor and medium sized corporations also have a lot to obtain by shelling out in a very good website. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. You've got invested a great cope of time developing and producing up the topic substance. The first thing you need to do is to choose the right web hosting plan. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. If you have any sort of questions pertaining to where and the best ways to use [http://deurl.de/wordpress_dropbox_backup_40988 wordpress backup], you can contact us at the internet site. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Millions of individuals and organizations are now successfully using this tool throughout the world. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. Word - Press is an open source content management system which is easy to use and offers many user friendly features. As for performing online business, websites and blogs are the only medium that are available to interact with customers and Word - Press perform this work with the help of cross-blog communication tools, comments and full user registration plug-ins.

	~~==Statement for group rings==~~
	~~When ''H''~~ is ~~a subgroup~~ of ~~finite [[Index~~ of ~~a subgroup\|index]] in ''G'', then~~ the ~~group ring ''R''[''G''] is finitely generated projective as a left~~ and ~~right ''R''[''H''] module,~~ so the ~~previous applies in a simple way. Let ''M'' be a finite dimensional representation~~ of ~~''G''~~ and ~~''N'' a finite dimensional representation of ''H''~~. ~~In this case, the module ''S'' ⊗~~<~~sub~~>~~''R''~~<~~/sub~~> ~~''N''~~ is ~~called~~ the ~~[[induced representation]] of ''N'' from ''H''~~ to ~~''G''~~, ~~and~~ <~~sub~~>~~''R''~~<~~/sub~~>~~''M'' is called the [[restricted representation]] of ''M'' from ''G''~~ to ~~''H''~~. ~~One has that:~~
	~~:<math>\operatorname{Ext}^n_G( M~~, ~~N\uparrow_H^G) \cong \operatorname{Ext}^n_H( M\downarrow_H^G~~, ~~N)</math>~~

	~~When ''n'' = 0~~, ~~this is called [[Frobenius reciprocity]] for completely reducible modules~~, and ~~Nakayama reciprocity in general~~. ~~See {{harv\|Benson\|1991\|p=42}}, which also contains these higher versions~~ of the ~~Mackey decomposition~~.

	~~==Statement for group cohomology==~~
	~~Specializing ''M''~~ to be the ~~trivial module produces the familiar Shapiro's lemma~~. ~~Let ''H'' be~~ a ~~subgroup of ''G''~~ and ~~''N'' a representation of ''H''~~. ~~For ''N''~~<~~sup~~>~~''G''~~<~~/sup~~> ~~the [[induced representation]]~~ of ~~''N'' from ''H''~~ to ~~''G'' using the [[tensor product]],~~ and ~~for H<sub>*</sub>~~ the [~~[group homology]]~~:
	~~:H<sub><~~/~~sub>(''G'', ''N''<sup>''G''<~~/~~sup>) = H<sub><~~/~~sub>(''H''~~, ~~''N'')~~
	~~Similarly~~, ~~for ''N''<sup>''G''</sup>~~ the ~~co-induced representation~~ of ~~''N'' from ''H''~~ to ~~''G'' using~~ the ~~[[Hom functor]]~~, and ~~for H~~<~~sup~~>*<~~/sup~~> the ~~[[group cohomology]]:~~
	~~:H<sup></sup>(''G~~'', ~~''N''<sup>''G''</sup>) = H<sup></sup>(''H''~~, ~~''N'')~~

	~~When ''H''~~ is ~~finite index in ''G''~~, ~~then the induced and coinduced representations coincide~~ and the ~~lemma is valid for both homology~~ and ~~cohomology.~~

	~~See {{harv\|Weibel\|\|p=172}}.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{Citation \| last1=Benson \| first1=D. J. \| title=Representations and cohomology. I \| publisher=[[Cambridge University Press]] \| series=Cambridge Studies in Advanced Mathematics \| isbn=978-~~0-521-36134-7 \| id={{MathSciNet \| id = 1110581}} \| year=1991 \| volume=30}}~~
	* {{Citation \| last1=Cartan \| first1=H. \|last2=Eilenberg \| first2=S. \| title=Homological Algebra \| publisher=[[Princeton University Press~~]] \| year=1956}}~~
	*{{citation
	~~\| last = Eckmann \| first = Beno \| authorlink = Beno Eckmann~~
	~~\| mr = 0058600~~
	~~\| journal = Annals~~ of ~~Mathematics \| series = 2nd ser.~~
	~~\| pages = 481–493~~
	~~\| title = Cohomology of groups~~ and ~~transfer~~
	~~\| volume = 58~~
	~~\| year = 1953~~
	~~\| doi = 10~~.~~2307/1969749~~
	~~\| issue = 3}}.~~
	* Page 59 of {{Neukirch et al. CNF}}
	* {{Weibel IHA}}

	~~[[Category:Homological algebra]]~~
	~~[[Category:Representation theory]]~~

37.109.54.194 at 21:08, 5 January 2014

2014-01-05T21:08:37Z

← Older revision		Revision as of 23:08, 5 January 2014
Line 1:		Line 1:
	~~Surely~~ the ~~second option would be more beneficial for any website. You may discover this probably~~ the ~~most time-consuming part~~ of ~~building~~ a ~~Word - Press MLM website~~. ~~The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. If you are using videos on your site then this is~~ the ~~plugin~~ to ~~use~~. It's ~~as simple as hiring a Wordpress plugin developer or learning how to create what~~ is ~~needed.~~ <br>~~<br>If you beloved this article~~ and ~~you also would like to collect more info with regards to [~~http://~~zpib~~.com/~~wordpress_dropbox_backup_514631 wordpress backup~~] ~~nicely visit the web site~~. ~~Any business enterprise that is certainly worth its name should really shell out a good deal~~ in ~~making sure that they have the most effective website that provides related info to its prospect~~. ~~Best of all~~, ~~you can still have all the functionality~~ that ~~you desire when you use the Word~~ - ~~Press platform~~. ~~Well Managed Administration The Word~~ - ~~Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly~~. By ~~purchasing Word - Press weblogs you can acquire your very own domain title and have total command~~ of ~~your web site. You can~~ also ~~get~~ a ~~free keyword tool that~~ is ~~to determine how strong other competing sites are and number of the searches on the most popular search sites.~~ <br><br>~~Digital photography~~ is a ~~innovative effort~~, ~~if you removethe stress to catch every position and viewpoint~~ of a ~~place~~, ~~you free yourself up to be more innovative~~ and ~~your outcomes will~~ be ~~much better. When~~ a ~~business benefits from its own domain name~~ and a ~~tailor-made blog~~, the ~~odds~~ of ~~ranking higher in~~ the ~~search engines and being visible~~ to ~~a greater number of people~~ is ~~more likely~~. ~~After age 35~~, ~~18%~~ of ~~pregnancies will end in miscarriage~~. ~~Storing write-ups in advance would have~~ to be ~~neccessary with~~ the ~~auto blogs~~. ~~Article Source:~~ ~~Stevens works in Internet~~ and ~~Network Marketing~~. <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. Find more information about Design To Wordpress here. Next you~~'~~ll go by way of~~ to ~~your simple Word - Press site. If you just want to share some picture~~ and ~~want to use it as a dairy~~, ~~that you want~~ to ~~share with your friends and family members~~, ~~then blogger would be an excellent choice. Word - Press offers constant updated services~~ and ~~products~~, ~~that too, absolutely free of cost.~~ <br><br>~~Every single module contains published data and guidelines~~, ~~usually a lot more than 1 video~~, and ~~when pertinent, incentive links~~ and ~~PDF files to assist you out~~. ~~Automated deal feed integration option to populate your blog with relevant deals~~. ~~However, you must also manually approve or reject comments so that your website does not promote parasitic behavior~~. ~~Page speed is an important factor~~ in ~~ranking, especially with Google~~. ~~I have never seen a plugin with such a massive array~~ of ~~features, this does everything that platinum SEO~~ and ~~All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife~~ of ~~Word - Press plugins~~.		In [[mathematics]], especially in the areas of [[abstract algebra]] dealing with [[group cohomology]] or relative homological algebra, '''Shapiro's lemma''', also known as the '''Eckmann–Shapiro lemma''', relates extensions of modules over one ring to extensions over another, especially the [[group ring]] of a [[group (mathematics)\|group]] and of a [[subgroup]]. It thus relates the [[group cohomology]] with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold Shapiro, who proved it in 1961;<ref>{{citation\|title=Differential algebra and algebraic groups\|volume=54\|series=Pure and applied mathematics\|first=Ellis Robert\|last=Kolchin\|publisher=Academic Press\|year=1973\|isbn=978-0-12-417650-8\|page=53\|url=http://books.google.com/books?id=yDCfhIjka-8C&pg=PA53}}.</ref> however, [[Beno Eckmann]] had discovered it earlier, in 1953.<ref>{{citation\|series=Lectures Notes in Mathematics\|volume=1758\|year=2001\|title=Continuous Bounded Cohomology of Locally Compact Groups\|first=Nicolas\|last=Monod\|publisher=Springer-Verlag\|contribution=Cohomological techniques\|pages=129–168\|doi=10.1007/3-540-44962-0_5\|isbn=978-3-540-42054-5}}.</ref>

			==Statement for rings==
			Let ''R'' → ''S'' be a [[ring homomorphism]], so that ''S'' becomes a left and right ''R''-module. Let ''M'' be a left ''S''-module and ''N'' a left ''R''-module. By restriction of scalars, ''M'' is also a left ''R''-module.
			* If ''S'' is projective as a right ''R''-module, then:
			:<math>\operatorname{Ext}^n_R(N, {}_R M) \cong \operatorname{Ext}^n_S(S \otimes_R N, M)</math>
			* If ''S'' is projective as a left ''R''-module, then:
			:<math>\operatorname{Ext}^n_R({}_R M,N) \cong \operatorname{Ext}^n_S(M,\operatorname{Hom}_R(S,N))</math>

			See {{harv\|Benson\|1991\|p=47}}. The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see {{harv\|Cartan\|Eilenberg\|1956\|p=118\|loc=VI.§5}}.

			==Statement for group rings==
			When ''H'' is a subgroup of finite [[Index of a subgroup\|index]] in ''G'', then the group ring ''R''[''G''] is finitely generated projective as a left and right ''R''[''H''] module, so the previous applies in a simple way. Let ''M'' be a finite dimensional representation of ''G'' and ''N'' a finite dimensional representation of ''H''. In this case, the module ''S'' ⊗<sub>''R''</sub> ''N'' is called the [[induced representation]] of ''N'' from ''H'' to ''G'', and <sub>''R''</sub>''M'' is called the [[restricted representation]] of ''M'' from ''G'' to ''H''. One has that:
			:<math>\operatorname{Ext}^n_G( M, N\uparrow_H^G) \cong \operatorname{Ext}^n_H( M\downarrow_H^G, N)</math>

			When ''n'' = 0, this is called [[Frobenius reciprocity]] for completely reducible modules, and Nakayama reciprocity in general. See {{harv\|Benson\|1991\|p=42}}, which also contains these higher versions of the Mackey decomposition.

			==Statement for group cohomology==
			Specializing ''M'' to be the trivial module produces the familiar Shapiro's lemma. Let ''H'' be a subgroup of ''G'' and ''N'' a representation of ''H''. For ''N''<sup>''G''</sup> the [[induced representation]] of ''N'' from ''H'' to ''G'' using the [[tensor product]], and for H<sub>*</sub> the [[group homology]]:
			:H<sub></sub>(''G'', ''N''<sup>''G''</sup>) = H<sub></sub>(''H'', ''N'')
			Similarly, for ''N''<sup>''G''</sup> the co-induced representation of ''N'' from ''H'' to ''G'' using the [[Hom functor]], and for H<sup>*</sup> the [[group cohomology]]:
			:H<sup></sup>(''G'', ''N''<sup>''G''</sup>) = H<sup></sup>(''H'', ''N'')

			When ''H'' is finite index in ''G'', then the induced and coinduced representations coincide and the lemma is valid for both homology and cohomology.

			See {{harv\|Weibel\|\|p=172}}.

			==Notes==
			{{reflist}}

			==References==
			* {{Citation \| last1=Benson \| first1=D. J. \| title=Representations and cohomology. I \| publisher=[[Cambridge University Press]] \| series=Cambridge Studies in Advanced Mathematics \| isbn=978-0-521-36134-7 \| id={{MathSciNet \| id = 1110581}} \| year=1991 \| volume=30}}
			* {{Citation \| last1=Cartan \| first1=H. \|last2=Eilenberg \| first2=S. \| title=Homological Algebra \| publisher=[[Princeton University Press]] \| year=1956}}
			*{{citation
			\| last = Eckmann \| first = Beno \| authorlink = Beno Eckmann
			\| mr = 0058600
			\| journal = Annals of Mathematics \| series = 2nd ser.
			\| pages = 481–493
			\| title = Cohomology of groups and transfer
			\| volume = 58
			\| year = 1953
			\| doi = 10.2307/1969749
			\| issue = 3}}.
			* Page 59 of {{Neukirch et al. CNF}}
			* {{Weibel IHA}}

			[[Category:Homological algebra]]
			[[Category:Representation theory]]

en>Tdoune: /* Solution */

2012-04-03T12:45:56Z

Solution

New page

Surely the second option would be more beneficial for any website. You may discover this probably the most time-consuming part of building a Word - Press MLM website. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. If you beloved this article and you also would like to collect more info with regards to [http://zpib.com/wordpress_dropbox_backup_514631 wordpress backup] nicely visit the web site. Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs. Article Source: Stevens works in Internet and Network Marketing. The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. Find more information about Design To Wordpress here. Next you'll go by way of to your simple Word - Press site. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost. Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Automated deal feed integration option to populate your blog with relevant deals. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. Page speed is an important factor in ranking, especially with Google. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.