Timsort - Revision history

en>John of Reading: Typo/general fixing, replaced: vice-versa → vice versa (2) using AWB

2015-01-08T09:32:54Z

Typo/general fixing, replaced: vice-versa → vice versa (2) using AWB

← Older revision		Revision as of 11:32, 8 January 2015
Line 1:		Line 1:
			I’m Erlinda from Boxholm studying Social Service. I did my schooling, secured 86% and hope to find someone with same interests in Climbing.<br><br>Here is my weblog - [http://kimstanleyrobinson.info/w/index.php5?title=User:MaxineFindley Substitute Teaching]

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178.78.227.14: Corrected space complexity of Smoothsort to O(1)

2014-02-13T09:14:38Z

Corrected space complexity of Smoothsort to O(1)

← Older revision		Revision as of 11:14, 13 February 2014
Line 1:		Line 1:
	In [[commutative algebra]], '''André–Quillen cohomology''' is a theory of [[cohomology]] for [[commutative ring]]s which is closely related to the [[cotangent complex]]. The first three cohomology groups were introduced by {{harvtxt\|Lichtenbaum\|Schlessinger\|1967}} and are sometimes called '''Lichtenbaum–Schlessinger functors''' ''T''<sup>0</sup>, ''T''<sup>1</sup>, ''T''<sup>2</sup>, and the higher groups were defined independently by M. André and by [[Daniel Quillen]] using methods of [[homotopy theory]]. It comes with a parallel homology theory called '''André–Quillen homology'''.

	~~==Motivation==~~
	Let ''A'' be a commutative ring, ''B'' be an ''A''-algebra, and ''M'' be a ''B''-module. André–Quillen cohomology is the derived functors of the [[derivation (abstract algebra)\|derivation]] functor Der<sub>''A''</sub>(''B'', ''M''). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings {{nowrap\|''A'' → ''B'' → ''C''}} and a ''C''-module ''M'', there is a three-term [[exact sequence]] of derivation modules:
	~~:<math>0 \to \operatorname{Der}_B(C, M) \to \operatorname{Der}_A(C, M) \to \operatorname{Der}_A(B, M).</math>~~
	This term can be extended to a six-term exact sequence using the functor [[Exalcomm]] of extensions of commutative algebras and a nine-term exact sequence using the [[Lichtenbaum–Schlessinger functor]]s. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.

	~~==Definition==~~		Surely the second option would be more beneficial for any website. When you cherished this informative article and also you desire to acquire more info relating to [http://xyz.ms/wordpress_backup_plugin_210303 wordpress backup] i implore you to visit the web page. You may discover this probably the most time-consuming part of building a Word - Press MLM website. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. s ultimately easy to implement and virtually maintenance free. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br>Creating a website from scratch can be such a pain. When you write a new post, you'll see a small bar that goes across the text input area. Which is perfect for building a mobile site for business use. Now, I want to anxiety that not every single query will be answered. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>This gives a clearer picture that online shoppers are familiar with the WP ecommerce system. s cutthroat competition prevailing in the online space won. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Provide the best and updated information to the web searchers and make use of these wonderful free themes and create beautiful websites. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. php file in the Word - Press root folder and look for this line (line 73 in our example):. Websites that do rank highly, do so becaue they use keyword-heavy post titles. So, we have to add our social media sharing buttons in website. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>Yet, overall, less than 1% of websites presently have mobile versions of their websites. I'm a large fan of using Word - Press to create pretty much any sort of web page. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. Extra investment in Drupal must be bypassed as value for money is what Drupal provides. The 2010 voting took place from July 7 through August 31, 2010.
	~~Let ''B''~~ be ~~an ''A''~~-~~algebra, and let ''M'' be~~ a ~~''B''~~-~~module~~. ~~Let ''P'' be~~ a ~~simplicial cofibrant ''A''-algebra resolution of ''B''~~. ~~André notates~~ the '~~'q''th cohomology group of ''B'' over ''A'' with coefficients in ''M'' by {{nowrap\|H~~<~~sup~~>~~''q''~~<~~/sup~~>~~(''A''~~, '~~'B'', ''M'')}}, while Quillen notates~~ the ~~same group as {{nowrap\|D<sup>''q''</sup>(''B''/''A'', ''M'')}}~~. ~~The ''q''th '''André–Quillen cohomology group'''~~ is:
	~~:<math>D^q(B/A~~, ~~M) = H^q(A~~, ~~B, M) \stackrel{\text{def}}{=} H^q(\operatorname{Der}_A(P, M))~~.<~~/math~~>
	~~The ''q''th '''André–Quillen homology group''' is:~~
	:<~~math~~>~~D_q(B/A~~, ~~M) = H_q(A~~, ~~B, M) \stackrel{\text{def}}{=} H_q(\Omega_{P/A} \otimes_B M)~~.<~~/math~~>

	~~Let {{nowrap\|L~~<~~sub~~>~~''B''/''A''</sub>}} denote the relative [[cotangent complex]] of ''B'' over ''A''~~. ~~Then we have~~ the ~~formulas:~~
	~~:<math>D^q~~(~~B/A, M~~) ~~= H^q(\operatorname{Hom}_B(L_{B/A}~~, ~~M))~~,<~~/math~~>
	:<~~math~~>~~D_q(B/A~~, ~~M) = H_q(L_{B/A} \otimes_B M).</math>~~

	~~==References==~~
	*{{Citation \| last1=André \| first1=M. \| title=Homologie des Algèbres Commutatives \| series=Grundlehren der mathematischen Wissenschaften \| volume=206 \| publisher=[[Springer-Verlag]] \| year=1974}}
	*{{Citation \| last1=Lichtenbaum \| first1=Stephen \|author1-link=Stephen Lichtenbaum \| last2=Schlessinger \| first2=M. \| title=The cotangent complex of ~~a morphism \| url=http://www.jstor.org/stable/1994516 \| id={{MR\|0209339}} \| year=1967 \| journal=[[Transactions~~ of ~~the American Mathematical Society]] \| issn=0002-9947 \| volume=128 \| pages=41–70}}~~
	* {{citation\| last1=Quillen \| first1=Daniel G. ~~\| title=Homology~~ of ~~commutative rings\|url=http://chromotopy.org/paste/quillen.djvu\|series=unpublished notes}}~~
	*{{Citation \| last1=Quillen \| first1=Daniel \| author1-~~link=Daniel Quillen \| title=On the (co-)homology~~ of ~~commutative rings \| series=Proc~~. ~~Symp~~. ~~Pure Mat~~. ~~\| volume=XVII \| publisher=[[American Mathematical Society]] \| year=1970}}~~
	*{{Citation \| last1=Weibel \| first1=Charles A. \| title=An introduction to homological algebra \| url=http://books.google.~~com/books?id=flm-dBXfZ_gC \| publisher=[[Cambridge University Press]] \| series=Cambridge Studies in Advanced Mathematics \| isbn=978-0-521-43500-0 \| id={{MR\|1269324}} \| year=1994 \| volume=38}}~~

	~~{{DEFAULTSORT:Andre-Quillen cohomology}}~~
	~~[[Category:Commutative algebra]]~~
	~~[[Category:Homotopy theory]]~~

en>BrotherE: /* Performance */ Replaced average with worst. Average case can be less than n log n because average depends on input distribution and for certain pathological distributions (for example if at most 1 element is misplaced), there are sorts that do better.

2013-12-19T20:03:16Z

Performance: Replaced average with worst. Average case can be less than n log n because average depends on input distribution and for certain pathological distributions (for example if at most 1 element is misplaced), there are sorts that do better.

← Older revision		Revision as of 22:03, 19 December 2013
Line 1:		Line 1:
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			==Motivation==
			Let ''A'' be a commutative ring, ''B'' be an ''A''-algebra, and ''M'' be a ''B''-module. André–Quillen cohomology is the derived functors of the [[derivation (abstract algebra)\|derivation]] functor Der<sub>''A''</sub>(''B'', ''M''). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings {{nowrap\|''A'' → ''B'' → ''C''}} and a ''C''-module ''M'', there is a three-term [[exact sequence]] of derivation modules:
			:<math>0 \to \operatorname{Der}_B(C, M) \to \operatorname{Der}_A(C, M) \to \operatorname{Der}_A(B, M).</math>
			This term can be extended to a six-term exact sequence using the functor [[Exalcomm]] of extensions of commutative algebras and a nine-term exact sequence using the [[Lichtenbaum–Schlessinger functor]]s. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.

			==Definition==
			Let ''B'' be an ''A''-algebra, and let ''M'' be a ''B''-module. Let ''P'' be a simplicial cofibrant ''A''-algebra resolution of ''B''. André notates the ''q''th cohomology group of ''B'' over ''A'' with coefficients in ''M'' by {{nowrap\|H<sup>''q''</sup>(''A'', ''B'', ''M'')}}, while Quillen notates the same group as {{nowrap\|D<sup>''q''</sup>(''B''/''A'', ''M'')}}. The ''q''th '''André–Quillen cohomology group''' is:
			:<math>D^q(B/A, M) = H^q(A, B, M) \stackrel{\text{def}}{=} H^q(\operatorname{Der}_A(P, M)).</math>
			The ''q''th '''André–Quillen homology group''' is:
			:<math>D_q(B/A, M) = H_q(A, B, M) \stackrel{\text{def}}{=} H_q(\Omega_{P/A} \otimes_B M).</math>

			Let {{nowrap\|L<sub>''B''/''A''</sub>}} denote the relative [[cotangent complex]] of ''B'' over ''A''. Then we have the formulas:
			:<math>D^q(B/A, M) = H^q(\operatorname{Hom}_B(L_{B/A}, M)),</math>
			:<math>D_q(B/A, M) = H_q(L_{B/A} \otimes_B M).</math>

			==References==
			*{{Citation \| last1=André \| first1=M. \| title=Homologie des Algèbres Commutatives \| series=Grundlehren der mathematischen Wissenschaften \| volume=206 \| publisher=[[Springer-Verlag]] \| year=1974}}
			*{{Citation \| last1=Lichtenbaum \| first1=Stephen \|author1-link=Stephen Lichtenbaum \| last2=Schlessinger \| first2=M. \| title=The cotangent complex of a morphism \| url=http://www.jstor.org/stable/1994516 \| id={{MR\|0209339}} \| year=1967 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=128 \| pages=41–70}}
			* {{citation\| last1=Quillen \| first1=Daniel G. \| title=Homology of commutative rings\|url=http://chromotopy.org/paste/quillen.djvu\|series=unpublished notes}}
			*{{Citation \| last1=Quillen \| first1=Daniel \| author1-link=Daniel Quillen \| title=On the (co-)homology of commutative rings \| series=Proc. Symp. Pure Mat. \| volume=XVII \| publisher=[[American Mathematical Society]] \| year=1970}}
			*{{Citation \| last1=Weibel \| first1=Charles A. \| title=An introduction to homological algebra \| url=http://books.google.com/books?id=flm-dBXfZ_gC \| publisher=[[Cambridge University Press]] \| series=Cambridge Studies in Advanced Mathematics \| isbn=978-0-521-43500-0 \| id={{MR\|1269324}} \| year=1994 \| volume=38}}

			{{DEFAULTSORT:Andre-Quillen cohomology}}
			[[Category:Commutative algebra]]
			[[Category:Homotopy theory]]

75.136.229.168: /* Minrun */ clarity, anaphora, idiomatic English

2012-08-21T03:14:14Z

Minrun: clarity, anaphora, idiomatic English

New page

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