Objectivity (frame invariance) - Revision history

en>Maurice Carbonaro: /* Euclidean transformation */Hyperlinked "(...) position vector (...)" to the "Position (vector)" article

2014-12-10T10:31:51Z

Euclidean transformation: Hyperlinked "(...) position vector (...)" to the "Position (vector)" article

← Older revision		Revision as of 12:31, 10 December 2014
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	~~In [[mathematics]]~~, ~~especially in the area of [[abstract algebra\|algebra]] known as [[module theory]]~~, ~~'''Schanuel's lemma''', named after [[Stephen Schanuel]], allows one~~ to ~~compare how far modules depart from being [[projective module\|projective]]~~. ~~It is useful~~ in ~~defining~~ the ~~Heller operator in~~ the ~~stable category~~, ~~and~~ in ~~giving elementary descriptions~~ of ~~[[projective dimension\|dimension shifting]]~~.		If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. If you are a positive thinker businessman then today you have to put your business online. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. To turn the Word - Press Plugin on, click Activate on the far right side of the list. There are plenty of tables that are attached to this particular database. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. If you liked this article and you also would like to acquire more info pertaining to [http://www.lvlywallpapers.com/profile/albarker wordpress backup] generously visit the web site. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. This includes enriching the content with proper key words, tactfully defining the tags and URL. <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. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Thus, Word - Press is a good alternative if you are looking for free blogging software. 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.

	~~==Statement==~~

	~~'''Schanuel's lemma'''~~ is the ~~following statement:~~

	If ~~0  →  ''K''  → ''P'' →  ''M'' →  0 and 0  → ''K'''  →  ''P'' '  →  ''M''  → 0~~ are ~~[[short exact sequence]]s~~ of ~~''R''~~-~~modules and ''P'' and ''P'' '~~ are ~~projective~~, then '~~'K'' &oplus; ''P'' '~~ is ~~[[Isomorphism\|isomorphic]]~~ to ~~''K'' ' &oplus; ''P.''~~

	~~==Proof==~~

	~~Define~~ the ~~following [[submodule]] of ''P'' &oplus; ''P'' ', where φ : ''P'' → ''M''~~ and ~~φ' : ''P'' ' → ''M'':~~

	:<~~math~~> ~~X = \{ (p,q) \in P \oplus P^\prime : \phi(p) = \phi^\prime(q) \}.~~ <~~/math~~>

	~~The map π~~ : ~~''X'' → ''P''~~, ~~where π is defined as~~ the ~~projection~~ of the ~~first coordinate~~ of ~~''X'' into ''P'', is surjective~~. ~~Since φ' is surjective, for any ''p''~~ <~~math~~>~~\in~~<~~/math~~> ~~''P''~~, ~~one may find~~ a ~~''q'' <math>\in</math> ''P'' ' such that φ~~(~~''p''~~) ~~= φ~~ '~~(''q'')~~. This ~~gives (''p'',''q'') <math>\~~in~~</math> ''X'' with π (''p'',''q'') = ''p''~~. ~~Now examine the~~ [~~[kernel (algebra)\|kernel]~~] of the ~~map π :~~

	~~<math>~~
	~~\begin{align}~~
	~~\text{ker} \; \pi &= \{ (0,q): (0,q) \~~in ~~X \} \\~~
	~~& = \{ (0~~,~~q): \phi^\prime(q) =0 \} \\~~
	~~& \cong \; \text{ker} \; \phi^\prime \cong K^\prime~~.
	~~\end{align}~~ <~~/math~~>

	~~We may conclude that there is a short exact sequence~~

	:<~~math~~> ~~0 \rightarrow K^\prime \rightarrow X \rightarrow P \rightarrow 0. </math>~~

	~~Since ''P'' is projective this sequence splits~~, ~~so ''X'' &cong; ''K'' ' &oplus; ''P'' . Similarly, we can write another map π : ''X'' → ''P'' '~~, and ~~the same argument as above shows that there is another short exact sequence~~

	~~:<math> 0 \rightarrow K \rightarrow X \rightarrow P^\prime \rightarrow 0~~, ~~</math>~~

	and ~~so ''X'' &cong; ''P'' ' &oplus; ''K''~~. ~~Combining the two equivalences~~ for ~~''X'' gives the desired result.~~

	~~==Long exact sequences==~~

	~~The above argument may also~~ be ~~generalized to [[long exact sequence]]s~~.~~<ref>{{cite book \| author = Lam~~, ~~T.Y. \| title = Lectures on Modules and Rings \| publisher = Springer \| year = 1999 \| id = ISBN 0~~-~~387~~-~~98428~~-~~3}} pgs~~. ~~165–167.</ref>~~

	~~==Origins==~~

	~~[[Stephen Schanuel]] discovered the argument in [[Irving Kaplansky]]'s [[homological algebra]] course at the [[University of Chicago]] in Autumn of 1958. Kaplansky writes:~~
	~~:''Early in the course~~ I ~~formed~~ a ~~one-step projective resolution~~ of ~~a module~~, and ~~remarked that if the kernel was projective~~ in ~~one resolution it was projective in all. I added that, although~~ the ~~statement was so simple~~ and ~~straightforward, it would be~~ a ~~while before we proved~~ it~~. Steve Shanuel spoke up and told me and~~ the class that it was quite easy, and thereupon sketched what has come to be known as "Schanuel's lemma." '' <ref>{{cite book \| author = Kaplansky, Irving. \| title = Fields and Rings \| publisher = University Of Chicago Press ~~\| year = 1972 \| id = ISBN 0-226-42451-0}} pgs~~. ~~165–168.</ref>~~

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

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

en>Rhododendrites: clean up using AWB

2013-12-20T04:37:22Z

clean up using AWB

← Older revision		Revision as of 06:37, 20 December 2013
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	~~Is there anything funnier than watching a toddler wibble-wobble~~ as ~~they quite take automobile steps? The moment~~ the ~~tears of laughter come~~ the ~~tears~~ of ~~delight~~. ~~The moment baby puts one foot ahead~~ of ~~the other for before represents no more one era~~ and the ~~starting~~ of ~~another.~~<br><br>~~If searching to get started in skateboarding~~, ~~you will have to to a few inexpensive sneaker. There are a lot~~ of ~~various organizations that market these, but sight within~~ the first ~~demand much more than other individuals. Now that skate footwear are becoming a much more popular~~, ~~some skate shoe companies promote shoes that even with skateboarding~~. ~~Anyone require regarding cautious when you get a associated with inexpensive athletic air jordan shoes. 1 firm that only sells true shoes~~ is ~~Adio Boot. Adio shoes are going~~ for ~~skateboarding~~, ~~understanding~~ that ~~is them~~. ~~Even so~~, ~~they still handle to look good.~~<br><br>~~You must become conversant~~ with the ~~lingo how~~ the ~~sellers use to sell their stuff. If they use terms like "variant"~~, ~~it radically~~, ~~and it is not the real Air Jordan released out of the factory and generally are~~ in ~~fact being sold without their consent. Must be enough signals anyone to keep away from individuals~~.<br><br>~~No doubt, thousands of types of shoes are within the whole~~. ~~Some people prefer to wear simple waders along with several want to receive on fashionable and stylish foot~~ is ~~wearing~~. ~~However~~, ~~most of~~ the ~~people feel discomfort their particular shoes after passing sometime. You may have problems with your waders~~ as ~~quite. On the other hand, Nike air jordan may well free of problem. That means you would never have hitches with this pair of shoes. It~~ is ~~mainly because they have grown to be easy to use and don't offer any style of ache at virtually.~~<br><br>~~Bearing in your~~ the two ~~certainties of Baby Footwear Science, appears like fitting (and rather cute) that Nike has designed a range of Air Jordan shoes~~ for ~~infants~~. ~~Comfy are created using the best of sports technology precision, and take under consideration the needs of developing babies~~.<br>~~<br>In issue time~~, ~~running is more effective than walk in shedding pounds~~ and ~~high~~-~~intensity running increases results than taking walks~~. ~~The performance on~~ the ~~exercise decide a huge~~ of ~~fat~~. If the ~~exercise and skills improve, your effect~~ of ~~weight-loss will be increased~~, ~~but since not, things will engage~~ in ~~the divergent~~. ~~It's easy to get~~ that ~~in the same time~~, the ~~long you run~~, the ~~better effect you will get~~, as ~~well as the more fast you run in the unit time, he better effect you get in your weight loosing~~.<br>~~<br>If you beloved this article~~ and ~~you would like to acquire more info pertaining to [http://www.plongeeo.com/chaussure~~-~~air~~-~~jordan~~-~~pas-cher~~/ ~~chaussure air jordan pas cher~~] ~~kindly visit our own page.~~		In [[mathematics]], especially in the area of [[abstract algebra\|algebra]] known as [[module theory]], '''Schanuel's lemma''', named after [[Stephen Schanuel]], allows one to compare how far modules depart from being [[projective module\|projective]]. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of [[projective dimension\|dimension shifting]].

			==Statement==

			'''Schanuel's lemma''' is the following statement:

			If 0  →  ''K''  → ''P'' →  ''M'' →  0 and 0  → ''K'''  →  ''P'' '  →  ''M''  → 0 are [[short exact sequence]]s of ''R''-modules and ''P'' and ''P'' ' are projective, then ''K'' &oplus; ''P'' ' is [[Isomorphism\|isomorphic]] to ''K'' ' &oplus; ''P.''

			==Proof==

			Define the following [[submodule]] of ''P'' &oplus; ''P'' ', where φ : ''P'' → ''M'' and φ' : ''P'' ' → ''M'':

			:<math> X = \{ (p,q) \in P \oplus P^\prime : \phi(p) = \phi^\prime(q) \}. </math>

			The map π : ''X'' → ''P'', where π is defined as the projection of the first coordinate of ''X'' into ''P'', is surjective. Since φ' is surjective, for any ''p'' <math>\in</math> ''P'', one may find a ''q'' <math>\in</math> ''P'' ' such that φ(''p'') = φ '(''q''). This gives (''p'',''q'') <math>\in</math> ''X'' with π (''p'',''q'') = ''p''. Now examine the [[kernel (algebra)\|kernel]] of the map π :

			<math>
			\begin{align}
			\text{ker} \; \pi &= \{ (0,q): (0,q) \in X \} \\
			& = \{ (0,q): \phi^\prime(q) =0 \} \\
			& \cong \; \text{ker} \; \phi^\prime \cong K^\prime.
			\end{align} </math>

			We may conclude that there is a short exact sequence

			:<math> 0 \rightarrow K^\prime \rightarrow X \rightarrow P \rightarrow 0. </math>

			Since ''P'' is projective this sequence splits, so ''X'' &cong; ''K'' ' &oplus; ''P'' . Similarly, we can write another map π : ''X'' → ''P'' ', and the same argument as above shows that there is another short exact sequence

			:<math> 0 \rightarrow K \rightarrow X \rightarrow P^\prime \rightarrow 0, </math>

			and so ''X'' &cong; ''P'' ' &oplus; ''K''. Combining the two equivalences for ''X'' gives the desired result.

			==Long exact sequences==

			The above argument may also be generalized to [[long exact sequence]]s.<ref>{{cite book \| author = Lam, T.Y. \| title = Lectures on Modules and Rings \| publisher = Springer \| year = 1999 \| id = ISBN 0-387-98428-3}} pgs. 165–167.</ref>

			==Origins==

			[[Stephen Schanuel]] discovered the argument in [[Irving Kaplansky]]'s [[homological algebra]] course at the [[University of Chicago]] in Autumn of 1958. Kaplansky writes:
			:''Early in the course I formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all. I added that, although the statement was so simple and straightforward, it would be a while before we proved it. Steve Shanuel spoke up and told me and the class that it was quite easy, and thereupon sketched what has come to be known as "Schanuel's lemma." '' <ref>{{cite book \| author = Kaplansky, Irving. \| title = Fields and Rings \| publisher = University Of Chicago Press \| year = 1972 \| id = ISBN 0-226-42451-0}} pgs. 165–168.</ref>

			==Notes==
			{{reflist}}

			[[Category:Homological algebra]]
			[[Category:Module theory]]

en>Helpful Pixie Bot: ISBNs (Build KH)

2012-05-12T00:26:24Z

ISBNs (Build KH)

New page

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