Empty lattice approximation - Revision history

en>AnomieBOT: Dating maintenance tags: {{Clarification}}

2014-11-10T01:06:47Z

Dating maintenance tags: {{Clarification}}

← Older revision		Revision as of 03:06, 10 November 2014
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	~~{{unreferenced\|date=January 2013}}~~		Today's wise client knows how advantageous using credit cards could be, but is additionally aware about the problems connected with unneccessary use. Even most thrifty of folks use their bank cards occasionally, and we all have training to learn from them! Keep reading for beneficial tips on using credit cards smartly.<br><br>
	~~In [[commutative algebra]]~~, ~~a '''ring~~ of ~~mixed characteristic''' is a commutative ring ''R'' having [[characteristic (algebra)\|characteristic]] zero~~ and ~~having an ideal ''I'' such that {{nowrap\|''R'' / ''I''}} has positive characteristic~~.

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	* The integers '''Z''' have characteristic zero, but ~~for~~ any ~~[[prime number]] ''p~~'', ~~'''Z'''/(''p'')~~ is ~~a [[finite field]]~~ with ~~''p'' elements and hence has characteristic ''p''~~.
	* The ring of integers of any [[number field]] is ~~of mixed characteristic~~
	* Fix a prime ''p'' and [~~[localization of a ring\|localize]~~] the ~~integers at~~ the ~~prime ideal (''p'')~~. ~~The resulting ring '''Z'''~~<~~sub~~>~~(''p'')~~<~~/sub~~> ~~has characteristic zero~~. ~~It has~~ a ~~unique maximal ideal ''p'''''Z'''~~<~~sub~~>~~(''p'')~~<~~/sub~~>, ~~and~~ the ~~quotient '''Z'''<sub>(''p'')</sub>/''p'''''Z'''<sub>(''p'')</sub> is~~ a ~~finite field with ''p'' elements~~. ~~In contrast~~ to ~~the previous example~~, ~~the~~ only ~~possible characteristics for rings of~~ the ~~form {{nowrap\|'''Z'''~~<~~sub~~>~~(''p'')~~</~~sub>~~ / ~~''I''}}~~ are ~~zero (~~when ~~''I'' is~~ the ~~zero ideal)~~ and ~~powers~~ of ~~''p'' (when ''I'' is any other non-unit ideal); it is~~ not ~~possible to have a quotient of any other characteristic~~.
	* If <~~math~~>P<~~/math~~> is a ~~non-zero prime ideal~~ of the ~~ring~~ <~~math~~>~~\mathcal{O}_K~~<~~/math~~> ~~of integers of~~ a ~~number field <math>K</math> then~~ the ~~[[localization~~ of a ~~ring\|localization]] of~~ <~~math~~>~~\mathcal{O}_K~~<~~/math> at <math>P</math~~> is ~~likewise of mixed characteristic~~.
	* The [[p-adic number\|''p''-adic integers]] '''Z'''<~~sub~~>~~''p''~~<~~/sub~~> ~~for any prime ''p''~~ are a ~~ring~~ of ~~characteristic zero. However~~, ~~they have an ideal generated by~~ the ~~image of~~ the ~~prime number ''p'' under~~ the ~~canonical map {{nowrap\|'''Z''' → '''Z'''<sub>''p''</sub>}}~~. ~~The quotient '''Z'''~~<~~sub~~>~~''p''~~<~~/sub~~>/'~~'p'''''Z'''<sub>''p''</sub>~~ is ~~again~~ the ~~finite field of ''p'' elements~~. ~~'''Z'''~~<~~sub~~>~~''p''~~<~~/sub~~> ~~is an example~~ of a [~~[completion (ring theory)\|complete]] [[discrete valuation ring]] of mixed characteristic~~.
	* The integers, the [[ring of integers]~~] of any number field~~, ~~and any localization~~ or ~~completion of one of these rings is a characteristic zero [[Dedekind domain]]~~.

	[~~[Category~~:~~Commutative algebra~~]]


	~~{{algebra-stub}}~~

en>Nanite: /* The crystal structures of metals */

2013-07-23T20:59:03Z

The crystal structures of metals

← Older revision		Revision as of 22:59, 23 July 2013
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	~~French luxury goods maker Hermes voiced its frustration at~~ having ~~arch-rival LVMH as its biggest external shareholder at its annual general meeting on Tuesday~~ and ~~once more called on the group to sell its stake~~.~~<br><br>LVMH~~, ~~the world~~'~~s biggest luxury group~~, ~~which owns 23 percent~~ of ~~Hermes, was fined 8 million euros by the French market watchdog AMF last year for failing to properly disclose its building~~ of a ~~stake before 2010~~.<br><br>~~Hermes~~, the ~~177[http:~~//~~dict.leo.org~~/~~?search=-year-old+maker -year-old maker] of Birkin and Kelly handbags which~~ is ~~more than 70 percent family-owned~~, ~~has been vehemently protesting~~ the ~~presence~~ of ~~LVMH in its shareholder capital ever since it learned of its surprise entry in 2010.~~<br>~~"We do not want shareholders that~~ are ~~rivals," Hermes Chief Executive Axel Dumas told~~ the ~~company~~'~~s annual shareholder meeting on Tuesday. "We want~~ to ~~preserve our independence~~."<br><br>~~After~~ the ~~meeting, Dumas told Reuters he was "not aware" whether LVMH would be willing to sell down its stake.~~<br>~~In an interview with Le Figaro newspaper published on Tuesday, Dumas said: "LVMH is totally free to sell its shares and to be honest, would be welcome to do so."~~<br>~~LVMH, owner~~ of ~~Louis Vuitton, Dior and~~ [~~http:~~//~~www~~.~~pcs~~-~~systems.co.uk/Images/celinebag.aspx Celine UK~~] ~~fashion brands, has repeatedly said it was "satisfied" being [https://Www.vocabulary.com/dictionary/Hermes%27+shareholder Hermes~~' ~~shareholder] and backed its management~~'~~s strategy.~~<br><br>~~"But satisfied does not mean friendly," Dumas told Le Figaro. And they (LVMH)~~ are ~~not particularly friendly with our management~~.~~"<br>Separately~~, ~~Dumas said Hermes was considering opening a shop in South Africa in~~ the ~~medium term, preferably in Johannesburg. He said he expected sales in Japan, one~~ of the ~~company~~'~~s biggest markets where sales rose 6.5 percent at constant exchange rates in 2013, would be similar this year~~.<br><br>~~Hermes shares - which have lost nearly 2 percent since Jan 1 after climbing nearly 17 percent in 2013 - were barely changed in midday trading at 259.2 euros, valuing~~ the ~~company at 27.4 billion euros~~.<br>~~That makes Hermes the third largest luxury group by market capitalisation behind LVMH and Richemont.~~<br>(~~Reporting by Pascale Denis~~ and ~~Astrid Wendlandt; Editing by Sophie Walker)~~		{{unreferenced\|date=January 2013}}
			In [[commutative algebra]], a '''ring of mixed characteristic''' is a commutative ring ''R'' having [[characteristic (algebra)\|characteristic]] zero and having an ideal ''I'' such that {{nowrap\|''R'' / ''I''}} has positive characteristic.

			== Examples ==
			* The integers '''Z''' have characteristic zero, but for any [[prime number]] ''p'', '''Z'''/(''p'') is a [[finite field]] with ''p'' elements and hence has characteristic ''p''.
			* The ring of integers of any [[number field]] is of mixed characteristic
			* Fix a prime ''p'' and [[localization of a ring\|localize]] the integers at the prime ideal (''p''). The resulting ring '''Z'''<sub>(''p'')</sub> has characteristic zero. It has a unique maximal ideal ''p'''''Z'''<sub>(''p'')</sub>, and the quotient '''Z'''<sub>(''p'')</sub>/''p'''''Z'''<sub>(''p'')</sub> is a finite field with ''p'' elements. In contrast to the previous example, the only possible characteristics for rings of the form {{nowrap\|'''Z'''<sub>(''p'')</sub> / ''I''}} are zero (when ''I'' is the zero ideal) and powers of ''p'' (when ''I'' is any other non-unit ideal); it is not possible to have a quotient of any other characteristic.
			* If <math>P</math> is a non-zero prime ideal of the ring <math>\mathcal{O}_K</math> of integers of a number field <math>K</math> then the [[localization of a ring\|localization]] of <math>\mathcal{O}_K</math> at <math>P</math> is likewise of mixed characteristic.
			* The [[p-adic number\|''p''-adic integers]] '''Z'''<sub>''p''</sub> for any prime ''p'' are a ring of characteristic zero. However, they have an ideal generated by the image of the prime number ''p'' under the canonical map {{nowrap\|'''Z''' → '''Z'''<sub>''p''</sub>}}. The quotient '''Z'''<sub>''p''</sub>/''p'''''Z'''<sub>''p''</sub> is again the finite field of ''p'' elements. '''Z'''<sub>''p''</sub> is an example of a [[completion (ring theory)\|complete]] [[discrete valuation ring]] of mixed characteristic.
			* The integers, the [[ring of integers]] of any number field, and any localization or completion of one of these rings is a characteristic zero [[Dedekind domain]].

			[[Category:Commutative algebra]]


			{{algebra-stub}}

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

2012-07-19T00:23:49Z

Robot - Moving category Fundamental physics concepts to Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

New page

French luxury goods maker Hermes voiced its frustration at having arch-rival LVMH as its biggest external shareholder at its annual general meeting on Tuesday and once more called on the group to sell its stake. LVMH, the world's biggest luxury group, which owns 23 percent of Hermes, was fined 8 million euros by the French market watchdog AMF last year for failing to properly disclose its building of a stake before 2010. Hermes, the 177[http://dict.leo.org/?search=-year-old+maker -year-old maker] of Birkin and Kelly handbags which is more than 70 percent family-owned, has been vehemently protesting the presence of LVMH in its shareholder capital ever since it learned of its surprise entry in 2010. "We do not want shareholders that are rivals," Hermes Chief Executive Axel Dumas told the company's annual shareholder meeting on Tuesday. "We want to preserve our independence." After the meeting, Dumas told Reuters he was "not aware" whether LVMH would be willing to sell down its stake. In an interview with Le Figaro newspaper published on Tuesday, Dumas said: "LVMH is totally free to sell its shares and to be honest, would be welcome to do so." LVMH, owner of Louis Vuitton, Dior and [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine UK] fashion brands, has repeatedly said it was "satisfied" being [https://Www.vocabulary.com/dictionary/Hermes%27+shareholder Hermes' shareholder] and backed its management's strategy. "But satisfied does not mean friendly," Dumas told Le Figaro. And they (LVMH) are not particularly friendly with our management." Separately, Dumas said Hermes was considering opening a shop in South Africa in the medium term, preferably in Johannesburg. He said he expected sales in Japan, one of the company's biggest markets where sales rose 6.5 percent at constant exchange rates in 2013, would be similar this year. Hermes shares - which have lost nearly 2 percent since Jan 1 after climbing nearly 17 percent in 2013 - were barely changed in midday trading at 259.2 euros, valuing the company at 27.4 billion euros. That makes Hermes the third largest luxury group by market capitalisation behind LVMH and Richemont. (Reporting by Pascale Denis and Astrid Wendlandt; Editing by Sophie Walker)