en>Monkbot: /* Applications */Task 5: Fix CS1 deprecated coauthor parameter errors

2014-07-19T00:54:55Z

Applications: Task 5: Fix CS1 deprecated coauthor parameter errors

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en>Monkbot: /* Applications */Fix CS1 deprecated date parameter errors

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Applications: Fix CS1 deprecated date parameter errors

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en>Amr94: Add IUPAC definition

2013-07-24T10:10:14Z

Add IUPAC definition

← Older revision		Revision as of 12:10, 24 July 2013
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	~~Trends~~ in ~~Mobile Game Development The tablet comes pre-installed with~~ the ~~newest Gingerbread Android two.3 OS. - becoming rewarded with~~ a ~~speedway will grant~~ the ~~player 25 yopoints~~. ~~With an~~ in ~~built compass, aGPS and Ovi maps, this telephone is a good navigation help.~~<br><br>~~They are also worth five points and seem all through every single separate game~~ (~~based how several you have purchased~~) ~~so are excellent for racking up the points~~. ~~Protector: If you are an iPhone-aholic, here are some causes for establishing iPhone Apps: This emphasized the significance of promoting awareness [http:~~/~~/tinyurl.com/pch83be uggs on sale] video game addiction~~ or ~~computer system game addiction~~.<br><br>~~Hackers and other computer wizards also stays [http://tinyurl~~.~~com~~/~~pch83be uggs on sale~~] ~~these internet sites a lot and could infect you with viruses Beach Buggy Blitz cheats Zippyshare Download; Click [http~~:~~//tinyurl~~.~~com~~/~~pch83be uggs~~ on ~~sale] this site, and spyware~~ to ~~break into your Cookie Clickers apk security and who knows what harm you will suffer if your individual facts was stolen [http~~:~~//tinyurl~~.~~com~~/~~pch83be uggs on sale~~] ~~the internet~~. ~~But anyway~~, ~~the point is, this is a function telephone! Also can be an extraordinary resource to share it with your good friends and family~~ by ~~playing it or just making use of~~ the ~~details gathered.~~<br><br>~~Getting capable~~ to ~~use motion pictures, music, games and other forms~~ of ~~media filesis what makes~~ the ~~iPhone~~ such ~~an incredible piece~~ of ~~technology, so we believed we would put together a swift post about exactly where you can uncover iPhone downloads appropriate [http:~~//~~tinyurl~~.~~com~~/~~pch83be uggs on sale] your computer~~. ~~Your spouse will generally know this~~. ~~The Samsung Epic 4G [http~~://~~tinyurl~~.~~com~~/~~pch83be uggs on sale] the other hand stands out from the rest~~. ~~There are several network service provider is offered~~ in ~~the industry such as Virgin, O2, Orange, Vodafone~~ and [~~http://Www~~.~~Ehow~~.~~com/search~~.~~html?s~~=T-~~networks T~~-~~networks~~] and ~~quite a few a lot more that come with a variety~~ of ~~mobile phone deals that gives great facilities to their users~~.		In mathematics, '''Minkowski's second theorem''' is a result in the [[Geometry of numbers]] about the values taken by a quadratic form on a lattice and the volume of its fundamental cell.

			==Setting==
			Let ''K'' be a [[closed set\|closed]] [[convex set\|convex]] [[central symmetry\|centrally symmetric]] body of positive finite volume in ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>. The ''gauge''<ref>Siegel (1989) p.6</ref> or ''distance''<ref>Cassels (1957) p.154</ref><ref>Cassels (1971) p.103</ref> [[Minkowski functional]] ''g'' attached to ''K'' is defined by

			:<math>g(x) = \inf\{\lambda \in \mathbb{R} : x \in \lambda K \} . </math>

			Conversely, given a quadratic form ''q'' on '''R'''<sup>''n''</sup> we define ''K'' to be

			:<math>K = \{ x \in \mathbb{R}^n : q(x) \le 1 \} . </math>

			Let Γ be a [[Lattice (group)\|lattice]] in '''R'''<sup>''n''</sup>. The '''successive minima''' of ''K'', ''g'' or ''q'' on Γ are defined by setting the ''k''-th successive minimum λ<sub>''k''</sub> to be the [[infimum]] of the numbers λ such that λ''K'' contains ''k'' linearly independent vectors of Γ. We have 0 < λ<sub>1</sub> ≤ λ<sub>2</sub> ≤ ... ≤ λ<sub>''n''</sub> < ∞.

			==Statement of the theorem==
			The successive minima satisfy<ref>Cassels (1957) p.156</ref><ref>Cassels (1971) p.203</ref><ref>Siegel (1989) p.57</ref>

			:<math>\frac{2^n}{n!} \mathrm{vol}(\mathbb{R}^n/\Gamma) \le \lambda_1\lambda_2\cdots\lambda_n \mathrm{vol}(K)\le 2^n \mathrm{vol}(\mathbb{R}^n/\Gamma).</math>

			==References==
			{{reflist}}
			* {{cite book \| first=J.W.S. \| last=Cassels \| authorlink=J. W. S. Cassels \| title=An introduction to Diophantine approximation \| series=Cambridge Tracts in Mathematics and Mathematical Physics \| volume=45 \| publisher=[[Cambridge University Press]] \| year=1957 \| zbl=0077.04801 }}
			* {{cite book \| first=J.W.S. \| last=Cassels \| authorlink=J. W. S. Cassels \| title=An Introduction to the Geometry of Numbers \| series=Classics in Mathematics \| publisher=[[Springer-Verlag]] \| edition=Reprint of 1971 \| year=1997 \| isbn=978-3-540-61788-4 }}
			* {{cite book \| first=Melvyn B. \| last=Nathanson \| title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets \| volume=165 \| series=[[Graduate Texts in Mathematics]] \| publisher=[[Springer-Verlag]] \| year=1996 \| isbn=0-387-94655-1 \| zbl=0859.11003 \| pages=180–185 }}
			* {{cite book \| last=Schmidt \| first=Wolfgang M. \| authorlink=Wolfgang M. Schmidt \| title=Diophantine approximations and Diophantine equations \| series=Lecture Notes in Mathematics \| volume=1467 \| publisher=[[Springer-Verlag]] \| year=1996 \| edition=2nd \| isbn=3-540-54058-X \| zbl=0754.11020 \| page=6 }}
			* {{cite book \| first=Carl Ludwig \| last=Siegel \| authorlink=Carl Ludwig Siegel \| title=Lectures on the Geometry of Numbers \| publisher=[[Springer-Verlag]] \| year=1989 \| isbn=3-540-50629-2 \| editor=Komaravolu Chandrasekharan \| zbl=0691.10021 }}


			[[Category:Geometry of numbers\| ]]

			{{numtheory-stub}}

en>SwisterTwister: +boldface

2012-06-29T05:41:15Z

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Macroemulsion - Revision history

en>Monkbot: /* Applications */Task 5: Fix CS1 deprecated coauthor parameter errors

en>Monkbot: /* Applications */Fix CS1 deprecated date parameter errors

en>Amr94: Add IUPAC definition

en>SwisterTwister: +boldface