Papyrus 62 - Revision history

en>Trappist the monk: /* Greek text of Matthew */Fix CS1 deprecated coauthor parameter errors; using AWB

2014-07-21T12:32:26Z

Greek text of Matthew: Fix CS1 deprecated coauthor parameter errors; using AWB

← Older revision		Revision as of 14:32, 21 July 2014
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	In [[graph theory]], a branch of mathematics, a '''Lévy family of graphs'''<!--, named after [[????? Lévy]],--> is a family of [[graph (mathematics)\|graphs]] ''G''<sub>''n''</sub>, ''n'' = 1, 2, 3, ..., which possess a certain type of "compactness" or "tangledness". Many naturally occurring families of graphs are Lévy families. Many mathematicians have noted this fact and have expressed surprise that it does not appear to have a ready explanation.

	~~Formally,~~ a ~~family~~ of ~~graphs ''G~~<~~sub~~>~~n''~~</~~sub>, ''n'' ~~=~~ 1~~,~~ 2, 3, ~~...~~, is~~ a ~~Lévy family if, for any~~ <~~math~~>~~\varepsilon~~>0<~~/math~~>		\ոMakіng sure a diet plan includеs the periodic veggie or piece of freѕh fгuitѕ is scarcely іtching the outеr lining of nourishment. It is a full way of life. Approprіate diet demands familiarity with your healthy requires and the best way to change yоur diet to satisfy individuɑls requirements. Үou think you will be up to it? No matter what your solution, here are some ideas to assist yoս.<br><br>Careful reading through of meals labels iѕ crucial. Reduced extra fat gooԁs can continue to consist [http://www.usellerfinance.com/auctions/item.php?id=42185235 free box of vigrx plus] ɦigh quantities of ѕalt, sѡeets or any other bad artificial additives. Remarkably ƿгocеsѕed foods are certainly not best for weight reduction. A lіst of elements in the brand must be simple for men and women to identify. 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	:<~~math~~> ~~\lim_{n\longrightarrow\infty}\alpha\left(G_n~~,~~\varepsilon\right) =0~~<~~/math~~>

	~~where~~

	: <~~math~~>~~\alpha(G,\varepsilon) = \max \left\{ 1-\frac{\left\|A_{(\varepsilon D)}\right\|}{\|G\|}\~~,:~~\, A\subseteq G, \|A\|>\|G\|~~/~~2 \right\}~~. </~~math>~~

	~~Here ''D'' is the [[graph diameter]~~] of ~~''G''~~, and ~~''A''~~<~~sub~~>~~(''n'')~~<~~/sub~~> ~~is the ''n''~~-~~[[graph neighborhood]] of ''A''~~. ~~Note that the maximization ranges over subsets ''A'' of ''G''~~, ~~subject to~~ '~~'A'' being over half~~ the ~~size of ''G''~~

	~~In words~~, ~~this means that one can take a subset of size at least half of ''G''~~, and ~~blow it up by only~~ <~~math~~>~~\epsilon~~<~~/math~~> of ~~the graph diameter~~, ~~and end up with nearly all the set~~.

	~~Long "stringy" (i.e. not "compact") families of graphs such~~ as ~~the [~~[~~cyclic graph]] of order ''n'' clearly don't have such a property~~: ~~one could consider a subset comprising the ''n~~/~~2'' neighborhood of a point (midnight to six o'clock, say)~~. ~~The graph has graph diameter ''D'' of about ''n~~/~~2''~~. ~~So the~~ <~~math~~>~~\epsilon D~~<~~/math~~>~~-neighborhood of the subset~~ is ~~only of size about ''n/2''~~. ~~A Levy family would have this neighborhood covering almost all the set. It should be clear~~ that a ~~Levy family must have a very special type~~ of ~~compact structure~~.

	* [[Hypercube graph]]s of order ''n'' are known to be a ~~Lévy family~~.
	* If ''S''<~~sub~~>~~''n''~~<~~/sub~~> is the ~~graph~~ with ~~points~~ that are ~~elements~~ of ~~the [[permutation group]]~~ of '~~'n'' elements~~, ~~with edges joining points that differ by a~~ [~~[Transposition (mathematics)\|transposition~~]~~], then~~ the ~~series ''S''~~<~~sub~~>~~''i''~~<~~/sub~~>, ~~''i=1~~,2,...'', is a ~~Lévy family~~.

	~~==References==~~

	* [[Béla Bollobás\|Bollobás]] (editor). ''Probabilistic combinatorics and ~~its applications~~.~~'' [[American Mathematical Society]]~~, ~~1991 (p63)~~

	~~{{DEFAULTSORT:Levy Family Of Graphs}}~~
	[~~[Category~~:~~Graph families]~~]

en>Leszek Jańczuk: /* External links */

2013-02-26T16:53:00Z

External links

← Older revision		Revision as of 18:53, 26 February 2013
Line 1:		Line 1:
			In [[graph theory]], a branch of mathematics, a '''Lévy family of graphs'''<!--, named after [[????? Lévy]],--> is a family of [[graph (mathematics)\|graphs]] ''G''<sub>''n''</sub>, ''n'' = 1, 2, 3, ..., which possess a certain type of "compactness" or "tangledness". Many naturally occurring families of graphs are Lévy families. Many mathematicians have noted this fact and have expressed surprise that it does not appear to have a ready explanation.

			Formally, a family of graphs ''G<sub>n''</sub>, ''n'' = 1, 2, 3, ..., is a Lévy family if, for any <math>\varepsilon>0</math>

	~~It depends on~~ the ~~quality~~ of the ~~Wordpress theme but even if it~~'~~s not a professional one you will be able to average 50~~-~~60$ EACH link~~. ~~You can either install Word - Press yourself or use free services offered on~~ the ~~web today. This is a service where people write articles using a specific keyword you have given them. Donor oocytes and menopausal pregnancy: Oocyte donation~~ to ~~women~~ of ~~advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. Understanding how Word - Press works~~ can be a ~~challenge~~, ~~but~~ it ~~is not too difficult when you learn more about it.~~ <br><br>~~As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. After all, Word - Press is free, many~~ of the ~~enhancements for Word - Press like themes and plugins are also free~~, and ~~there is plenty~~ of ~~free information online about how to use Word - Press. If you liked this posting and you would like to obtain much more details with regards to~~ [~~http~~:~~//2am.eu/wordpressbackupplugin299036 wordpress dropbox backup] kindly take~~ a ~~look at~~ the web-page. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a ~~site easier~~ to ~~configure~~. ~~Many times the camera is following Mia, taking in her point~~ of ~~view in almost every frame~~. <br><br>~~But before choosing any one~~ of ~~these, let~~'~~s compare between the two~~. ~~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. Every single Theme might~~ be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. 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>~~There has been a huge increase in~~ the ~~number~~ of ~~developers releasing free premium Word - Press themes over~~ the ~~years. I have compiled~~ a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Some examples of its additional features include; code inserter (~~for use with adding Google Analytics, Adsense section targeting etc~~) ~~Webmaster verification assistant~~, ~~Link Mask Generator~~, ~~Robots. It~~'~~s now become a great place to sell it thanks to Woo - Commerce. Fortunately~~, ~~Word - Press Customization Service is available these days~~, ~~right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project~~. ~~<br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors~~. ~~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~~. ~~By the time you get the Gallery Word - Press Themes~~, ~~the first thing that you should know~~ is ~~on how to install it~~. ~~Page speed is an important factor in ranking, especially with Google~~. ~~Get started today so that people searching for your type of business will be directed to you~~.		:<math> \lim_{n\longrightarrow\infty}\alpha\left(G_n,\varepsilon\right) =0</math>

			where

			: <math>\alpha(G,\varepsilon) = \max \left\{ 1-\frac{\left\|A_{(\varepsilon D)}\right\|}{\|G\|}\,:\, A\subseteq G, \|A\|>\|G\|/2 \right\}. </math>

			Here ''D'' is the [[graph diameter]] of ''G'', and ''A''<sub>(''n'')</sub> is the ''n''-[[graph neighborhood]] of ''A''. Note that the maximization ranges over subsets ''A'' of ''G'', subject to ''A'' being over half the size of ''G''

			In words, this means that one can take a subset of size at least half of ''G'', and blow it up by only <math>\epsilon</math> of the graph diameter, and end up with nearly all the set.

			Long "stringy" (i.e. not "compact") families of graphs such as the [[cyclic graph]] of order ''n'' clearly don't have such a property: one could consider a subset comprising the ''n/2'' neighborhood of a point (midnight to six o'clock, say). The graph has graph diameter ''D'' of about ''n/2''. So the <math>\epsilon D</math>-neighborhood of the subset is only of size about ''n/2''. A Levy family would have this neighborhood covering almost all the set. It should be clear that a Levy family must have a very special type of compact structure.

			* [[Hypercube graph]]s of order ''n'' are known to be a Lévy family.
			* If ''S''<sub>''n''</sub> is the graph with points that are elements of the [[permutation group]] of ''n'' elements, with edges joining points that differ by a [[Transposition (mathematics)\|transposition]], then the series ''S''<sub>''i''</sub>, ''i=1,2,...'', is a Lévy family.

			==References==

			* [[Béla Bollobás\|Bollobás]] (editor). ''Probabilistic combinatorics and its applications.'' [[American Mathematical Society]], 1991 (p63)

			{{DEFAULTSORT:Levy Family Of Graphs}}
			[[Category:Graph families]]

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Diaeresis

2012-06-26T19:21:57Z

Repairing links to disambiguation pages - You can help! - Diaeresis

New page

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