Papyrus 56 - Revision history

en>Leszek Jańczuk: removed Category:Manuscripts of the Austrian National Library; added Category:Biblical manuscripts of the Austrian National Library using HotCat

2014-08-18T18:47:32Z

removed Category:Manuscripts of the Austrian National Library; added Category:Biblical manuscripts of the Austrian National Library using HotCat

← Older revision		Revision as of 20:47, 18 August 2014
Line 1:		Line 1:
	In the ~~[[mathematics\|mathematical]] study of [[functional analysis]]~~, ~~the '''Banach–Mazur distance''' is~~ a ~~way~~ to ~~define~~ a ~~[[distance function\|distance]] on the set ''Q''(''n'')~~ of ~~''n''-dimensional [[normed space]]s~~. ~~If ''X''~~ and ~~''Y'' are two finite-dimensional normed spaces with~~ the same ~~dimension, let GL(''X'',''Y'') denote the collection of all linear isomorphisms ''T'' : ''X'' → ''Y''~~. ~~The Banach–Mazur distance between ''X''~~ and ~~''Y'' is defined by~~		If you present photography effectively, it helps you look much more properly at the globe around you. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Should you go with simple HTML or use a platform like Wordpress. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>If you have any kind of questions relating to where and how to make use of [http://shortener.us/wordpress_backup_634439 wordpress backup], you could call us at our web-site. Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. 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. There are number of web services that offer Word press development across the world. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Word - Press is the most popular personal publishing platform which was launched in 2003. Press CTRL and the numbers one to six to choose your option.

	~~: <math>\delta(X, Y) = \log \Bigl( \inf \{ \\|T\\| \\|T^{-1}\\| : T \in \operatorname{GL}(X, Y) \} \Bigr).</math>~~

	~~Equipped with~~ the ~~metric ''δ'', the space ''Q''(''n'')~~ is ~~a [[compact metric space]], called the '''Banach–Mazur compactum'''~~.

	~~Many authors prefer to work with the ''multiplicative Banach–Mazur distance''~~
	:<~~math~~>~~d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \{ \\|T\\| \\|T^{-1}\\| : T \in \operatorname{GL}(X, Y) \},~~<~~/math~~>
	~~for which ''d''(''X'', ''Z'') ≤ ''d''(''X'', ''Y'') ''d''(''Y'', ''Z'')~~ and ~~''d''(''X'', ''X'') = 1.~~
	[~~[John ellipsoid\|F. John's theorem]] on the maximal ellipsoid contained in a convex body gives the estimate:~~

	~~: <math>d(X, \ell^n_2) \le \sqrt{n}, \,</math> <ref>~~http://~~users~~.~~uoa.gr/~apgiannop/cube.ps</ref>~~
	~~where ℓ<sub>''n''</sub><sup>2</sup> denotes '''R'''<sup>''n''</sup> with the Euclidean norm (see the article on [[Lp space\|''L''<sup>''p''<~~/~~sup> spaces~~]]).
	~~From this it follows that ''d''(''X''~~, ~~''Y'') ≤ ''n''~~ for ~~every couple (''X''~~, ~~''Y'') in ''Q''(''n'')~~. ~~However~~, ~~for~~ the ~~classical spaces, this upper bound~~ for ~~the diameter of ''Q''(''n'') is far from being approached. For example~~, ~~the distance between ℓ<sub>''n''</sub><sup>1</sup>~~ and ~~ℓ<sub>''n''</sub><sup>∞</sup>~~ is ~~(only)~~ of ~~order ''n''<sup>1/2</sup> (up~~ to ~~a multiplicative constant independent from~~ the ~~dimension ''n'')~~.

	~~A major achievement in~~ the ~~direction~~ of ~~estimating the diameter of ''Q''(''n'')~~ is ~~due to E. Gluskin, who proved~~ in ~~1981 that~~ the ~~(multiplicative) diameter of~~ the ~~Banach–Mazur compactum is bounded below~~ by ~~''c'' ''n''~~, ~~for some universal ''c'' > 0~~.

	~~Gluskin's method introduces a class of random symmetric polytopes ''P''(''ω'') in '''R'''~~<~~sup~~>~~''n''~~<~~/sup~~>, and the ~~normed spaces ''X''(''ω'') having ''P''(''ω'') as unit ball (~~the ~~vector space is~~ '~~''R'''<sup>''n''</sup>~~ and the ~~norm is~~ the ~~[[Norm (mathematics)\|gauge]] of ''P''(''ω''))~~. ~~The proof consists in showing that~~ the ~~required estimate is true with large probability for two independent copies of~~ the ~~normed space ''X''(''ω'')~~.

	~~''Q''(2) is an [[absolute extensor]]~~.<~~ref~~>~~[http://www~~.~~iop.org/EJ/article/0036~~-~~0279/53/1/L06/RMS_53_1_L06~~.~~pdf?request-id=e60e8254-2945-4600-b648-67deecf06a15 The Banach–Mazur compactum~~ is ~~not homeomorphic~~ to the ~~Hilbert cube]~~<~~/ref~~> ~~On the other hand, ''Q''(2) is not homeomorphic to a [[Hilbert cube]].~~

	~~==Notes==~~
	<~~references/~~>

	~~== References ==~~
	* {{springer\|title=Banach–Mazur compactum\|id=B/b110100\|first=A.A.\|last=Giannopoulos}}
	* {{cite journal
	~~\| last = Gluskin~~
	~~\| first = Efim D.~~
	~~\| title = The diameter~~ of the ~~Minkowski compactum is roughly equal~~ to ~~''n'' (Russian)~~
	~~\| journal = Funktsional~~. ~~Anal. i Prilozhen.~~
	~~\| volume = 15~~
	~~\| year = 1981~~
	~~\| issue = 1~~
	~~\| pages = 72–73~~
	~~\| mr = 0609798~~
	}}
	* {{cite book
	~~\| last = Tomczak-Jaegermann~~
	~~\| first = Nicole \| authorlink = Nicole Tomczak-Jaegermann~~
	~~\| title = Banach-Mazur distances~~ and ~~finite~~-~~dimensional operator ideals~~
	~~\| series = Pitman Monographs and Surveys in Pure and Applied Mathematics 38~~
	~~\| publisher = Longman Scientific & Technical~~, ~~Harlow; copublished in the United States with John Wiley & Sons~~, ~~Inc.~~, ~~New York~~
	~~\| year = 1989~~
	~~\| pages = xii+395~~
	~~\| isbn = 0-582-01374-7~~
	~~\| mr = 0993774~~
	}}
	*http://planetmath.org/encyclopedia/BanachMazurCompactum.html
	*[http://users.uoa.gr/~apgiannop/cube.ps A note on ~~the Banach-Mazur distance to the cube]~~
	*[http://at.yorku.ca/p/a~~/a/o/26~~.~~htm The Banach~~-~~Mazur compactum~~ is the ~~Alexandroff compactification of a Hilbert cube manifold]~~

	~~{{DEFAULTSORT:Banach-Mazur compactum}}~~
	~~[[Category:Functional analysis]]~~
	~~[[Category:Metric geometry]]~~

en>Leszek Jańczuk: /* Further reading */

2013-02-26T16:54:51Z

Further reading

← Older revision		Revision as of 18:54, 26 February 2013
Line 1:		Line 1:
	~~i acquired this situation~~ [~~http://www.youtube.com/watch?v=gRc1VeLQxec tv built in dvd~~] ~~With~~ [~~http://www.youtube.com/watch?v=XajQ4YxWRLA 32 inch tv dvd combo~~] ~~Player Built In, [http://www.youtube.com/watch?v=obDRZxxQFGI Www.Youtube.Com~~], ~~programs as~~ a ~~result of checking out great lead and additionally CNET research attached~~ to ~~it. this task showed up well packaged and also in good condition after ebay, delivered pack in~~ a ~~pack by means of back between~~ the ~~exact packaging. trouble free~~ set-~~up and that I was off and running within just around 30 minutes~~. ~~The most important wireless local area network, tuner, visualize,~~ and ~~additionally everything been successful rather nicely. I came across the specific separated in order to become sensitive as well as mild and that I just like [http://www.youtube.com/watch?v=GJ~~-~~cGj3kueI 19 flat screen tv~~ with ~~dvd player]~~ the ~~minor mouse click when forcing the specific buttons. used every little thing to make certain it actually was operating clearly~~, ~~streamed a motion picture view~~ the ~~most important show and would be really happy. grabbed within the following day and also turned the product regarding, each and every thing is quality, position to obtain a mug~~ of ~~coffee, followed back plus the screen was black colored. Deader compared with asleep, but nevertheless attained appear to be. this is from where the narrative will useful. I called customer care, need~~ all ~~the way through for an english language-discussing North american, and additionally ran your few assessments inside the tech~~'~~s directions, final that will it wanted to be changed~~. The Specific support stated e would be working with a dub after one more Visio man inside 48 hours, but had been a piece skeptical, then again within about 8 many I managed to get a proof call detail in my circumstances the direction they might be giving the (never refurb) fix substitute my own. Again an English-discussing American. Yesterday the acquired a good manners email suggesting myself the new unit had gotten been chose up through shipper, as well as today i received your service contact through the shipper suggesting us to expect the dub to arrange supply and additionally design. really from using a Thursday evening to Wednesday day. I terribly lack the brand new preset however, and it is basically bad about the 1st definitely one was unable at once, then again drastically this is top client service i'~~ve possible practiced inside my your life! So, i merged attitude regarding allowing the exact set up the 3 star rating. I~~'~~m upbeat your brand new set cannot fold for the reason that right now there were so many great product reviews after other people. <br><br>~~		In the [[mathematics\|mathematical]] study of [[functional analysis]], the '''Banach–Mazur distance''' is a way to define a [[distance function\|distance]] on the set ''Q''(''n'') of ''n''-dimensional [[normed space]]s. If ''X'' and ''Y'' are two finite-dimensional normed spaces with the same dimension, let GL(''X'',''Y'') denote the collection of all linear isomorphisms ''T'' : ''X'' → ''Y''. The Banach–Mazur distance between ''X'' and ''Y'' is defined by

	~~Bought this situation to exchange your Moms' ancient~~-~~design tube Television~~. ~~During create once affixing among~~ the ~~many prime air services~~, the ~~only method for connecting in order to~~ the ~~ancient Television had been by RGB link~~. ~~A negative half effect related~~ with ~~this hookup way ended up being there would be a "standard appear to be" challenge~~ in which ~~ads~~ and ~~rap played with a very noticeable upper volume~~ (~~besides the fact that this challenge is really removed with federal commission~~). ~~Anyways, immediately after studying Amazons~~' ~~offerings,~~ the ~~chose~~ the ~~specific LG 32LN5300 model~~. ~~The HIGH-DEFINITION MULTIMEDIA INTERFACE meet removed the sound challenge possesses an amazing photograph~~. ~~She is very happy together~~ with ~~her new Television~~. ~~Any mention to share with you: should you be travelling to employ incorporated stand~~, ~~render sure your Phillips screwdriver is magnets~~. ~~Hanging~~ the ~~exact will~~ the ~~television can be very overwhelming else once~~ the ~~a couple of top cheat positions can be major~~ and ~~do not nourish each anchoring screws inside her or his installing slots such as the bottow pair should.~~ <br><br> ~~Here our company~~ is a ~~month or more after plus~~ the ~~unique lay arrived secure, and will work good~~. ~~we moved from your Samsung plasma~~ to ~~the~~, ~~whilst still being utilize~~ the ~~Samsung as part~~ of the ~~place~~, ~~thus I will assess consumers during an continual schedule~~. ~~I assume it does not material much basically because plasmas are now being phased out~~, ~~yet~~ the ~~pic to~~ the ~~Samsung plasma in fact~~ is ~~much better than~~ the ~~Guided photograph from~~ the ~~VIZIO. It~~'~~s only a a lot deeper, simpler, blacker look~~. ~~With~~ that ~~said,~~ the ~~exact VIZIO~~ is ~~ok to view - each styles are typically great, some~~ of the ~~pungency~~ is ~~very useful, reflect~~ is ~~way less versus plasma~~, ~~the exact audio definitely~~ is not ~~so negative, plus~~ the ~~piano tuner appears~~ to ~~duty well alongside simultaneously basic~~ and ~~raised~~-~~def electronic data precisely from using~~ the ~~connection (virtually no wire box)~~. ~~I'd spend money~~ on ~~it yet again~~.		: <math>\delta(X, Y) = \log \Bigl( \inf \{ \\|T\\| \\|T^{-1}\\| : T \in \operatorname{GL}(X, Y) \} \Bigr).</math>

			Equipped with the metric ''δ'', the space ''Q''(''n'') is a [[compact metric space]], called the '''Banach–Mazur compactum'''.

			Many authors prefer to work with the ''multiplicative Banach–Mazur distance''
			:<math>d(X, Y) := \mathrm{e}^{\delta(X, Y)} = \inf \{ \\|T\\| \\|T^{-1}\\| : T \in \operatorname{GL}(X, Y) \},</math>
			for which ''d''(''X'', ''Z'') ≤ ''d''(''X'', ''Y'') ''d''(''Y'', ''Z'') and ''d''(''X'', ''X'') = 1.
			[[John ellipsoid\|F. John's theorem]] on the maximal ellipsoid contained in a convex body gives the estimate:

			: <math>d(X, \ell^n_2) \le \sqrt{n}, \,</math> <ref>http://users.uoa.gr/~apgiannop/cube.ps</ref>
			where ℓ<sub>''n''</sub><sup>2</sup> denotes '''R'''<sup>''n''</sup> with the Euclidean norm (see the article on [[Lp space\|''L''<sup>''p''</sup> spaces]]).
			From this it follows that ''d''(''X'', ''Y'') ≤ ''n'' for every couple (''X'', ''Y'') in ''Q''(''n''). However, for the classical spaces, this upper bound for the diameter of ''Q''(''n'') is far from being approached. For example, the distance between ℓ<sub>''n''</sub><sup>1</sup> and ℓ<sub>''n''</sub><sup>∞</sup> is (only) of order ''n''<sup>1/2</sup> (up to a multiplicative constant independent from the dimension ''n'').

			A major achievement in the direction of estimating the diameter of ''Q''(''n'') is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by ''c'' ''n'', for some universal ''c'' > 0.

			Gluskin's method introduces a class of random symmetric polytopes ''P''(''ω'') in '''R'''<sup>''n''</sup>, and the normed spaces ''X''(''ω'') having ''P''(''ω'') as unit ball (the vector space is '''R'''<sup>''n''</sup> and the norm is the [[Norm (mathematics)\|gauge]] of ''P''(''ω'')). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space ''X''(''ω'').

			''Q''(2) is an [[absolute extensor]].<ref>[http://www.iop.org/EJ/article/0036-0279/53/1/L06/RMS_53_1_L06.pdf?request-id=e60e8254-2945-4600-b648-67deecf06a15 The Banach–Mazur compactum is not homeomorphic to the Hilbert cube]</ref> On the other hand, ''Q''(2) is not homeomorphic to a [[Hilbert cube]].

			==Notes==
			<references/>

			== References ==
			* {{springer\|title=Banach–Mazur compactum\|id=B/b110100\|first=A.A.\|last=Giannopoulos}}
			* {{cite journal
			\| last = Gluskin
			\| first = Efim D.
			\| title = The diameter of the Minkowski compactum is roughly equal to ''n'' (Russian)
			\| journal = Funktsional. Anal. i Prilozhen.
			\| volume = 15
			\| year = 1981
			\| issue = 1
			\| pages = 72–73
			\| mr = 0609798
			}}
			* {{cite book
			\| last = Tomczak-Jaegermann
			\| first = Nicole \| authorlink = Nicole Tomczak-Jaegermann
			\| title = Banach-Mazur distances and finite-dimensional operator ideals
			\| series = Pitman Monographs and Surveys in Pure and Applied Mathematics 38
			\| publisher = Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York
			\| year = 1989
			\| pages = xii+395
			\| isbn = 0-582-01374-7
			\| mr = 0993774
			}}
			*http://planetmath.org/encyclopedia/BanachMazurCompactum.html
			*[http://users.uoa.gr/~apgiannop/cube.ps A note on the Banach-Mazur distance to the cube]
			*[http://at.yorku.ca/p/a/a/o/26.htm The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold]

			{{DEFAULTSORT:Banach-Mazur compactum}}
			[[Category:Functional analysis]]
			[[Category:Metric geometry]]

en>Leszek Jańczuk: ref

2011-08-26T19:05:26Z

ref

New page

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