Pressure-volume curves - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10067)

2014-03-29T16:12:40Z

WP:CHECKWIKI error fixes + other fixes using AWB (10067)

← Older revision		Revision as of 18:12, 29 March 2014
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	In the mathematical theory of [[free probability]], the notion of '''free independence''' was introduced by [[Dan Voiculescu (mathematician)\|Dan Voiculescu]].<ref>D. Voiculescu, K. Dykema, A. Nica, "Free Random Variables", CIRM Monograph Series, AMS, Providence, RI, 1992</ref> The definition of free independence is parallel to the classical definition of [[independence (probability)\|independence]], except that the role of Cartesian products of [[measure space]]s (corresponding to [[tensor product]]s of their function algebras) is ~~played by the notion of~~ a ~~[[free product]] of (non~~-~~commutative) probability spaces~~.		Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. Step-4 Testing: It is the foremost important of your Plugin development process. If you loved this article and you simply would like to receive more info about [http://247url.info/wordpress_backup_555756 wordpress dropbox backup] nicely visit the web site. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br>These folders as well as files have to copied and the saved. The higher your blog ranks on search engines, the more likely people will find your online marketing site. 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>Minor and medium sized corporations also have a lot to obtain by shelling out in a very good website. As of now, Pin2Press is getting ready to hit the market. You've got invested a great cope of time developing and producing up the topic substance. The first thing you need to do is to choose the right web hosting plan. After that the developer adds the unordered list for navigations. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. 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. However, you may not be able to find a theme that is in sync with your business. Contact Infertility Clinic Providing One stop Fertility Solutions at:. 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>Instead, you can easily just include it with our bodies integration field in e - Panel. I don't want that for my visitors and I'm quite sure they don't either. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.

	~~In the context of Voiculescu's free probability theory~~, ~~many classical~~-~~probability theorems or phenomena have free probability analogs~~: the ~~same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion~~ of ~~independence is replaced by free independence~~. ~~Examples of~~ this ~~include~~: the ~~free central limit theorem; notions~~ of ~~[[free convolution]]; existence~~ of ~~[[free stochastic calculus]] and so on.~~

	~~Let <math>(A~~,~~\phi)</math> be a [[non-commutative probability space]]~~, i.~~e. a [[Identity element\|unital]] [[Algebra over a field\|algebra]] <math>A</math> over <math>\mathbb{C}</math> equipped with a [[Unital map\|unital]] [[linear functional]] <math>\phi:A\~~to~~\mathbb{C}</math>. As an example, one could take,~~ for a ~~probability measure <math>\mu</math>,~~

	: ~~<math>A = L^\infty(\mathbb{R},\mu),\phi(f) = \int f(t)\,d\mu(t)~~.<~~/math~~>

	~~Another example may be~~ <~~math>A=M_N</math~~>, the ~~algebra of <math>N\times N</math> matrices with~~ the ~~functional given by~~ the ~~normalized trace <math>\phi=\frac{1}{N}Tr</math>~~. ~~Even more generally~~, <~~math~~>A<~~/math~~> ~~could be~~ a ~~[[von Neumann algebra]] and <math>\phi</math>~~ a ~~state on <math>A</math>~~. ~~A final example~~ is the ~~[[group algebra]] <math>A=\mathbb{C}\Gamma</math>~~ of ~~a (discrete) [[Group (mathematics)\|group]] <math>\Gamma</math> with~~ the ~~functional~~ <~~math~~>~~\phi~~<~~/math~~> ~~given by the group trace <math>\phi (g) = \delta_{g=e}~~,~~g\in \Gamma</math>~~.

	~~Let <math>\{A_i : i\in~~ I~~\}</math>~~ be a ~~family of unital subalgebras of <math>A</math>~~.

	~~'''Definition'''~~. ~~The family <math>\{A_i : i\in I\}</math>~~ is ~~called ''freely independent'' if~~
	~~<math>\phi(x_1 x_2 \cdots x_n) =0 </math>~~
	~~whenever <math>\phi(x_j)=0</math>~~, ~~<math>x_j \in A_{i(j)}</math>~~ and ~~<math>i(1)\neq i(2)~~, ~~i(2)\neq i(3),\dots</math>~~.

	If <~~math~~>~~X_i\in A~~<~~/math~~>, ~~<math>i\~~in I~~</math> is a family of elements of <math>A</math> (these can be thought of as random variables in <math>A</math>), they are called~~

	''~~freely independent~~'~~' if the algebras <math>A_i</math> generated by <math>1</math> and <math>X_i</math> are freely independent~~.

	~~== Examples~~ of ~~free independence ==~~

	* Let <math>\Gamma</math> be the [[free product]] of ~~groups <math>\Gamma_i~~,~~i\in I</math>~~, ~~let <math>A=\mathbb{C}\Gamma</math> be the group algebra~~, ~~<math>\phi(g)=\delta_{g=e}</math> be~~ the ~~group trace~~, and ~~set <math>A_i=\mathbb{C}\Gamma_i\subset A</math>. Then <math>A_i:i\~~in ~~I</math> are freely independent~~.

	* Let <math>U_i(N),~~i=1~~,~~2</math> be <math>N\times N</math> unitary [[random matrices]], taken independently at random~~ from the <math>N\times N</math> [[unitary group]] (with respect to the [[Haar measure]]). Then <math>U_1(N),U_2(N)</math> become asymptotically freely independent as <math>N\to\infty</math>. (Asymptotic freeness means that the definition of freeness holds in the limit as <math>N\to\infty</math>).

	* More generally, independent [[random matrices]] tend to ~~be asymptotically freely independent, under certain conditions~~.

	~~== References ==~~

	~~{{Reflist}}~~

	~~[[Category:Functional analysis]]~~
	~~[[Category:Free probability theory]]~~
	~~[[Category:Free algebraic structures]]~~

en>DoctorKubla: Replacing deprecated {{wikify}} with {{underlinked}}

2012-12-17T18:06:07Z

Replacing deprecated {{wikify}} with {{underlinked}}

@@ Line 1: / Line 1: @@
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+In the mathematical theory of [[free probability]], the notion of '''free independence''' was introduced by [[Dan Voiculescu (mathematician)|Dan Voiculescu]].<ref>D. Voiculescu, K. Dykema, A. Nica, "Free Random Variables", CIRM Monograph Series, AMS, Providence, RI, 1992</ref> The definition of free independence is parallel to the classical definition of [[independence (probability)|independence]], except that the role of Cartesian products of [[measure space]]s (corresponding to [[tensor product]]s of their function algebras) is played by the notion of a [[free product]] of (non-commutative) probability spaces.
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+In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability analogs: the same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion of independence is replaced by free independence.  Examples of this include: the free central limit theorem; notions of [[free convolution]]; existence of [[free stochastic calculus]] and so on.
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+Let <math>(A,\phi)</math> be a [[non-commutative probability space]], i.e. a [[Identity element|unital]] [[Algebra over a field|algebra]] <math>A</math> over <math>\mathbb{C}</math> equipped with a [[Unital map|unital]] [[linear functional]] <math>\phi:A\to\mathbb{C}</math>.  As an example, one could take, for a probability measure <math>\mu</math>,
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+: <math>A = L^\infty(\mathbb{R},\mu),\phi(f) = \int f(t)\,d\mu(t).</math>
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+Another example may be <math>A=M_N</math>, the algebra of <math>N\times N</math> matrices with the functional given by the normalized trace <math>\phi=\frac{1}{N}Tr</math>.  Even more generally, <math>A</math> could be a [[von Neumann algebra]] and <math>\phi</math> a state on <math>A</math>.  A final example is the [[group algebra]] <math>A=\mathbb{C}\Gamma</math> of a (discrete) [[Group (mathematics)|group]] <math>\Gamma</math> with the functional <math>\phi</math> given by the group trace <math>\phi (g) = \delta_{g=e},g\in \Gamma</math>.
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+Let <math>\{A_i : i\in I\}</math>  be a family of unital subalgebras of <math>A</math>.

en>Jesse V.: various fixes, added orphan, wikify tags using AWB

2012-04-13T15:12:20Z

various fixes, added orphan, wikify tags using AWB

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