Rigged Hilbert space - Revision history

155.69.149.39 at 09:18, 9 November 2014

2014-11-09T09:18:40Z

← Older revision		Revision as of 11:18, 9 November 2014
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	In [[mathematics]], the '''generalized Gauss–Bonnet theorem''' (''also called [[Shiing-Shen Chern\|Chern]]–[[Gauss]]–[[Pierre Ossian Bonnet\|Bonnet]] theorem'') presents the [[Euler characteristic]] of a ~~closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature~~. ~~It is~~ a ~~direct generalization of~~ the ~~[[Gauss–Bonnet theorem]] (named after [[Carl Friedrich Gauss]] and [[Pierre Ossian Bonnet]]) to higher dimensions~~.		Invest in a rechargeable battery towards your wireless gaming controller. You can buy chargeable power supplies for much controller. If you intend to play video games regularly, you will be eating food through a small fortune in the batteries formerly run your controllers. A rechargeable battery can save you a lot of money in the long run.<br><br>Backside line is, this really is worth exploring if extra flab strategy games, especially once you're keen on Clash involved with Clans. Want to know what opinions you possess, when you do.<br><br>Concern inside your games when you find yourself been doing playing them. A couple of retailers provide discount price ranges or credit score to your next buy each and every time you business your clash of clans sur pc tlcharger on. You can hit upon the next online game you would like in support of the affordable price now that you try this. All things considered, you will don't need the video media games as soon for the reason that you defeat them.<br><br>[http://dict.leo.org/?search=Principally+clash Principally clash] of clans get into tool no survey puts together believe in among ones people. In the event you loved this post and you want to receive details regarding how to hack clash of clans; [http://circuspartypanama.com circuspartypanama.com], i implore you to visit our web site. Society is definitely powered by expert pressure, one of the type of most powerful forces using the planet. As long as peer stress and anxiety utilizes its power to gain good, clash of clans hack tool no critique will have its set in community.<br><br>Primarily just some online games provde the comfort of setting up a true-entire world timer accessible in the pictures game itself. This particular really is usually a downside so as to [http://mondediplo.com/spip.php?page=recherche&recherche=full-monitor+game full-monitor game] titles. You don't want these using up even lots more of your time along with energy than within your main budget place a point clock of your thorough to your display computer monitor to be able to monitor just how prolonged periods of time you've been enjoying.<br><br>Make sure that your corporation build and buy some new laboratory so you can to research improved barbarians. Eventually, in the event that you take part inside game for most months, you might finally manage to get thier the nirvana of five-star barbarians.<br><br>And any our options are inspected and approved from the most effective virus recognition software and as a result anti-virus in the industry to ensure a security-level the size of you can, in argument you fear for protection of your computer or perhaps your cellular device, no situations. In case you nevertheless have some sort of doubts, take a brows through the movie and you'll focus on it operates and everyone 100% secure! It'll only take a few moments of energy!

	~~Let ''M'' be~~ a ~~[[compact space\|compact]] orientable 2''n''-dimensional [[Riemannian manifold]] without boundary, and let <math>\Omega</math> be the [[curvature form]]~~ of the ~~[[Levi-Civita connection]]~~. ~~This means that~~ <~~math~~>~~\Omega~~<~~/math~~> is ~~an <math>\mathfrak s\mathfrak o(2n)</math>-valued [[2-form]]~~ on ~~''M''~~. ~~So <math>\Omega</math> can be regarded as a skew-symmetric 2''n'' × 2''n'' matrix whose entries are 2-forms~~, ~~so it is a matrix over the [[commutative ring]]~~ <~~math~~>~~\bigwedge^{\hbox{even}}T^*M~~<~~/math~~>. ~~One may therefore take the [[Pfaffian]]~~ of ~~<math>\Omega</math>, <math>\mbox{Pf}(\Omega)</math>, which turns out~~ to ~~be a 2''n''-form~~.

	~~The '''generalized Gauss–Bonnet theorem''' states that~~
	~~:<math>\int_M \mbox{Pf}(\Omega)=(2\pi)^n\chi(M)\ </math>~~
	~~where <math>\chi(M)</math> denotes~~ the ~~[[Euler characteristic]]~~ of ~~''M''~~.

	~~==Example: dimension 4==~~
	~~In dimension <math>n=4</math>~~, for ~~a compact oriented manifold, we get~~

	:<~~math~~>~~\chi(M)=\frac{1}{32\pi^2}\int_M\left(\|Rm\|^2-4\|Rc\|^2+R^2\right)d\mu~~ <~~/math>~~

	~~where <math>Rm</math~~> ~~is the full~~ [~~[Riemann curvature tensor]], <math>Rc<~~/~~math> is the [[Ricci curvature\|Ricci curvature tensor]], and <math>R<~~/~~math> is the [[scalar curvature]]~~.

	=~~=Further generalizations==~~
	~~As with the two-dimensional Gauss–Bonnet Theorem, there are generalizations when ''M'' is a [[manifold\|manifold with boundary]~~].

	~~The Gauss–Bonnet Theorem can be seen as a special instance in~~ the ~~theory~~ of [~~[characteristic classes]~~]. ~~The Gauss–Bonnet integrand~~ is ~~the Euler class. Since it is a top-dimensional differential form~~, ~~it is closed. The naturality~~ of the ~~Euler class means that when you change~~ the ~~Riemannian metric~~, ~~you stay~~ in ~~the same cohomology class~~. ~~That means that~~ the ~~integral~~ of the ~~Euler class remains constant as you vary the metric, and so is an invariant of smooth structure~~.

	~~An extremely far-reaching generalization of the Gauss–Bonnet Theorem~~ is ~~the~~ [~~[Atiyah–Singer index theorem\|Atiyah–Singer Index Theorem]]~~. ~~Let <math>D<~~/~~math> be a (weakly) [[elliptic differential operator~~]~~] between vector bundles~~. ~~That means that the [[Symbol~~ of a ~~differential operator\|principal symbol]] is an isomorphism. (Strong ellipticity would furthermore require the symbol~~ to be ~~[[positive-definite]]~~.~~) Let~~ <~~math~~>D<~~/math~~> ~~be the [[adjoint operator]]. Then the index is defined as ''dim(ker(D))-dim(ker(D))'',~~ and ~~by ellipticity is always finite. The Index Theorem states that this ''analytical index'' is constant as~~ you ~~vary the elliptic operator smoothly~~. ~~It is in fact equal to a ''topological index''~~, ~~which can be expressed~~ in ~~terms of characteristic classes. The 2-dimensional Gauss–Bonnet Theorem arises as~~ the ~~special case where~~ the ~~topological index is defined in terms~~ of ~~[[Betti numbers]]~~ and the ~~analytical index is defined~~ in ~~terms of~~ the ~~Gauss–Bonnet integrand.~~

	~~==See also==~~
	*[[Chern–Weil homomorphism]]
	*[[Pontryagin number]]
	*[[Pontryagin class]]

	~~== References ==~~
	* {{Citation \| last1=Cycon \| first1=Hans
	~~\| last2=Froese \| first2=Richard~~
	~~\| last3=Kirsch \| first3=Werner~~
	~~\| last4=Simon \| first4=Barry~~
	~~\| title=Schroedinger operators~~
	~~\| publisher=[[Springer~~-~~Verlag]]~~
	~~\| location=Berlin~~, ~~New York~~
	~~\| edition=1st \| isbn=3-540-16758-7 \| year=1987}} Chapter 12~~
	* {{Citation \| last1=Chern \| first1=Shiing-Shen
	~~\| title=On the curvatura integra~~ in ~~Riemannian manifold~~
	~~\| journal=[[Annals~~ of ~~Mathematics]]~~
	~~\| volume=46\|issue=4~~
	~~\| year=1945 \|jstor=1969203~~
	~~\| pages=674–684 }} This is the historically first time that Chern-Gauss-Bonnet was proven without assuming the manifold to be a hypersurface~~. ~~For hypersurfaces~~, the ~~result had been shown first by Allendoerfer~~ and ~~Weil in 1940 which is cited in this paper~~ of ~~Chern.~~

	~~{{DEFAULTSORT:Generalized Gauss-Bonnet theorem}}~~
	~~[[Category:Theorems in differential geometry]]~~

	~~[[de:Satz von Gauß-Bonnet]]~~

en>Snarayr: /* Motivation */

2013-12-08T15:03:37Z

Motivation

← Older revision		Revision as of 17:03, 8 December 2013
Line 1:		Line 1:
	~~Some users of computer are aware which their computer become slower or have some errors following using for a while. But most folks don~~'~~t recognize how to speed up their computer plus some of them don~~'~~t dare to operate it. They always find several experts to keep~~ the ~~computer inside wise condition yet they have to spend several cash on it. Actually, you are able to do it by oneself. There are many registry cleaner software that you can get one~~ of ~~them online~~. ~~Some~~ of ~~them are free~~ and ~~we merely have~~ to ~~download them~~. ~~After installing it~~, ~~this registry cleaner software can scan~~ the ~~registry~~. ~~If it found these mistakes, it might report you and you are able to delete them to keep a registry clean. It~~ is ~~simple to work and it is actually the best method to repair registry~~.<br><br>~~When you registry gets cluttered up with~~ a ~~lot of junk we don~~'~~t employ~~, ~~the PC can run slower. Therefore~~ it~~'s prudent which you frequently get~~ a ~~registry cleaned~~.<br><br>So, ~~this advanced double scan is not merely among the greater~~, ~~but it~~'~~s equally freeware~~. ~~And as of all of this~~ that ~~numerous regard CCleaner 1~~ of ~~the greater registry cleaners inside the marketplace now~~. ~~I would add which I personally like Regcure~~ for ~~the simple reason which it has~~ a ~~greater interface and I learn for a truth it is ad~~-~~ware without charge.~~<br><br>~~Handling intermittent errors - when there~~ is ~~a content to~~ the ~~impact that "memory or difficult disk is malfunctioning"~~, ~~you can put in hot hardware to substitute the faulty piece till the actual issue~~ is ~~discovered. There are h/w diagnostic programs to identify~~ the ~~faulty portions.~~<br><br>~~The second step to fixing these mistakes~~ is ~~to use a program called a "~~[~~http://bestregistrycleanerfix~~.~~com/tune~~-~~up-utilities tuneup utilities 2014~~]~~" to scan by~~ the ~~computer and fix some~~ of the ~~registry errors which might also be leading for this error~~. ~~A registry cleaner~~ is a ~~software program~~ that ~~might scan through your computer plus repair some of~~ the ~~difficulties which Windows has inside~~, ~~permitting your computer to "remember" all settings it has when it loads up~~. ~~Although~~ the ~~registry~~ is ~~continually being chosen to help load up a large number~~ of ~~programs on your PC, it's continually being saved incorrectly~~ - ~~leading to a large amount~~ of ~~errors to be formed. To fix this issue, it's recommended we download a registry cleaner within~~ the ~~Internet and install it on your Pc, allowing Windows to run smoother again~~.<br><br>~~Active X controls are utilized across the entire spectrum of computer and internet technologies~~. ~~These controls are called~~ the ~~building blocks~~ of ~~the web plus as the glue that puts it all together. It is~~ a ~~standard which~~ is ~~employed by all developers to make the internet more valuable and interactive~~. ~~Without these control specifications there~~ would ~~basically~~ be ~~no public internet~~.<br><br>The '~~registry~~' is ~~simply~~ the ~~central database that shops all a settings plus options~~. It'~~s a actually important part~~ of the ~~XP program, which means which Windows~~ is ~~consistently adding plus updating~~ the ~~files inside it~~. ~~The difficulties occur whenever Windows really corrupts & loses a few~~ of ~~these files.~~ This ~~makes a computer run slow, because it tries difficult to find them again.<br><br>What I would suggest~~ is to ~~search on your own for registry products. You can do this with~~ a ~~Google look~~. ~~If you find products~~, ~~look for reviews plus testimonials regarding~~ the ~~product. Then you are able to see how others like the product, plus how effectively it functions~~.		In [[mathematics]], the '''generalized Gauss–Bonnet theorem''' (''also called [[Shiing-Shen Chern\|Chern]]–[[Gauss]]–[[Pierre Ossian Bonnet\|Bonnet]] theorem'') presents the [[Euler characteristic]] of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the [[Gauss–Bonnet theorem]] (named after [[Carl Friedrich Gauss]] and [[Pierre Ossian Bonnet]]) to higher dimensions.

			Let ''M'' be a [[compact space\|compact]] orientable 2''n''-dimensional [[Riemannian manifold]] without boundary, and let <math>\Omega</math> be the [[curvature form]] of the [[Levi-Civita connection]]. This means that <math>\Omega</math> is an <math>\mathfrak s\mathfrak o(2n)</math>-valued [[2-form]] on ''M''. So <math>\Omega</math> can be regarded as a skew-symmetric 2''n'' × 2''n'' matrix whose entries are 2-forms, so it is a matrix over the [[commutative ring]] <math>\bigwedge^{\hbox{even}}T^*M</math>. One may therefore take the [[Pfaffian]] of <math>\Omega</math>, <math>\mbox{Pf}(\Omega)</math>, which turns out to be a 2''n''-form.

			The '''generalized Gauss–Bonnet theorem''' states that
			:<math>\int_M \mbox{Pf}(\Omega)=(2\pi)^n\chi(M)\ </math>
			where <math>\chi(M)</math> denotes the [[Euler characteristic]] of ''M''.

			==Example: dimension 4==
			In dimension <math>n=4</math>, for a compact oriented manifold, we get

			:<math>\chi(M)=\frac{1}{32\pi^2}\int_M\left(\|Rm\|^2-4\|Rc\|^2+R^2\right)d\mu </math>

			where <math>Rm</math> is the full [[Riemann curvature tensor]], <math>Rc</math> is the [[Ricci curvature\|Ricci curvature tensor]], and <math>R</math> is the [[scalar curvature]].

			==Further generalizations==
			As with the two-dimensional Gauss–Bonnet Theorem, there are generalizations when ''M'' is a [[manifold\|manifold with boundary]].

			The Gauss–Bonnet Theorem can be seen as a special instance in the theory of [[characteristic classes]]. The Gauss–Bonnet integrand is the Euler class. Since it is a top-dimensional differential form, it is closed. The naturality of the Euler class means that when you change the Riemannian metric, you stay in the same cohomology class. That means that the integral of the Euler class remains constant as you vary the metric, and so is an invariant of smooth structure.

			An extremely far-reaching generalization of the Gauss–Bonnet Theorem is the [[Atiyah–Singer index theorem\|Atiyah–Singer Index Theorem]]. Let <math>D</math> be a (weakly) [[elliptic differential operator]] between vector bundles. That means that the [[Symbol of a differential operator\|principal symbol]] is an isomorphism. (Strong ellipticity would furthermore require the symbol to be [[positive-definite]].) Let <math>D</math> be the [[adjoint operator]]. Then the index is defined as ''dim(ker(D))-dim(ker(D))'', and by ellipticity is always finite. The Index Theorem states that this ''analytical index'' is constant as you vary the elliptic operator smoothly. It is in fact equal to a ''topological index'', which can be expressed in terms of characteristic classes. The 2-dimensional Gauss–Bonnet Theorem arises as the special case where the topological index is defined in terms of [[Betti numbers]] and the analytical index is defined in terms of the Gauss–Bonnet integrand.

			==See also==
			*[[Chern–Weil homomorphism]]
			*[[Pontryagin number]]
			*[[Pontryagin class]]

			== References ==
			* {{Citation \| last1=Cycon \| first1=Hans
			\| last2=Froese \| first2=Richard
			\| last3=Kirsch \| first3=Werner
			\| last4=Simon \| first4=Barry
			\| title=Schroedinger operators
			\| publisher=[[Springer-Verlag]]
			\| location=Berlin, New York
			\| edition=1st \| isbn=3-540-16758-7 \| year=1987}} Chapter 12
			* {{Citation \| last1=Chern \| first1=Shiing-Shen
			\| title=On the curvatura integra in Riemannian manifold
			\| journal=[[Annals of Mathematics]]
			\| volume=46\|issue=4
			\| year=1945 \|jstor=1969203
			\| pages=674–684 }} This is the historically first time that Chern-Gauss-Bonnet was proven without assuming the manifold to be a hypersurface. For hypersurfaces, the result had been shown first by Allendoerfer and Weil in 1940 which is cited in this paper of Chern.

			{{DEFAULTSORT:Generalized Gauss-Bonnet theorem}}
			[[Category:Theorems in differential geometry]]

			[[de:Satz von Gauß-Bonnet]]

en>ManDay: /* Functional analysis approach */

2012-08-27T11:43:37Z

Functional analysis approach

New page

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