Eccentric anomaly: Difference between revisions

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In [[mathematics]], the '''Chern–Weil homomorphism''' is a basic construction in the [[Chern–Weil theory]], relating for a [[smooth manifold]] ''M'' the [[curvature]] of ''M'' to the [[de Rham cohomology]] groups of ''M'', i.e., geometry to topology. This theory of [[Shiing-Shen Chern]] and [[André Weil]] from the 1940s was an important step in the theory of [[characteristic class]]es.  It is a generalization of the [[Chern–Gauss–Bonnet theorem]].
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This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.<br><br>The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.<br><br>99                  &#9733;&#9733;&#9733;&#9733;&#9733;Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�[http://Browse.Deviantart.com/?q=s+famous s famous] examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.<br><br>And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �<br>This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.<br><br>Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx� <br>The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�<br>�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!<br><br>She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.<br><br>The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum<br>But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.<br><br>In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.<br><br>Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3[http://www.sharkbayte.com/keyword/D+printer D printer].<br><br>Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.<br><br>WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES<br>The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals <br>With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. &#9733;&#9733;&#9733;&#9733;&#9733;<br><br>If you have any kind of questions pertaining to where and ways to utilize [http://nouveauclashofclanstriche.blogspot.com/ http://nouveauclashofclanstriche.blogspot.com/], you can call us at our own web site.
 
Denote by <math>\mathbb K</math> either the [[real number|real field]] or [[complex field]]. Let ''G'' be a real or complex [[Lie group]] with [[Lie algebra]] <math>\mathfrak g</math>; and let
 
:<math>\mathbb K(\mathfrak g^*)</math>
 
denote the algebra of <math>\mathbb K</math>-valued [[Polynomials on vector spaces|polynomial]]s on <math>\mathfrak g</math>. Let <math>\mathbb K(\mathfrak g^*)^{Ad(G)}</math> be the subalgebra of fixed points in <math> \mathbb K(\mathfrak g^*)</math> under the [[adjoint representation of a Lie group|adjoint action]] of ''G'', so that for instance
 
:<math>f(t_1,\dots,t_k)=f(Ad_g t_1,\dots, Ad_g t_k) \, </math>
 
for all <math>f\in\mathbb K(\mathfrak g^*)^{Ad(G)}. \, </math>
 
The '''Chern–Weil homomorphism''' is a homomorphism of <math>\mathbb K</math>-algebras from <math>\mathbb K(\mathfrak g^*)^{Ad(G)}</math> to the cohomology algebra <math>H^*(M,\mathbb K)</math>.   Such a homomorphism exists and is uniquely defined for every [[principal bundle|principal G-bundle]] ''P'' on ''M''.   If ''G'' is compact, then under the homomorphism, the cohomology ring of the [[classifying space]] for ''G''-bundles ''B''<sup>''G''</sup> is isomorphic to the algebra <math>\mathbb K(\mathfrak g^*)^{Ad(G)}</math> of invariant polynomials:
:<math>H^*(B^G, \mathbb{K}) \cong \mathbb K(\mathfrak g^*)^{Ad(G)}.</math>
For non-compact groups like SL(''n'','''R'''), there may be cohomology classes that are not represented by invariant polynomials.
 
==Definition of the homomorphism==
Choose any [[connection form]] w in P, and let <math>\Omega</math> be the associated [[curvature]] 2-form. If <math>f\in\mathbb K(\mathfrak g^*)^{Ad(G)}</math> is a homogeneous polynomial of degree&nbsp;''k'', let <math>f(\Omega)</math> be the 2''k''-form on ''P'' given by
 
:<math>f(\Omega)(X_1,\dots,X_{2k})=\frac{1}{(2k)!}\sum_{\sigma\in\mathfrak S_{2k}}\epsilon_\sigma f(\Omega(X_{\sigma(1)},X_{\sigma(2)}),\dots,\Omega(X_{\sigma(2k-1)}, X_{\sigma(2k)}))</math>
 
where <math>\epsilon_\sigma</math> is the sign of the permutation <math>\sigma</math> in the symmetric group on 2''k'' numbers <math>\mathfrak S_{2k}</math>.
 
(see [[Pfaffian]]).
 
One can then show that
 
:<math>f(\Omega)</math>
 
is a [[closed differential form|closed form]], so that
 
:<math>df(\Omega)=0, \, </math>
 
and that the [[de Rham cohomology]] class of
 
:<math>f(\Omega) \, </math>
 
is independent of the choice of connection on ''P'', so it depends only upon the principal bundle.
 
Thus letting
 
:<math>\phi(f) \, </math>
 
be the cohomology class obtained in this way from f, we obtain an algebra homomorphism
 
:<math>\phi:\mathbb K(\mathfrak g^*)^{Ad(G)}\rightarrow H^*(M,\mathbb K). \, </math>
 
==References==
* {{citation|last=Bott|first=R.|authorlink=Raoul Bott|title=On the Chern–Weil homomorphism and the continuous cohomology of Lie groups|journal=Advances in Math| year=1973|volume=11|pages=289–303|doi=10.1016/0001-8708(73)90012-1}}.
* {{citation|last=Chern|first=S.-S.|authorlink=S.-S. Chern|title=Topics in Differential Geometry|publisher=Institute for Advanced Study, mimeographed lecture notes|year=1951}}.
* Shiing-Shen Chern, ''Complex Manifolds Without Potential Theory'' (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.
*:The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.
* {{citation|last1=Chern|first1=S.-S.|authorlink1=S.-S. Chern|last2=Simons|first2=J|authorlink2=James Harris Simons|title=Characteristic forms and geometric invariants|journal=The Annals of Mathematics. Second Series|volume=99|number=1|year=1974|pages=48–69|jstor=1971013}}.
* {{citation|last1=Kobayashi|first1=S.|last2=Nomizu|first2=K.|title=[[Foundations of Differential Geometry]], Vol. 2|publisher=Wiley-Interscience|year=1963|publication-date= new ed. 2004}}.
* {{citation|last1=Narasimhan|first1=M.|last2=Ramanan|first2=S.|title=Existence of universal connections|journal=Amer. J. Math.|volume=83|year=1961|pages=563–572|jstor=2372896|doi=10.2307/2372896}}.
* {{citation|last1=Morita|first1=Shigeyuki|title=Geometry of Differential Forms|journal=A.M.S monograph|volume=201|year=2000}}.
 
{{DEFAULTSORT:Chern-Weil homomorphism}}
[[Category:Differential geometry]]
[[Category:Characteristic classes]]

Latest revision as of 02:18, 8 January 2015

He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is.

I keep conjuring up images of him in 1983, trying to reignite the passion.
I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry.

He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk.

The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen.

99 ★★★★★Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�s famous examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329.

And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. �
This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you.

Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx�
The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message�
�The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road!

She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week.

The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum
But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7.

In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth.

Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3D printer.

Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too.

WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES
The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals
With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. ★★★★★

If you have any kind of questions pertaining to where and ways to utilize http://nouveauclashofclanstriche.blogspot.com/, you can call us at our own web site.

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