212.44.26.236: SL2 can mean Situational Leadership II

2014-10-01T14:56:10Z

SL2 can mean Situational Leadership II

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	~~In differential geometry, the '''integration along fibers''' of a [[differential form\|''k''-form]] yields a~~ <~~math~~>~~(k-m)~~</~~math>~~-~~form where~~ ''~~m'' is the dimension~~ of the ~~fiber, via "integration". More precisely, let <math>\pi: E \~~to ~~B</math> be a fiber bundle over a manifold with compact oriented fibers~~. ~~If <math>\alpha</math> is~~ a ~~''k''-form on ''E''~~, ~~then let:~~		<br><br>[http://hostgator.com/hostgator-coupons hostgator.com]Let's have an appearance on world's webhosting business: Thousands and hundreds of webhosting spread around the world and their company purpose is basic - to get acknowledged and make profit. With a lot noise around the web hosting market, it is a tough task for us to get the right hosting. Nonetheless, it's not too difficult for me to detect Hostgator - a hosting service that stood apart from the rest with exceptional features. What I am going to perform in this short article is to examine Hostgator hosting in addition to disclosing the secret of Hostgator success.<br><br>In any web company, we depend on excellent web hosts to provide stable and trusted internet hosting services. Hostgator offers prompt and promising qualities.<br><br>Hostgator provides a bunch of packages to consumers. Almost all the packages provided are cost effective. Hostgator shared hosting are charged $4.95 regular monthly for the standard shared hosting packaged. At costs which begin from $9.95 yearly, you can delight in much more compact bundles which actually include a bunch of things. Although there still have a lot bundles which offering less expensive rates, however if you start to research and attempt Hostgator, you will feel that every dollar and cent you invested is simply not wasted. Do your very own research on the attributes included in every bundles Hostgator providing, then you can distinguish the differences with others bundles provided by other host.<br><br>Hostgator also provides a lot of various other functions. For exaple, Hostgator assists you to back up the web websites immediately and safely. You have the option to host unrestricted domains, sub domains and FTP accounts. Of cause, the function which pointed out simply now will not be readily available if you are the basic hosting bundle subscriber. For basic package, Hostgator still provides you single domain which packaged together with twenty sub domains. In my viewpoint, you can begin your trial with Hostgator's fundamental plan. When business gets mature and thinking for growth, you can anytime upgrade your plan to even more advance one.<br><br>Hostgator offers control panel. If you enjoyed this write-up and you would certainly such as to get additional info regarding [http://dreamindianetwork.hillpost.in/groups/the-stairs-of-entanglement-hosting-domain-registration/ Hostgator Coupon] kindly go to the web site. This feature helps users to handle their domains and sub domains in even more flexible manner. Besides, the control board trial which is available in conventional offers the valuable help devices to new individuals. Out of so lots of host around the globe, Hostgator's cpanel can be said succeeding the leading position amongst them.<br><br>In regard to consumer supports, Hostgator has a very strong consumer care group which working 24/7 to assist you take on all types of troubles pertinent to their items. You can either contact them with phone or live talk with them with internet. You can likewise email your issues to them. They will feedback promptly without taking any [http://www.Wonderhowto.com/search/charges/ charges]. This is why previously, Hostgator still the finest option for clients.<br><br>A last words for those who are beginning - don't be scare by what you are reviewing as constructing a good website is not that hard. The work is in truth rather simple and direct as long as you're getting the right info. To get begun, you'll initially require a domain and a reputable hosting service.

	~~: <math>(\pi_* \alpha)_b(w_1, \dots, w_{k-m}) = \int_{\pi^{-1}(b)} \beta</math>~~

	~~where <math>\beta</math>~~ is the ~~induced top~~-~~form on~~ the ~~fiber <math>\pi^{-1}(b)</math>; i~~.~~e., an <math>m</math>-form given by~~

	~~:<math>\beta(v_1, \dots, v_m) = \alpha(\widetilde{w_1}, \dots, \widetilde{w_{k-m}}, v_1, \dots, v_m), \quad \widetilde{w_i}\text{~~ the ~~lifts~~ of ~~} w_i~~.<~~/math~~>

	~~(To see~~ <~~math~~>~~b \mapsto (\pi_* \alpha)_b</math> is smooth~~, ~~work it out in coordinates; cf~~. ~~an example below~~.)

	<~~math~~>\pi_*<~~/math~~> ~~is then~~ a ~~linear map <math>\Omega^k(E) \~~to ~~\Omega^{k-m}(B)</math>~~, which ~~is in fact surjective~~. ~~By Stokes' formula~~, if the ~~fibers have no boundaries~~, the ~~map descends to [[de Rham cohomology]]:~~

	:<~~math~~>~~\pi_*: \operatorname{H}^k(E) \to \operatorname{H}^{k-m}(B).~~<~~/math~~>

	~~This is~~ also ~~called the fiber integration. Now, suppose <math>\pi</math> is~~ a ~~sphere bundle; i.e~~., the ~~typical fiber is a sphere~~. ~~Then there is an exact sequence <math>0 \~~to ~~K \to \Omega^(E) \overset{\pi_}\to \Omega^*(B) \to 0</math>~~, ~~''K''~~ the ~~kernel,~~
	which ~~leads to a long exact sequence, using <math>\operatorname{H}^k(B) \simeq \operatorname{H}^{k+m}(K)</math>:~~
	:<math>\dots \rightarrow \operatorname{H}^k(B) \overset{\delta}\to \operatorname{H}^{k+m+1}(B) \overset{\pi^} \rightarrow \operatorname{H}^{k+m+1}(E) \overset{\pi_} \rightarrow \operatorname{H}^{k+1}(B) \rightarrow \dots</math>,
	~~called~~ the ~~[[Gysin sequence]]~~.

	~~== Example ==~~
	~~Let <math>\pi: M \times [0~~, ~~1] \to M</math> be an obvious projection~~. ~~For simplicity~~, ~~assume <math>M = \mathbb{R}^n</math>~~ with ~~coordinates <math>x_j</math>~~ and ~~consider a ''k''-form:~~

	~~:<math>\alpha = f \~~, ~~dx_{i_1} \wedge \dots \wedge dx_{i_k} + g \, dt \wedge dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}~~.<~~/math~~>

	~~Then, at each point in ''M'',~~

	:<~~math~~>~~\pi_(\alpha) = \pi_(g \, dt \wedge dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}) = \left( \int_0^1 g(\cdot, t) \, dt \right) \, {dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}}~~.~~</math>~~

	~~From~~ this ~~the next formula follows easily~~: ~~if <math>\alpha<~~/~~math> is any ''k''-form on <math>M \times I,<~~/~~math>~~

	~~:<math>\pi_(d \alpha) = \alpha_1 - \alpha_0 - d \pi_(\alpha)<~~/~~math>~~

	~~where <math>\alpha_i<~~/~~math> is~~ the ~~restriction~~ of ~~<math>\alpha<~~/~~math>~~ to ~~<math>M \times \{i\}</math>~~. This ~~formula~~ is ~~a special case~~ of ~~Stokes' formula. As an application~~ of ~~this~~, ~~let <math>f: M \times [0, 1] \to N</math>~~ be ~~a smooth map (thought of as a homotopy)~~. ~~Then the composition~~ <~~math~~>h = \pi_* \circ f^*<~~/math~~> ~~is a [[homotopy operator]]:~~

	~~:<math>d \circ h + h \circ d = f_1^* - f_0^*: \Omega^k(N) \~~to ~~\Omega^k(M)~~,~~</math>~~

	which ~~implies <math>f_1, f_0<~~/~~math> induces the same map~~ on ~~cohomology~~. ~~For example, let ''U'' be an open ball~~ with ~~center at the origin and let <math>f_t~~: ~~U \to U, x \mapsto tx<~~/~~math>~~. ~~Then <math>\operatorname{H}^k(U) = \operatorname{H}^k(pt)<~~/~~math>, the fact known as the [[Poincaré lemma]~~].
	~~<!--~~
	~~== Calculus formulae ==~~
	~~The usual formulae from calculus (in particular~~, ~~Stokes' formula) continue to hold~~ for ~~integration along fibers~~.

	~~Let~~ <~~math~~>~~\pi: E \to N~~<~~/math~~> be a ~~fiber bundle~~.

	~~Stokes~~' ~~formula says:~~
	~~:<math>\pi_(d \alpha) - d (\pi_ \alpha) = (-1)^{k+r+1} \pi~~'~~_* (i^* \alpha)</math>-->~~

	~~== See also ==~~
	*[[Gysin homomorphism]]
	*[[Transgression map]]

	~~== References ==~~
	*[[Michele Audin]], Torus actions on symplectic manifolds, Birkhauser, 2004

	~~[[Category:Differential geometry]]~~

173.174.157.72 at 01:41, 20 November 2013

2013-11-20T01:41:39Z

New page

In differential geometry, the '''integration along fibers''' of a [[differential form|''k''-form]] yields a <math>(k-m)</math>-form where ''m'' is the dimension of the fiber, via "integration". More precisely, let <math>\pi: E \to B</math> be a fiber bundle over a manifold with compact oriented fibers. If <math>\alpha</math> is a ''k''-form on ''E'', then let:

: <math>(\pi_* \alpha)_b(w_1, \dots, w_{k-m}) = \int_{\pi^{-1}(b)} \beta</math>

where <math>\beta</math> is the induced top-form on the fiber <math>\pi^{-1}(b)</math>; i.e., an <math>m</math>-form given by

:<math>\beta(v_1, \dots, v_m) = \alpha(\widetilde{w_1}, \dots, \widetilde{w_{k-m}}, v_1, \dots, v_m), \quad \widetilde{w_i}\text{ the lifts of } w_i.</math>

(To see <math>b \mapsto (\pi_* \alpha)_b</math> is smooth, work it out in coordinates; cf. an example below.)

<math>\pi_*</math> is then a linear map <math>\Omega^k(E) \to \Omega^{k-m}(B)</math>, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to [[de Rham cohomology]]:

:<math>\pi_*: \operatorname{H}^k(E) \to \operatorname{H}^{k-m}(B).</math>

This is also called the fiber integration. Now, suppose <math>\pi</math> is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence <math>0 \to K \to \Omega^*(E) \overset{\pi_*}\to \Omega^*(B) \to 0</math>, ''K'' the kernel,
which leads to a long exact sequence, using <math>\operatorname{H}^k(B) \simeq \operatorname{H}^{k+m}(K)</math>:
:<math>\dots \rightarrow \operatorname{H}^k(B) \overset{\delta}\to \operatorname{H}^{k+m+1}(B) \overset{\pi^*} \rightarrow \operatorname{H}^{k+m+1}(E) \overset{\pi_*} \rightarrow \operatorname{H}^{k+1}(B) \rightarrow \dots</math>,
called the [[Gysin sequence]].

== Example ==
Let <math>\pi: M \times [0, 1] \to M</math> be an obvious projection. For simplicity, assume <math>M = \mathbb{R}^n</math> with coordinates <math>x_j</math> and consider a ''k''-form:

:<math>\alpha = f \, dx_{i_1} \wedge \dots \wedge dx_{i_k} + g \, dt \wedge dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}.</math>

Then, at each point in ''M'',

:<math>\pi_*(\alpha) = \pi_*(g \, dt \wedge dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}) = \left( \int_0^1 g(\cdot, t) \, dt \right) \, {dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}}.</math>

From this the next formula follows easily: if <math>\alpha</math> is any ''k''-form on <math>M \times I,</math>

:<math>\pi_*(d \alpha) = \alpha_1 - \alpha_0 - d \pi_*(\alpha)</math>

where <math>\alpha_i</math> is the restriction of <math>\alpha</math> to <math>M \times \{i\}</math>. This formula is a special case of Stokes' formula. As an application of this, let <math>f: M \times [0, 1] \to N</math> be a smooth map (thought of as a homotopy). Then the composition <math>h = \pi_* \circ f^*</math> is a [[homotopy operator]]:

:<math>d \circ h + h \circ d = f_1^* - f_0^*: \Omega^k(N) \to \Omega^k(M),</math>

which implies <math>f_1, f_0</math> induces the same map on cohomology. For example, let ''U'' be an open ball with center at the origin and let <math>f_t: U \to U, x \mapsto tx</math>. Then <math>\operatorname{H}^k(U) = \operatorname{H}^k(pt)</math>, the fact known as the [[Poincaré lemma]].


== See also ==
*[[Gysin homomorphism]]
*[[Transgression map]]

== References ==
*[[Michele Audin]], Torus actions on symplectic manifolds, Birkhauser, 2004

[[Category:Differential geometry]]

SL2 - Revision history

212.44.26.236: SL2 can mean Situational Leadership II

173.174.157.72 at 01:41, 20 November 2013