Heavy traffic approximation - Revision history

en>Jodosma: /* Heavy traffic condition */ rm repetition of "as"

2014-04-24T07:42:41Z

Heavy traffic condition: rm repetition of "as"

← Older revision		Revision as of 09:42, 24 April 2014
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	In mathematics, the '''base change map''' relates the [[Direct image functor\|direct image]] and the [[Pullback (category theory)\|pull-back]] of [[sheaf (mathematics)\|sheaves]]. More precisely, it is the following [[natural transformation]] of sheaves:		On a different show, she might be a standard issue political lobbyist.<br>The costs of poor control are huge. $174 billion in total costs to the system is related to diabetes.3 times higher to take care of a patient with diabetes than to take care of a patient without diabetes.<br>http://www.restaurantcalcuta.com/outlet/ugg.asp?p=239 <br /> http://www.restaurantcalcuta.com/outlet/ugg.asp?p=68 <br /> http://www. If you adored this information and you would like to obtain additional information relating to [http://peterlongogolfshow.com/coach/ Coach Outlet Online] kindly browse through the internet site. restaurantcalcuta.com/outlet/ugg.asp?p=296 <br /> http://www.restaurantcalcuta.com/outlet/ugg.asp?p=254 <br /> http://www.restaurantcalcuta.com/outlet/ugg.asp?p=258 <br />
	~~:<math>g^(R^r f_ \mathcal{F}) \to R^r f^'_(g'^\mathcal{F})</math>~~
	~~where <math>f: X \to S, f':X' \to S'~~, ~~g':X' \to X, g:S' \to S</math> are continuous maps between topological spaces that form a [[Cartesian_square_(category_theory)\|Cartesian square]]<!-- there really should~~ be a ~~diagram~~. ~~--> and~~ <~~math~~>~~\mathcal{F}</math> is a sheaf on ''X''~~.

	~~In [[general topology]],~~ the ~~map~~ is ~~an isomorphism under some mild technical conditions~~. ~~An analogous result holds for étale cohomologies (~~with ~~topological spaces replaced by sites), though more difficult. See [[proper base change theorem]].~~

	~~== General topology ==~~
	~~If ''X'' is a Hausdorff topological space, ''S'' is~~ a ~~locally compact Hausdorff space and ''f'' is universally closed (i.e~~., <~~math~~>~~X \times_S T \to T<~~/~~math> is closed for any continuous map <math>T \to S<~~/~~math>), then~~
	~~the base change map is an isomorphism~~.~~<ref>{{harvnb\|Milne, Theorem 17~~.~~3}}<~~/~~ref> Indeed, we have: for <math>s \in S~~</~~math~~>,
	:~~<math>(R^r f_* \mathcal{F})_s = \varinjlim H^r(U, \mathcal{F}) = H^r(X_s, \mathcal{F}), \quad X_s = f^{-1}(s)<~~/~~math>~~
	~~and so for <math>s = g(t)<~~/~~math>~~
	~~:<math>g^* (R^r f_* \mathcal{F})_t = H^r(X_s, \mathcal{F}) = H^r(X'_t, g'^* \mathcal{F}) = R^r f'_* (g'^* \mathcal{F})_t~~.</~~math>~~

	~~== Derivation =~~=
	Since <math>g'^</~~math~~> ~~is left adjoint to <math>g'_<~~/~~math>, we have:~~
	~~:<math>\operatorname{id} \to g'_* \circ g'^*<~~/~~math>~~
	and so
	:<math>R^r f_* \to R^r f_* \circ g'_* \circ g'^*.</~~math>~~
	~~The [[Grothendieck spectral sequence~~]~~] then gives~~ the ~~first map and the last map (they are edge maps) in:~~
	:<math>R^r f_* \circ g'_* \circ g'^* \to R^r(f \circ g')_* \circ g'^* = ~~R^r(g \circ f')_* \circ g'^* \to g_* \circ R^r f'_* \circ g'^*.~~</~~math~~>
	~~Combining this with the above we get~~
	:<math>R^r f_* \to g_* \circ R^r f'_* \circ g'^*.</~~math~~>
	~~Again using the adjoint relation we get the desired map.~~

	~~== See also ==~~
	*[[Theorem on formal functions]]

	~~== References ==~~
	~~{{reflist}}~~
	* [[James Milne (mathematician)\|J. S. Milne]] (2012). "[http://www.~~jmilne~~.~~org~~/~~math~~/~~CourseNotes~~/~~LEC.pdf Lectures on Étale Cohomology]~~






	~~{{geometry-stub}}~~

	~~[[Category:Geometry]]~~

en>Gareth Jones: /* Heavy traffic condition */ add QED regime as alternative name for 3rd limiting regime

2014-01-06T15:01:03Z

Heavy traffic condition: add QED regime as alternative name for 3rd limiting regime

← Older revision		Revision as of 17:01, 6 January 2014
Line 1:		Line 1:
			In mathematics, the '''base change map''' relates the [[Direct image functor\|direct image]] and the [[Pullback (category theory)\|pull-back]] of [[sheaf (mathematics)\|sheaves]]. More precisely, it is the following [[natural transformation]] of sheaves:
			:<math>g^(R^r f_ \mathcal{F}) \to R^r f^'_(g'^\mathcal{F})</math>
			where <math>f: X \to S, f':X' \to S', g':X' \to X, g:S' \to S</math> are continuous maps between topological spaces that form a [[Cartesian_square_(category_theory)\|Cartesian square]]<!-- there really should be a diagram. --> and <math>\mathcal{F}</math> is a sheaf on ''X''.

			In [[general topology]], the map is an isomorphism under some mild technical conditions. An analogous result holds for étale cohomologies (with topological spaces replaced by sites), though more difficult. See [[proper base change theorem]].

	~~The stars of British fashion will descend tonight~~ to ~~celebrate~~ the ~~in-house talent at the British Fashion Awards~~.<br><br>~~Click here to see an archive gallery of the British Designer of Year throughout the years~~<br>~~The annual event celebrates the creativity and success of the world'~~s ~~best fashion talents~~, ~~including designers~~, ~~models~~ and ~~writers.~~ <br><br>~~This year the event expands and targets a wider audience whose able to attend a night packed with fashion icons~~, ~~stars of the moment~~, ~~emerging talent and fashion-conscious celebrities. It will take place at the English National Opera~~.<br>~~British Designer of the Year~~ is ~~one of the most emblematic awards of the night. It [http~~:/~~/www~~.~~pcs-systems.co.uk~~/~~Images/celinebag.aspx Cheap Celine Bags~~] ~~recognises an internationally acclaimed, leading British designer who directed~~ the ~~shape of fashion in~~ the ~~past year~~ in ~~both the UK and internationally~~. ~~Stella McCartney bagged the award last year, together~~ with ~~Designer Brand of~~ the ~~Year, but 2013 sees different contenders.~~<br><br>~~The nominees are: Christopher Kane, Phoebe Philo and Sarah Burton for Womenswear Designer of~~ the ~~Yearand Christopher Bailey, Kim Jones and Tom Ford for~~ [http://~~Www~~.~~Squidoo~~.~~com~~/~~search~~/~~results?q=Menswear Menswear~~].<br>As we wait for the winner, we take a look at the designers that have been awarded British Designer of Year since the 1980s.<br>Click here to see an archive gallery of the British Designer of Year throughout the years		== General topology ==
			If ''X'' is a Hausdorff topological space, ''S'' is a locally compact Hausdorff space and ''f'' is universally closed (i.e., <math>X \times_S T \to T</math> is closed for any continuous map <math>T \to S</math>), then
			the base change map is an isomorphism.<ref>{{harvnb\|Milne, Theorem 17.3}}</ref> Indeed, we have: for <math>s \in S</math>,
			:<math>(R^r f_* \mathcal{F})_s = \varinjlim H^r(U, \mathcal{F}) = H^r(X_s, \mathcal{F}), \quad X_s = f^{-1}(s)</math>
			and so for <math>s = g(t)</math>
			:<math>g^* (R^r f_* \mathcal{F})_t = H^r(X_s, \mathcal{F}) = H^r(X'_t, g'^* \mathcal{F}) = R^r f'_* (g'^* \mathcal{F})_t.</math>

			== Derivation ==
			Since <math>g'^</math> is left adjoint to <math>g'_</math>, we have:
			:<math>\operatorname{id} \to g'_* \circ g'^*</math>
			and so
			:<math>R^r f_* \to R^r f_* \circ g'_* \circ g'^*.</math>
			The [[Grothendieck spectral sequence]] then gives the first map and the last map (they are edge maps) in:
			:<math>R^r f_* \circ g'_* \circ g'^* \to R^r(f \circ g')_* \circ g'^* = R^r(g \circ f')_* \circ g'^* \to g_* \circ R^r f'_* \circ g'^*.</math>
			Combining this with the above we get
			:<math>R^r f_* \to g_* \circ R^r f'_* \circ g'^*.</math>
			Again using the adjoint relation we get the desired map.

			== See also ==
			*[[Theorem on formal functions]]

			== References ==
			{{reflist}}
			* [[James Milne (mathematician)\|J. S. Milne]] (2012). "[http://www.jmilne.org/math/CourseNotes/LEC.pdf Lectures on Étale Cohomology]






			{{geometry-stub}}

			[[Category:Geometry]]

24.53.78.52 at 12:49, 27 August 2012

2012-08-27T12:49:57Z

New page

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