Natural ventilation - Revision history

en>Danhatton: Placement of brackets (i.e. order of operations) in equation (1) corrected; terms in equation (1) explicitly defined.

2014-09-18T22:34:43Z

Placement of brackets (i.e. order of operations) in equation (1) corrected; terms in equation (1) explicitly defined.

← Older revision		Revision as of 00:34, 19 September 2014
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	~~{{no footnotes\|date=January 2011}}~~		I am Oscar and I totally dig that title. North Dakota is our birth place. What I love doing is playing baseball but I haven't produced a dime with it. Managing people is what I do and the salary has been really satisfying.<br><br>My weblog ... [http://rivoli.enaiponline.com/user/view.php?id=438251&course=1 std testing at home]
	~~[[File:MovingSingularity.png\|right\|thumb\|390px\|Solutions to the differential equation <math>\frac{dy}{dx} = \frac{1}{2y}</math> subject to the initial conditions y(0)=0, 1~~ and ~~2 (red, green and blue curves respectively)~~. ~~The positions of the moving singularity at x= 0, -1 and -4~~ is ~~indicated by the vertical lines~~.]]

	~~In the theory of [[ordinary differential equation]]s,~~ a ~~'''movable singularity'''~~ is ~~a point where the solution of the equation [[well-behaved\|behaves badly]]~~ and ~~which is "movable" in~~ the ~~sense that its location depends on the [[initial conditions]] of the differential equation~~.<~~ref name=BenderOrszag7~~>

	{{Cite book \| last = Bender \| first = Carl M. \| authorlink = \| coauthors = Orszag, Steven A. \| title = Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Series \| publisher = Springer \| date = 1999 \| location = \| pages = 7 }}<~~/ref~~>
	~~Suppose we have an [[ordinary differential equation]] in the complex domain~~. ~~Any given solution ''y''(''x'') of this equation may well have singularities at various points (i~~.e. ~~points at which it is not a regular [[holomorphic function]], such as~~ [[branch points]], [[Essential singularity\|essential singularities]] or [[Pole (complex analysis)\|poles]]). A singular point is said to be '''movable''' if its location depends on the particular solution we have chosen, rather than being fixed by the equation itself.

	~~For example the equation~~

	:~~<math> \frac{dy}{dx} = \frac{1}{2y}<~~/~~math>~~

	~~has solution <math>y=\sqrt{x-c}<~~/~~math> for any constant ''c''~~. ~~This solution has a branchpoint at <math>x=c<~~/~~math>, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant ''c'').~~

	~~It is a basic feature of linear ordinary differential equations that singularities of solutions occur only at singularities of the equation, and so linear equations do not have movable singularities.~~

	When attempting to look for 'good' nonlinear differential equations it is this property of linear equations that one would like to see: asking for no movable singularities is often too stringent, instead one often asks for the so-called [[Painlevé transcendents\|Painlevé property]]: 'any movable singularity should be a pole', first used by [[Sofia Kovalevskaya]].

	== ~~References ==~~
	~~{{reflist}}~~
	* Einar Hille (1997), ''Ordinary Differential Equations in the Complex Domain'', Dover. ISBN 0-486-69620-0

	~~[[Category:Complex analysis]]~~
	~~[[Category:Ordinary differential equations]~~]

201.141.146.65 at 18:23, 26 January 2014

2014-01-26T18:23:03Z

← Older revision		Revision as of 20:23, 26 January 2014
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	~~Hi there~~, ~~I am Alyson Pomerleau~~ and ~~I believe it seems fairly good when you say it~~. ~~North Carolina~~ is where we'~~ve been living for years and will by no means move~~. ~~Playing badminton~~ is a ~~factor that he~~ is ~~completely addicted~~ to. ~~Credit authorising is where my main income comes from~~.<br><br>~~my site;~~ [~~http~~:~~//mybrandcp~~.~~com/xe/board_XmDx25/107997 love psychics~~]		{{no footnotes\|date=January 2011}}
			[[File:MovingSingularity.png\|right\|thumb\|390px\|Solutions to the differential equation <math>\frac{dy}{dx} = \frac{1}{2y}</math> subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). The positions of the moving singularity at x= 0, -1 and -4 is indicated by the vertical lines.]]

			In the theory of [[ordinary differential equation]]s, a '''movable singularity''' is a point where the solution of the equation [[well-behaved\|behaves badly]] and which is "movable" in the sense that its location depends on the [[initial conditions]] of the differential equation.<ref name=BenderOrszag7>

			{{Cite book \| last = Bender \| first = Carl M. \| authorlink = \| coauthors = Orszag, Steven A. \| title = Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Series \| publisher = Springer \| date = 1999 \| location = \| pages = 7 }}</ref>
			Suppose we have an [[ordinary differential equation]] in the complex domain. Any given solution ''y''(''x'') of this equation may well have singularities at various points (i.e. points at which it is not a regular [[holomorphic function]], such as [[branch points]], [[Essential singularity\|essential singularities]] or [[Pole (complex analysis)\|poles]]). A singular point is said to be '''movable''' if its location depends on the particular solution we have chosen, rather than being fixed by the equation itself.

			For example the equation

			:<math> \frac{dy}{dx} = \frac{1}{2y}</math>

			has solution <math>y=\sqrt{x-c}</math> for any constant ''c''. This solution has a branchpoint at <math>x=c</math>, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant ''c'').

			It is a basic feature of linear ordinary differential equations that singularities of solutions occur only at singularities of the equation, and so linear equations do not have movable singularities.

			When attempting to look for 'good' nonlinear differential equations it is this property of linear equations that one would like to see: asking for no movable singularities is often too stringent, instead one often asks for the so-called [[Painlevé transcendents\|Painlevé property]]: 'any movable singularity should be a pole', first used by [[Sofia Kovalevskaya]].

			== References ==
			{{reflist}}
			* Einar Hille (1997), ''Ordinary Differential Equations in the Complex Domain'', Dover. ISBN 0-486-69620-0

			[[Category:Complex analysis]]
			[[Category:Ordinary differential equations]]

en>Arcandam: clean up, typos fixed: fire place → fireplace, mid-point → midpoint, United Stated → United States (2) using AWB

2012-06-22T18:04:56Z

clean up, typos fixed: fire place → fireplace, mid-point → midpoint, United Stated → United States (2) using AWB

New page

Hi there, I am Alyson Pomerleau and I believe it seems fairly good when you say it. North Carolina is where we've been living for years and will by no means move. Playing badminton is a factor that he is completely addicted to. Credit authorising is where my main income comes from.<br><br>my site; [http://mybrandcp.com/xe/board_XmDx25/107997 love psychics]