Atmospheric tide - Revision history

84.130.53.50: Edit for clarity

2014-11-17T16:35:36Z

Edit for clarity

← Older revision		Revision as of 18:35, 17 November 2014
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	~~In [[mathematics]], in the field of [[ordinary differential equation]]~~s~~, the '''Sturm–Picone comparison theorem''', named after [[Jacques Charles François Sturm]] and [[Mauro Picone]],~~ is ~~a classical theorem which provides criteria for the [[oscillation theory\|oscillation]]~~ and ~~[[oscillation theory\|non-oscillation]] of solutions of certain [[linear differential equation]]s in the real domain~~.		The author's name is Andera and she believes it sounds fairly great. What me and my family members love is bungee leaping but I've been taking on new issues recently. Distributing manufacturing has been his profession for some time. For many years she's been residing in Kentucky but her husband wants them to transfer.<br><br>My web-site - [http://black7.mireene.com/aqw/5741 clairvoyance]

	~~Let <math>p_i,\, q_i,\,</math>~~ '~~'i'' = 1, 2, be real-valued continuous functions~~ on ~~the interval [''a'', ''b''] and let~~
	~~#<math>(p_1(x) y^\prime)^\prime + q_1(x) y = 0 \,</math>~~
	~~#<math>(p_2(x) y^\prime)^\prime + q_2(x) y = 0 \,</math>~~
	~~be two homogeneous linear second order differential equations in [[self-adjoint form]] with~~
	~~:<math>0 < p_2(x) \le p_1(x)\,</math>~~
	~~and~~
	~~:<math>q_1(x) \le q_2(x)~~.~~\,</math>~~

	~~Let ''u'' be a non-trivial solution of (1) with successive roots at ''z''<sub>1</sub> and ''z''<sub>2</sub> and let ''v'' be a non-trivial solution of (2)~~. ~~Then one of the following properties holds.~~
	*There exists an '~~'x'' in [''z''<sub>1</sub>, ''z''<sub>2</sub>] such that ''v''(''x'') = 0; or~~
	*there exists a λ in ~~'''R''' such that ''v''(''x'') = λ ''u''(''x'').~~

	NOTE: The first part of the conclusion is due to Sturm (1836).<ref>C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186</ref> The second (alternative) part of this theorem is due to ~~Picone (1910)<ref>M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine, Ann. Scuola Norm. Pisa 11 (1909), 1–141~~.<~~/ref~~><~~ref~~>{{cite doi\|10.1007/3-7643-7359-8_1}}</ref> whose simple proof was given using his now famous [[Picone identity]]. In the special case where both equations are identical one obtains the [[Sturm separation theorem]]. For an extension of this important theorem to a comparison theorem involving three or more real second order equations see the [[Hartman–Mingarelli comparison theorem]] where a simple proof was given using the [[Mingarelli identity]].

	~~== References ==~~

	~~{{Reflist}}~~
	*Diaz, J. B.; [[Joyce McLaughlin\|McLaughlin, Joyce R.]] ''Sturm comparison theorems for ordinary and partial differential equations''. Bull. Amer. Math. Soc. 75 1969 335–339 [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183530292 pdf]
	* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 79, Krieger, Huntington ISBN 0-~~88275~~-~~368-1 .~~
	*{{cite book\| last = Teschl\| given = G.\|authorlink=Gerald Teschl\| title = Ordinary Differential Equations and Dynamical Systems\| publisher=[[American Mathematical Society]]\| place = [[~~Providence, Rhode Island\|Providence]]\| year = 2012\| isbn= 978-0-8218-8328-0\| url =~~ http://~~www~~.~~mat~~.~~univie.ac.at~~/~~~gerald~~/~~ftp/book-ode/}}~~

	~~{{DEFAULTSORT:Sturm-Picone comparison theorem}}~~
	~~[[Category:Ordinary differential equations]]~~
	~~[[Category:Theorems in analysis]~~]

158.39.149.68: /* General Solution of Laplaces Equation */

2013-12-02T17:15:56Z

General Solution of Laplaces Equation

← Older revision		Revision as of 19:15, 2 December 2013
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	~~The writer~~ is ~~called Wilber Pegues~~. ~~Alaska is where I~~'~~ve always been living~~. ~~To perform lacross~~ is ~~something I really enjoy performing~~. ~~My working day occupation~~ is ~~an information officer but I~~'~~ve already applied for another 1~~.<br><br>~~Stop by my page :: psychic readings online~~ - [http://~~brazil~~.~~amor-amore~~.~~com~~/~~irboothe brazil~~.~~amor~~-~~amore~~.~~com~~],		In [[mathematics]], in the field of [[ordinary differential equation]]s, the '''Sturm–Picone comparison theorem''', named after [[Jacques Charles François Sturm]] and [[Mauro Picone]], is a classical theorem which provides criteria for the [[oscillation theory\|oscillation]] and [[oscillation theory\|non-oscillation]] of solutions of certain [[linear differential equation]]s in the real domain.

			Let <math>p_i,\, q_i,\,</math> ''i'' = 1, 2, be real-valued continuous functions on the interval [''a'', ''b''] and let
			#<math>(p_1(x) y^\prime)^\prime + q_1(x) y = 0 \,</math>
			#<math>(p_2(x) y^\prime)^\prime + q_2(x) y = 0 \,</math>
			be two homogeneous linear second order differential equations in [[self-adjoint form]] with
			:<math>0 < p_2(x) \le p_1(x)\,</math>
			and
			:<math>q_1(x) \le q_2(x).\,</math>

			Let ''u'' be a non-trivial solution of (1) with successive roots at ''z''<sub>1</sub> and ''z''<sub>2</sub> and let ''v'' be a non-trivial solution of (2). Then one of the following properties holds.
			*There exists an ''x'' in [''z''<sub>1</sub>, ''z''<sub>2</sub>] such that ''v''(''x'') = 0; or
			*there exists a λ in '''R''' such that ''v''(''x'') = λ ''u''(''x'').

			NOTE: The first part of the conclusion is due to Sturm (1836).<ref>C. Sturm, Mémoire sur les équations différentielles linéaires du second ordre, J. Math. Pures Appl. 1 (1836), 106–186</ref> The second (alternative) part of this theorem is due to Picone (1910)<ref>M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine, Ann. Scuola Norm. Pisa 11 (1909), 1–141.</ref><ref>{{cite doi\|10.1007/3-7643-7359-8_1}}</ref> whose simple proof was given using his now famous [[Picone identity]]. In the special case where both equations are identical one obtains the [[Sturm separation theorem]]. For an extension of this important theorem to a comparison theorem involving three or more real second order equations see the [[Hartman–Mingarelli comparison theorem]] where a simple proof was given using the [[Mingarelli identity]].

			== References ==

			{{Reflist}}
			*Diaz, J. B.; [[Joyce McLaughlin\|McLaughlin, Joyce R.]] ''Sturm comparison theorems for ordinary and partial differential equations''. Bull. Amer. Math. Soc. 75 1969 335–339 [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183530292 pdf]
			* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 79, Krieger, Huntington ISBN 0-88275-368-1 .
			*{{cite book\| last = Teschl\| given = G.\|authorlink=Gerald Teschl\| title = Ordinary Differential Equations and Dynamical Systems\| publisher=[[American Mathematical Society]]\| place = [[Providence, Rhode Island\|Providence]]\| year = 2012\| isbn= 978-0-8218-8328-0\| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

			{{DEFAULTSORT:Sturm-Picone comparison theorem}}
			[[Category:Ordinary differential equations]]
			[[Category:Theorems in analysis]]

88.77.151.251 at 15:51, 7 August 2012

2012-08-07T15:51:51Z

New page

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