Ducci sequence - Revision history

82.1.125.96 at 20:12, 22 April 2014

2014-04-22T20:12:53Z

← Older revision		Revision as of 22:12, 22 April 2014
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	~~Friends call him Royal~~. ~~Delaware is our beginning location~~. Interviewing is ~~what~~ I ~~do for~~ a residing ~~but~~ I ~~strategy on changing it~~. The preferred hobby for my children and me is ~~dancing~~ and now I'm attempting to ~~earn cash~~ with it.<br><br>~~Feel free to surf to my blog post:~~ [http://~~Miketoerner~~.com/~~ActivityFeed/MyProfile~~/tabid/60/userId/~~35329~~/Default.aspx ~~car warranty~~]		Hello friend. extended auto [http://www.hk-301.net/clansphere/index.php?mod=users&action=view&id=3833 extended car warranty] auto warranty warranty Let me [http://Extended-car-Warranty-services-review.Toptenreviews.com/ introduce] myself. I am Ron but extended auto warranty I don't like when people use my full name. [http://www.nbcnews.com/id/23147777/ns/business-consumer_news/t/auto-warranty-firms-launch-sleazy-scam/ Interviewing] is how I make a residing and it's some thing I truly enjoy. For many years she's been living in Kansas. [http://ktva-online.com/index.php?mod=users&action=view&id=12078 car warranty] The preferred hobby for my children and me is taking part in crochet and now I'm attempting to make money with it.<br><br>My web page; [http://mwarrenassociates.com/UserProfile/tabid/253/userId/228462/Default.aspx mwarrenassociates.com]

en>りっき: /* Examples */ adopting the reference to Wikis guideline.

2014-02-18T10:56:22Z

Examples: adopting the reference to Wikis guideline.

← Older revision		Revision as of 12:56, 18 February 2014
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	~~In [[mathematics]], in the field of [[ordinary differential equation]]s, a nontrivial solution to an ordinary differential equation~~		Friends call him Royal. Delaware is our beginning location. Interviewing is what I do for a residing but I strategy on changing it. The preferred hobby for my children and me is dancing and now I'm attempting to earn cash with it.<br><br>Feel free to surf to my blog post: [http://Miketoerner.com/ActivityFeed/MyProfile/tabid/60/userId/35329/Default.aspx car warranty]

	~~:<math>F(x,y,y',\ \dots,\ y^{(n-1)})=y^{(n)} \quad x \in [0,+\infty)</math>~~

	is ~~called '''oscillating''' if it has an infinite number of [[root of~~ a ~~function\|root]]s, otherwise~~ it ~~is called '''non-oscillating'''~~. The ~~differential equation~~ is ~~called~~ '~~''oscillating''' if~~ it ~~has an oscillating solution.~~
	~~The number of roots carries also information on the [[Spectrum (functional analysis)\|spectrum]] of associated [[boundary value problem]]s~~.

	~~== Examples ==~~
	~~The differential equation~~

	:<~~math~~>~~y'' + y = 0\~~ <~~/math~~>

	~~is oscillating as sin(''x'') is a solution.~~

	~~== Connection with spectral theory ==~~
	Oscillation theory was initiated by [[Jacques Charles François Sturm]] in his investigations of [[Sturm–Liouville theory\|Sturm–Liouville problems]] from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional [[Schrödinger equation]] the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum.

	~~== Relative oscillation theory ==~~
	In 1996 [[Fritz Gesztesy\|Gesztesy]]–[[Barry Simon\|Simon]]–[[Gerald Teschl\|Teschl]] showed that the number of roots of the [[Wronski determinant]] of two eigenfunctions of a Sturm–Liouville problem gives the number of eigenvalues between the corresponding eigenvalues. It was later on generalized by Krüger–Teschl to the case of two eigenfunctions of two different Sturm–Liouville problems. The investigation of the number of roots of the Wronski determinant of two solutions is known as relative oscillation theory.

	~~== See also ==~~
	~~Classical results in oscillation theory are~~:

	* [[Kneser's theorem (differential equations)]]
	* [[Sturm–Picone comparison theorem]]
	* [[Sturm separation theorem]]

	~~==References==~~
	*[[Frederick_Valentine_Atkinson\|Atkinson, F.V.]] (1964). Discrete and Continuous Boundary Problems, Academic Press.
	*Gesztesy, F.; Simon, B.; Teschl, G. (1996). Zeros of the Wronskian and renormalized oscillation theory, Am. J. Math. 118, 571–594.
	*Kreith, K. (1973). Oscillation Theory, Lecture Notes in Mathematics 324, Springer.
	*Krüger, H; Teschl G. (2009). Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function, Commun. Math. Phys. 287, 613–640.
	*Sturm, J.C.F. (1836). Memoire sur les equations diferentielles lineaires du second ordre, J. Math. Pures Appl. 1, 106–186.
	*Swanson, C.A. (1968). Comparison and Oscillation Theory of Linear Differential Equations, Academic Press.
	*{{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~~/}}
	*Weidmann, J. (1987). Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258, Springer.

	~~{{DEFAULTSORT:Oscillation (Differential Equation)}}~~
	~~[[Category:Ordinary differential equations]~~]


	~~{{mathanalysis-stub}}~~

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3608894

2013-03-16T20:20:04Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3608894

← Older revision		Revision as of 22:20, 16 March 2013
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	~~Nice~~ to ~~satisfy you~~, ~~my name~~ is ~~Ling and I totally dig that name. Managing individuals~~ is ~~what I do in my working day occupation~~. ~~Delaware~~ is ~~our birth location~~. ~~Camping is some thing that I've done for many years~~.<br><br>~~Also visit my webpage~~: [http://www.~~nadelstich~~-~~mecky~~.~~de/index~~.~~php?mod=users&action=view&id=4380 extended car warranty~~]		In [[mathematics]], in the field of [[ordinary differential equation]]s, a nontrivial solution to an ordinary differential equation

			:<math>F(x,y,y',\ \dots,\ y^{(n-1)})=y^{(n)} \quad x \in [0,+\infty)</math>

			is called '''oscillating''' if it has an infinite number of [[root of a function\|root]]s, otherwise it is called '''non-oscillating'''. The differential equation is called '''oscillating''' if it has an oscillating solution.
			The number of roots carries also information on the [[Spectrum (functional analysis)\|spectrum]] of associated [[boundary value problem]]s.

			== Examples ==
			The differential equation

			:<math>y'' + y = 0\ </math>

			is oscillating as sin(''x'') is a solution.

			== Connection with spectral theory ==
			Oscillation theory was initiated by [[Jacques Charles François Sturm]] in his investigations of [[Sturm–Liouville theory\|Sturm–Liouville problems]] from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional [[Schrödinger equation]] the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum.

			== Relative oscillation theory ==
			In 1996 [[Fritz Gesztesy\|Gesztesy]]–[[Barry Simon\|Simon]]–[[Gerald Teschl\|Teschl]] showed that the number of roots of the [[Wronski determinant]] of two eigenfunctions of a Sturm–Liouville problem gives the number of eigenvalues between the corresponding eigenvalues. It was later on generalized by Krüger–Teschl to the case of two eigenfunctions of two different Sturm–Liouville problems. The investigation of the number of roots of the Wronski determinant of two solutions is known as relative oscillation theory.

			== See also ==
			Classical results in oscillation theory are:

			* [[Kneser's theorem (differential equations)]]
			* [[Sturm–Picone comparison theorem]]
			* [[Sturm separation theorem]]

			==References==
			*[[Frederick_Valentine_Atkinson\|Atkinson, F.V.]] (1964). Discrete and Continuous Boundary Problems, Academic Press.
			*Gesztesy, F.; Simon, B.; Teschl, G. (1996). Zeros of the Wronskian and renormalized oscillation theory, Am. J. Math. 118, 571–594.
			*Kreith, K. (1973). Oscillation Theory, Lecture Notes in Mathematics 324, Springer.
			*Krüger, H; Teschl G. (2009). Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function, Commun. Math. Phys. 287, 613–640.
			*Sturm, J.C.F. (1836). Memoire sur les equations diferentielles lineaires du second ordre, J. Math. Pures Appl. 1, 106–186.
			*Swanson, C.A. (1968). Comparison and Oscillation Theory of Linear Differential Equations, Academic Press.
			*{{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/}}
			*Weidmann, J. (1987). Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258, Springer.

			{{DEFAULTSORT:Oscillation (Differential Equation)}}
			[[Category:Ordinary differential equations]]


			{{mathanalysis-stub}}

en>Xqbot: robot Modifying: no:Ducci-sekvens

2011-06-15T11:56:50Z

robot Modifying: no:Ducci-sekvens

New page

Nice to satisfy you, my name is Ling and I totally dig that name. Managing individuals is what I do in my working day occupation. Delaware is our birth location. Camping is some thing that I've done for many years.<br><br>Also visit my webpage: [http://www.nadelstich-mecky.de/index.php?mod=users&action=view&id=4380 extended car warranty]