Kaczmarz method - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10458)

2014-09-13T08:06:38Z

WP:CHECKWIKI error fixes using AWB (10458)

← Older revision		Revision as of 10:06, 13 September 2014
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	~~Oscar is how he's called~~ and ~~he totally enjoys this name~~. ~~Years in the past we moved to North Dakota~~. ~~One of the things she loves most is to do aerobics~~ and ~~now she~~ is ~~attempting~~ to ~~make money~~ with it. ~~Hiring has been my occupation for some time but~~ I've ~~already utilized for another 1~~.<br><br>~~Feel free to visit~~ my ~~web blog -~~ over the counter std test ([http://~~facehack~~.~~ir/index~~.~~php?do=~~/~~profile-110~~/~~info/ Discover More~~])		Hello and welcome. My title is Numbers Wunder. The preferred pastime for my children and me is to play baseball but I haven't produced a dime with it. South Dakota is exactly where I've always been residing. For many years he's been working as a meter reader and it's something he truly appreciate.<br><br>Here is my webpage; over the counter std test [[http://www.youporntime.com/user/KBiaggini go now]]

64.134.66.166 at 02:00, 11 February 2014

2014-02-11T02:00:19Z

← Older revision		Revision as of 04:00, 11 February 2014
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	~~[[Image:PonceletPorism.gif\|thumb\|right\|Illustration of Poncelet's porism for ''n'' = 3, a triangle that~~ is ~~inscribed in one circle and circumscribes another.]]~~		Oscar is how he's called and he totally enjoys this name. Years in the past we moved to North Dakota. One of the things she loves most is to do aerobics and now she is attempting to make money with it. Hiring has been my occupation for some time but I've already utilized for another 1.<br><br>Feel free to visit my web blog - over the counter std test ([http://facehack.ir/index.php?do=/profile-110/info/ Discover More])

	~~In [[geometry]], '''Poncelet's porism''' (sometimes referred to as '''Poncelet~~'s ~~closure theorem'''), named after French engineer~~ and ~~mathematician [[Jean-Victor Poncelet]], states the following: Let ''C'' and ''D'' be two plane [[conic]]s~~. ~~If it is possible to find, for a given ''n'' > 2, one ''n''-sided [[polygon]] that is simultaneously inscribed~~ in ~~''C'' and circumscribed around ''D'' (i.e., a [[bicentric polygon]]), then it is possible to find infinitely many of them.~~

	~~Poncelet's [[porism]] can be understood in terms of an [[elliptic curve]].~~

	~~==Sketch of proof==~~
	View ''C'' and ''D'' as projective curves in '''P'''<sup>2</sup>. For simplicity, assume that ''C'' and ''D'' meet transversely. Then by [[Bézout's theorem]], ''C'' ∩ ''D'' consists of 4 (complex) points. For ''d'' in ''D'', let ''ℓ''<sub>''d''</sub> be the ~~tangent line~~ to ~~''D'' at ''d''~~. ~~Let ''X'' be the subvariety~~ of ~~''C'' × ''D'' consisting of (''c'',''d'') such that ''ℓ''<sub>''d''</sub> passes through ''c''. Given ''c'',~~ the ~~number of ''d'' with (''c'',''d'') ∈ ''X''~~ is ~~1 if ''c'' ∈ ''C'' ∩ ''D''~~ and ~~2 otherwise. Thus the projection ''X'' → ''C'' ≃ '''P'''<sup>1</sup> presents ''X'' as a degree 2 cover ramified above 4 points, so ''X''~~ is ~~an elliptic curve (once we fix a base point on ''X''). Let <math>\sigma</math> be the involution of ''X'' sending a general (''c'',''d'')~~ to ~~the other point (''c'',''d''′)~~ with ~~the same first coordinate~~. ~~Any involution of an elliptic curve with a fixed point, when expressed in the group law,~~ has ~~the form ''x'' → ''p'' − ''x''~~ for some ''p'', so <math>\sigma</math> has this form. Similarly, the projection ''X'' → ''D'' is a degree 2 morphism ramified over the contact points on ''D'' of the four lines tangent to both ''C'' and ''D'', and the corresponding involution <math>\tau</math> has the form ''x'' → ''q'' − ''x'' for ~~some ''q''~~. ~~Thus the composition <math>\tau \sigma~~<~~/math~~> ~~is a translation on ''X''. If a power of~~ <~~math~~>\tau \sigma</math> has a fixed point, that power must be the identity. Translated back into the language of ''C'' and ''D'', this means that if one point ''c'' ∈ ''C'' (equipped with a corresponding ''d'') gives rise to ~~an orbit that closes up (i.e., gives an ''n''~~-~~gon), then so does every point. The degenerate cases in which ''C'' and ''D'' are not transverse follow from a limit argument.~~

	~~== See also ==~~
	* [[Hartshorne ellipse]]
	* [[Steiner's porism]]
	* [[Tangent lines to circles]]

	~~==References==~~
	*[[Bos, H. J. M.]]; Kers, C.; Oort, F.; Raven, D. W. Poncelet's closure theorem. Expositiones Mathematicae '''5''' (~~1987), no. 4, 289–364.~~

	~~== External links ==~~
	* [http://~~sbseminar~~.~~wordpress.com~~/~~2007/07/16/poncelets-porism/ David Speyer on Poncelet's Porism]~~
	*D. ~~Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''~~
	* [http://www.borcherds.co.uk/PonceletsPorism.html Java applet] by Michael Borcherds showing the cases ''n'' =~~ 3, 4, 5, 6, 7, 8 (including the convex cases for ''n'' = 7, 8) made using [http:~~//~~www.geogebra.org/webstart/ GeoGebra].~~
	* [http://www.borcherds.co.uk/geogebra/PonceletsPorismEllipseParabolaOrder3.html Java applet] by Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using [http://www.geogebra.org/webstart/ GeoGebra].
	* [http://www.borcherds.co.uk/geogebra/PonceletsPorismEllipsesOrder3.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using [http://www.geogebra.org/webstart/ GeoGebra].
	* [http://www.borcherds.co.uk/geogebra/~~PonceletsPorismTwoConicsOrder5.html Java applet~~] ~~by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5~~) ~~made using [http://www.geogebra.org/webstart/ GeoGebra].~~
	* [http://www.borcherds.co.uk/geogebra/PonceletsPorismTwoConicsOrder6.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using [http://www.geogebra.org/webstart/ GeoGebra].
	* [http://poncelet.math.nthu.edu.tw/disk3/cabrijava/poncelet3-exterior2.html Java applet] showing the exterior case for n = 3 at National Tsing Hua University.
	* [http://mathworld.wolfram.com/PonceletsPorism.html Article on Poncelet's Porism] at Mathworld.

	~~[[Category:Conic sections]]~~
	~~[[Category:Elliptic curves]]~~

en>Megaloxantha at 00:00, 10 January 2014

2014-01-10T00:00:15Z

← Older revision		Revision as of 02:00, 10 January 2014
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	~~Greetings~~. ~~Allow me begin by telling you~~ the ~~writer~~'s ~~title~~ - ~~Phebe~~. ~~Hiring~~ is ~~her day occupation now but she~~'s ~~always wanted her own company~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my spouse reside~~. ~~To collect coins~~ is ~~what her family~~ and ~~her enjoy~~.<br><br>~~Look~~ into ~~my site std home test~~; [http://~~drupal~~.~~12thirty4~~.com/~~gmaps~~/~~node~~/~~4991 mouse click~~ the ~~next webpage~~],		[[Image:PonceletPorism.gif\|thumb\|right\|Illustration of Poncelet's porism for ''n'' = 3, a triangle that is inscribed in one circle and circumscribes another.]]

			In [[geometry]], '''Poncelet's porism''' (sometimes referred to as '''Poncelet's closure theorem'''), named after French engineer and mathematician [[Jean-Victor Poncelet]], states the following: Let ''C'' and ''D'' be two plane [[conic]]s. If it is possible to find, for a given ''n'' > 2, one ''n''-sided [[polygon]] that is simultaneously inscribed in ''C'' and circumscribed around ''D'' (i.e., a [[bicentric polygon]]), then it is possible to find infinitely many of them.

			Poncelet's [[porism]] can be understood in terms of an [[elliptic curve]].

			==Sketch of proof==
			View ''C'' and ''D'' as projective curves in '''P'''<sup>2</sup>. For simplicity, assume that ''C'' and ''D'' meet transversely. Then by [[Bézout's theorem]], ''C'' ∩ ''D'' consists of 4 (complex) points. For ''d'' in ''D'', let ''ℓ''<sub>''d''</sub> be the tangent line to ''D'' at ''d''. Let ''X'' be the subvariety of ''C'' × ''D'' consisting of (''c'',''d'') such that ''ℓ''<sub>''d''</sub> passes through ''c''. Given ''c'', the number of ''d'' with (''c'',''d'') ∈ ''X'' is 1 if ''c'' ∈ ''C'' ∩ ''D'' and 2 otherwise. Thus the projection ''X'' → ''C'' ≃ '''P'''<sup>1</sup> presents ''X'' as a degree 2 cover ramified above 4 points, so ''X'' is an elliptic curve (once we fix a base point on ''X''). Let <math>\sigma</math> be the involution of ''X'' sending a general (''c'',''d'') to the other point (''c'',''d''′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form ''x'' → ''p'' − ''x'' for some ''p'', so <math>\sigma</math> has this form. Similarly, the projection ''X'' → ''D'' is a degree 2 morphism ramified over the contact points on ''D'' of the four lines tangent to both ''C'' and ''D'', and the corresponding involution <math>\tau</math> has the form ''x'' → ''q'' − ''x'' for some ''q''. Thus the composition <math>\tau \sigma</math> is a translation on ''X''. If a power of <math>\tau \sigma</math> has a fixed point, that power must be the identity. Translated back into the language of ''C'' and ''D'', this means that if one point ''c'' ∈ ''C'' (equipped with a corresponding ''d'') gives rise to an orbit that closes up (i.e., gives an ''n''-gon), then so does every point. The degenerate cases in which ''C'' and ''D'' are not transverse follow from a limit argument.

			== See also ==
			* [[Hartshorne ellipse]]
			* [[Steiner's porism]]
			* [[Tangent lines to circles]]

			==References==
			*[[Bos, H. J. M.]]; Kers, C.; Oort, F.; Raven, D. W. Poncelet's closure theorem. Expositiones Mathematicae '''5''' (1987), no. 4, 289–364.

			== External links ==
			* [http://sbseminar.wordpress.com/2007/07/16/poncelets-porism/ David Speyer on Poncelet's Porism]
			*D. Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''
			* [http://www.borcherds.co.uk/PonceletsPorism.html Java applet] by Michael Borcherds showing the cases ''n'' = 3, 4, 5, 6, 7, 8 (including the convex cases for ''n'' = 7, 8) made using [http://www.geogebra.org/webstart/ GeoGebra].
			* [http://www.borcherds.co.uk/geogebra/PonceletsPorismEllipseParabolaOrder3.html Java applet] by Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using [http://www.geogebra.org/webstart/ GeoGebra].
			* [http://www.borcherds.co.uk/geogebra/PonceletsPorismEllipsesOrder3.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using [http://www.geogebra.org/webstart/ GeoGebra].
			* [http://www.borcherds.co.uk/geogebra/PonceletsPorismTwoConicsOrder5.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5) made using [http://www.geogebra.org/webstart/ GeoGebra].
			* [http://www.borcherds.co.uk/geogebra/PonceletsPorismTwoConicsOrder6.html Java applet] by Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using [http://www.geogebra.org/webstart/ GeoGebra].
			* [http://poncelet.math.nthu.edu.tw/disk3/cabrijava/poncelet3-exterior2.html Java applet] showing the exterior case for n = 3 at National Tsing Hua University.
			* [http://mathworld.wolfram.com/PonceletsPorism.html Article on Poncelet's Porism] at Mathworld.

			[[Category:Conic sections]]
			[[Category:Elliptic curves]]

en>Colonies Chris: minor fixes

2012-05-14T15:19:31Z

minor fixes

New page

Greetings. Allow me begin by telling you the writer's title - Phebe. Hiring is her day occupation now but she's always wanted her own company. South Dakota is exactly where me and my spouse reside. To collect coins is what her family and her enjoy.<br><br>Look into my site std home test; [http://drupal.12thirty4.com/gmaps/node/4991 mouse click the next webpage],