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en>Yobot: WP:CHECKWIKI error fixes using AWB (10479)

2014-10-08T07:38:39Z

WP:CHECKWIKI error fixes using AWB (10479)

← Older revision		Revision as of 09:38, 8 October 2014
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	~~{{more footnotes\|date=December 2010}}~~		Wilber Berryhill is the title his parents gave him and he totally digs that title. Distributing production is where her main income comes from. Some time in the past she selected to reside in Alaska and her mothers and fathers reside close by. One of the things she cheap [http://kard.dk/?p=24252 psychic phone] readings - [http://checkmates.co.za/index.php?do=/profile-56347/info/ pop over to this website] - enjoys most is canoeing and she's been performing it for quite a whilst.<br><br>My blog post: [http://Fashionlinked.com/index.php?do=/profile-13453/info/ accurate psychic readings]
	~~[[Image:Cyc7~~.~~png\|thumb\| <math>\|z^6+z^5+z^4+z^3+ </math>~~
	~~<math> z^2+z+1\|=1</math>]]~~
	~~In mathematics, a '''polynomial lemniscate''' or ''polynomial level curve''~~ is ~~a [[algebraic curve\|plane algebraic curve]] of degree 2n, constructed~~ from ~~a polynomial ''p'' with complex coefficients of degree ''n''~~.

	~~For any such polynomial ''p'' and positive real number ''c'', we may define a set of complex numbers by <math>\|p(z)\| = c.</math> This set of numbers may be equated to points~~ in the ~~real Cartesian plane, leading~~ to ~~an algebraic curve ''ƒ''(''x'', ''y'') = ''c''<sup>2</sup> of degree 2''n'', which results from expanding out <math>p(z) \bar p(\bar z)</math>~~ in ~~terms of ''z'' = ''x'' + ''iy''~~.

	~~When ''p'' is a polynomial~~ of ~~degree 1 then~~ the ~~resulting curve is simply a circle whose center is the zero of ''p''. When ''p'' is a polynomial of degree 2 then the curve is a~~ [~~[Cassini oval]].~~

	~~== Erdős lemniscate ==~~

	~~[[Image~~:~~Erdos5~~.~~png\|thumb\|Erdős lemniscate of degree ten and genus six]]~~
	A conjecture of [[Paul Erdős\|Erdős]] which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ''&fnof;''(''x'', ''y'') = 1 of degree 2''n'' when ''p~~'' is [[Polynomial#Classifications\|monic]], which Erdős conjectured was attained when ''p''(''z'') ~~=~~ z<sup>''n''</sup> − 1.~~
	~~This is still not proved but Fryntov and [[Fedor Nazarov\|Nazarov~~]~~] proved that ''p'' gives a~~
	~~local maximum.<ref>~~
	~~{{cite journal\|~~
	~~first1=A\|~~
	~~last1=Fryntov\|~~
	~~first2=F\|~~
	~~last2=Nazarov\|~~
	~~title=New estimates for the length of the Erdos~~-~~Herzog-Piranian lemniscate\|~~
	~~year=2008\|~~
	~~journal=Linear and Complex Analysis\|~~
	~~volume=226\|~~
	~~pages=49–60\|~~
	~~url=~~http://~~arxiv~~.~~org~~/~~abs/0808~~.~~0717}}<~~/~~ref> In the case when ''n'' = 2, the Erdős lemniscate is the [[Lemniscate of Bernoulli]]~~

	~~:<math>(x^2+y^2)^2=2(x^2~~-~~y^2)\,<~~/~~math>~~

	~~and it has been proven that~~ this ~~is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary ''n''~~-~~fold points, one of which~~ is ~~at the origin,~~ and ~~a [[geometric genus\|genus]] of (~~'~~'n'' − 1)(''n'' − 2)/2. By [[inversive geometry\|inverting]] the Erdős lemniscate in the unit circle, one obtains~~ a ~~nonsingular curve of degree ''n''~~.

	~~== Generic polynomial lemniscate ==~~

	~~In general, a polynomial lemniscate will not touch at the origin, and will have only two ordinary ''n''-fold singularities, and hence a genus of (''n'' − 1)~~<~~sup~~>2<~~/sup~~>~~. As a real curve, it can have a number of disconnected components. Hence, it will not look like a lemniscate, making the name something of a misnomer.~~
	~~[[Image~~:~~Mandelcurve2.png\|thumb\|Mandelbrot curve M<sub>2</sub> of degree eight and genus nine]]~~

	~~An interesting example of such polynomial lemniscates are the Mandelbrot curves.~~
	If we set ''p''<sub>0</sup> = ''z'', and ''p''<sub>''n''</sub> = ''p''<sub>''n''−1</sub><sup>2</sup> + ''z'', then the corresponding polynomial lemniscates M<sub>n</sub> defined by \|''p''<sub>''n''</sub>(''z'')\| = 1 converge to the boundary of the [[~~Mandelbrot set]].~~
	The Mandelbrot curves are of degree 2<sup>n+1</sup>.<ref>{{citation\|title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction\|publisher=Springer\|year=2007\|isbn=9781402054563\|page=492\|url=http://~~books.google~~.com/~~books~~?~~id=mbtCAAAAQBAJ&pg=PA492\|first1=Vladimir G.\|last1~~=~~Ivancevic\|first2=Tijana T.\|last2=Ivancevic}}.<~~/~~ref>~~

	~~== Notes ==~~
	~~{{Reflist}}~~

	~~==References==~~
	*[[Alexandre Eremenko]] and [[Walter Hayman]], ''On the length of lemniscates'', Michigan Math. J., (1999), '''46''', no. 2, 409–415 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1030132418]
	*O. S. Kusnetzova and V. G. Tkachev, ''Length functions of lemniscates'', Manuscripta Math., (2003), '''112''', 519–538 [http://~~arxiv.org/abs/math.CV/0306327]~~
	*[http://www.mathcurve.com/courbes2d/cassinienne/cassinienne.shtml "Cassinian curve" at Encyclopédie des Formes Mathématiques Remarquables]

	~~{{DEFAULTSORT:Polynomial Lemniscate}}~~
	~~[[Category:Curves]]~~
	~~[[Category:Algebraic curves]~~]

en>AnomieBOT: Dating maintenance tags: {{Citation needed}}

2013-12-19T17:06:25Z

Dating maintenance tags: {{Citation needed}}

← Older revision		Revision as of 19:06, 19 December 2013
Line 1:		Line 1:
	~~She~~ is ~~known~~ by [http://~~Www~~.~~Animecontent~~.~~com~~/~~blog~~/~~349119 over~~ the ~~counter std test~~] name of ~~Myrtle Shryock~~. ~~For years I~~'~~ve been operating as a payroll clerk~~. ~~Her family members lives in Minnesota~~. ~~To gather cash is what her family members~~ and ~~her enjoy~~.		{{more footnotes\|date=December 2010}}
			[[Image:Cyc7.png\|thumb\| <math>\|z^6+z^5+z^4+z^3+ </math>
			<math> z^2+z+1\|=1</math>]]
			In mathematics, a '''polynomial lemniscate''' or ''polynomial level curve'' is a [[algebraic curve\|plane algebraic curve]] of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''.

			For any such polynomial ''p'' and positive real number ''c'', we may define a set of complex numbers by <math>\|p(z)\| = c.</math> This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ''ƒ''(''x'', ''y'') = ''c''<sup>2</sup> of degree 2''n'', which results from expanding out <math>p(z) \bar p(\bar z)</math> in terms of ''z'' = ''x'' + ''iy''.

			When ''p'' is a polynomial of degree 1 then the resulting curve is simply a circle whose center is the zero of ''p''. When ''p'' is a polynomial of degree 2 then the curve is a [[Cassini oval]].

			== Erdős lemniscate ==

			[[Image:Erdos5.png\|thumb\|Erdős lemniscate of degree ten and genus six]]
			A conjecture of [[Paul Erdős\|Erdős]] which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ''&fnof;''(''x'', ''y'') = 1 of degree 2''n'' when ''p'' is [[Polynomial#Classifications\|monic]], which Erdős conjectured was attained when ''p''(''z'') = z<sup>''n''</sup> − 1.
			This is still not proved but Fryntov and [[Fedor Nazarov\|Nazarov]] proved that ''p'' gives a
			local maximum.<ref>
			{{cite journal\|
			first1=A\|
			last1=Fryntov\|
			first2=F\|
			last2=Nazarov\|
			title=New estimates for the length of the Erdos-Herzog-Piranian lemniscate\|
			year=2008\|
			journal=Linear and Complex Analysis\|
			volume=226\|
			pages=49–60\|
			url=http://arxiv.org/abs/0808.0717}}</ref> In the case when ''n'' = 2, the Erdős lemniscate is the [[Lemniscate of Bernoulli]]

			:<math>(x^2+y^2)^2=2(x^2-y^2)\,</math>

			and it has been proven that this is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary ''n''-fold points, one of which is at the origin, and a [[geometric genus\|genus]] of (''n'' − 1)(''n'' − 2)/2. By [[inversive geometry\|inverting]] the Erdős lemniscate in the unit circle, one obtains a nonsingular curve of degree ''n''.

			== Generic polynomial lemniscate ==

			In general, a polynomial lemniscate will not touch at the origin, and will have only two ordinary ''n''-fold singularities, and hence a genus of (''n'' − 1)<sup>2</sup>. As a real curve, it can have a number of disconnected components. Hence, it will not look like a lemniscate, making the name something of a misnomer.
			[[Image:Mandelcurve2.png\|thumb\|Mandelbrot curve M<sub>2</sub> of degree eight and genus nine]]

			An interesting example of such polynomial lemniscates are the Mandelbrot curves.
			If we set ''p''<sub>0</sup> = ''z'', and ''p''<sub>''n''</sub> = ''p''<sub>''n''−1</sub><sup>2</sup> + ''z'', then the corresponding polynomial lemniscates M<sub>n</sub> defined by \|''p''<sub>''n''</sub>(''z'')\| = 1 converge to the boundary of the [[Mandelbrot set]].
			The Mandelbrot curves are of degree 2<sup>n+1</sup>.<ref>{{citation\|title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction\|publisher=Springer\|year=2007\|isbn=9781402054563\|page=492\|url=http://books.google.com/books?id=mbtCAAAAQBAJ&pg=PA492\|first1=Vladimir G.\|last1=Ivancevic\|first2=Tijana T.\|last2=Ivancevic}}.</ref>

			== Notes ==
			{{Reflist}}

			==References==
			*[[Alexandre Eremenko]] and [[Walter Hayman]], ''On the length of lemniscates'', Michigan Math. J., (1999), '''46''', no. 2, 409–415 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1030132418]
			*O. S. Kusnetzova and V. G. Tkachev, ''Length functions of lemniscates'', Manuscripta Math., (2003), '''112''', 519–538 [http://arxiv.org/abs/math.CV/0306327]
			*[http://www.mathcurve.com/courbes2d/cassinienne/cassinienne.shtml "Cassinian curve" at Encyclopédie des Formes Mathématiques Remarquables]

			{{DEFAULTSORT:Polynomial Lemniscate}}
			[[Category:Curves]]
			[[Category:Algebraic curves]]

99.236.216.63 at 05:02, 8 March 2012

2012-03-08T05:02:38Z

New page

She is known by [http://Www.Animecontent.com/blog/349119 over the counter std test] name of Myrtle Shryock. For years I've been operating as a payroll clerk. Her family members lives in Minnesota. To gather cash is what her family members and her enjoy.