Effective medium approximations - Revision history

190.188.236.80: /* Formula */

2014-12-06T19:47:17Z

Formula

← Older revision		Revision as of 21:47, 6 December 2014
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	~~The filled-in Julia set <math>\ K(f) </math> of a polynomial <math>\ f </math> is :~~		Hi there. Allow me start by introducing the author, her name is Sophia. Office supervising is where her primary earnings comes from but she's currently applied for another one. Playing badminton is a factor that he is totally addicted to. Ohio is exactly where her house is.<br><br>my web-site :: good psychic ([http://www.khuplaza.com/dent/14869889 simply click the up coming post])
	* a [[Julia set]] and its [[Interior (topology)\|interior]],
	* [[escaping set\|non-escaping set]]

	~~==Formal definition==~~

	The filled-in [[Julia set]] <math>\ K(f) </math> of a polynomial <math>\ f </math> is defined as the set of all points <math>z\,</math> of the dynamical plane that have [[Bounded sequence\|bounded]] [[Orbit (dynamics)\|orbit]] with respect to <math>\ f </math>

	~~<math> \ K(f) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \not\to \infty\ as\ k \to \infty \} </math>~~
	~~<br>~~
	~~where :~~

	~~<math>\mathbb{C}</math> is~~ the ~~[[Complex number\|set of complex numbers]]~~

	~~<math> \ f^{(k)} (z) </math> is the <math>\ k</math> -fold [[Function composition\|composition]] of <math>f \~~,~~</math> with itself = [[Iterated function\|iteration of function]] <math>f \,</math>~~

	~~==Relation to the Fatou set==~~
	~~The filled-in Julia set~~ is ~~the [[Complement (set theory)\|(absolute) complement]] of the [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]]~~. ~~<br />~~
	~~<math>K(f) = \mathbb{C} \setminus A_{f}(\infty)</math>~~

	~~The [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]]~~ is one ~~of the [[Classification of Fatou components\|components of the Fatou set]]~~.~~<br />~~
	~~<math>A_{f}(\infty) = F_\infty </math>~~

	~~In other words, the filled-in Julia set~~ is ~~the [[Complement (set theory)\|complement]] of the unbounded [[Classification of Fatou components\|Fatou component]]: <br />~~
	~~<math>K(f) = F_\infty^C.</math>~~

	~~==Relation between Julia, filled-in Julia set and attractive basin of infinity==~~
	~~{{Wikibooks\|Fractals }}~~

	~~The [[Julia set]]~~ is ~~the common [[Boundary (topology)\|boundary]] of the filled-in Julia set and the [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]] <br>~~
	~~<br />~~
	~~<math>J(f)\, = \partial K(f) =\partial A_{f}(\infty)</math><br />~~
	~~<br />~~
	~~where : <br />~~
	<math>A_{f}(\infty)</math> denotes the [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]] = exterior of filled-in Julia set = set of escaping points for <math>f</math><br />
	~~<br />~~
	~~<math>A_{f}(\infty) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \to \infty\ as\ k \~~to ~~\infty \}~~. ~~</math>~~

	If the filled-in Julia set has no [[Interior (topology)\|interior]] then the [[Julia set]] coincides with the filled-in Julia set. This happens when all the critical points of <math>f</math> are pre-periodic. Such critical points are often called [[Misiurewicz point]]s.

	~~==Spine ==~~

	~~The most studied polynomials are probably those of the form <math>f(z)=z^2 + c</math>, which are often denoted by <math>f_c</math>,~~ where ~~<math>c</math>~~ is ~~any complex number~~. ~~In this case, the spine~~ <~~math>S_c\,</math~~> ~~of the filled Julia set~~ <~~math>\ K \,</math> is defined as [[Arc (projective geometry)\|arc]] between <math>\beta\,</math>[[Periodic points of complex quadratic mappings\|-fixed point]] and <!-- its preimage --> <math~~>-~~\beta\,</math>,~~

	~~<math>S_c = \left [ - \beta , \beta \right ]\,</math>~~

	~~with such properties:~~
	*spine lies inside <math>\ K \,</math>.<ref>[http:~~//www.math.rochester.edu/u/faculty/doug/oldcourses/215s98/lecture10.html Douglas C. Ravenel~~ : ~~External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester]</ref> This makes sense when <math>K\,</math> is connected and full <ref>~~[http://www.~~emis.de/journals/EM/expmath/volumes/13/13.1/Milnor.pdf John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)]</ref>~~
	* spine is invariant under 180 degree rotation,
	* spine is a finite topological tree,
	*[[Complex quadratic polynomial\|Critical point]] <math> z_{cr} = 0 \,</math> always belongs to the spine.<ref>[http://arxiv.org/abs/math/9801148 Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case]</ref>
	*<math>\beta\,</math>[[Periodic points of complex quadratic mappings\|-fixed point]] is a landing point of [[external ray]] of angle zero <math>\mathcal{R}^K _0</math>,
	*<math>-\beta\,</math> is landing point of [[external ray]] <math>\mathcal{R}^K _{1/2}</math>.

	~~Algorithms for constructing the spine:~~
	*[[b:Fractals/Iterations in the complex plane/Julia set\|detailed version]] is described by A. Douady<ref>A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. ~~Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986.</ref>~~

	*Simplified version of algorithm:
	**connect <math>- \beta\,</~~math> and <math> \beta\,<~~/~~math> within <math>K\,</math> by an arc,~~
	**when <math>K\,</math> has empty interior then arc is unique,
	**otherwise take the shortest way that contains <math>0</math>.<ref>[http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521547666 K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257]~~</ref>~~

	~~Curve <math>R\,</math> :~~

	~~<math>R\ \overset{\underset{\mathrm{def}}{}}{=} \ R_{1/2}\ \cup\ S_c\ \cup \ R_0 \,</math>~~

	~~divides dynamical plane into two components.~~

	~~==Images==~~
	~~<gallery>~~
	~~Image:Time escape Julia set from coordinate (phi-2, 0~~)~~.jpg\|Filled Julia set for f<sub>c</sub>, c=φ−2=-0.38..., where φ means [[Golden ratio]]~~
	~~Image:Julia_IIM_1.jpg\| Filled Julia with no interior = Julia set. It is for c=i.~~
	~~Image:Filled.jpg\| Filled Julia set for c=-1+0.1*i. Here Julia set is the boundary of filled-in Julia set.~~
	~~Image:ColorDouadyRabbit1.jpg\|[[Douady rabbit]]~~
	~~</gallery>~~

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

	~~==References==~~
	~~# Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.~~
	~~# Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, [http://www2.mat.dtu.dk/publications/uk?id=122 MAT-Report no. 1996-42].~~

	~~{{DEFAULTSORT:Filled Julia Set}}~~
	~~[[Category:Fractals]]~~
	~~[[Category:Limit sets]]~~
	~~[[Category:Complex dynamics]]~~

en>Monkbot: /* Circular and spherical inclusions */Fix CS1 deprecated date parameter errors

2014-01-17T19:07:45Z

Circular and spherical inclusions: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 21:07, 17 January 2014
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	The ~~author~~ is ~~known~~ by the ~~name~~ of ~~Figures Lint~~. ~~Years ago we moved to North Dakota~~. ~~His wife doesn't like it~~ the ~~way he does but what he really likes performing~~ is ~~to do aerobics~~ and ~~he's been performing it for fairly~~ a ~~while~~. ~~Bookkeeping~~ is ~~her day job now.~~<br><br>~~Also visit my web page~~ ... [http://~~nationlinked~~.~~com~~/~~index~~.~~php~~?do=/~~profile~~-~~35688~~/~~info~~/ ~~std home test~~]		The filled-in Julia set <math>\ K(f) </math> of a polynomial <math>\ f </math> is :
			* a [[Julia set]] and its [[Interior (topology)\|interior]],
			* [[escaping set\|non-escaping set]]

			==Formal definition==

			The filled-in [[Julia set]] <math>\ K(f) </math> of a polynomial <math>\ f </math> is defined as the set of all points <math>z\,</math> of the dynamical plane that have [[Bounded sequence\|bounded]] [[Orbit (dynamics)\|orbit]] with respect to <math>\ f </math>

			<math> \ K(f) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \not\to \infty\ as\ k \to \infty \} </math>
			<br>
			where :

			<math>\mathbb{C}</math> is the [[Complex number\|set of complex numbers]]

			<math> \ f^{(k)} (z) </math> is the <math>\ k</math> -fold [[Function composition\|composition]] of <math>f \,</math> with itself = [[Iterated function\|iteration of function]] <math>f \,</math>

			==Relation to the Fatou set==
			The filled-in Julia set is the [[Complement (set theory)\|(absolute) complement]] of the [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]]. <br />
			<math>K(f) = \mathbb{C} \setminus A_{f}(\infty)</math>

			The [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]] is one of the [[Classification of Fatou components\|components of the Fatou set]].<br />
			<math>A_{f}(\infty) = F_\infty </math>

			In other words, the filled-in Julia set is the [[Complement (set theory)\|complement]] of the unbounded [[Classification of Fatou components\|Fatou component]]: <br />
			<math>K(f) = F_\infty^C.</math>

			==Relation between Julia, filled-in Julia set and attractive basin of infinity==
			{{Wikibooks\|Fractals }}

			The [[Julia set]] is the common [[Boundary (topology)\|boundary]] of the filled-in Julia set and the [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]] <br>
			<br />
			<math>J(f)\, = \partial K(f) =\partial A_{f}(\infty)</math><br />
			<br />
			where : <br />
			<math>A_{f}(\infty)</math> denotes the [[Basin of attraction\|attractive basin]] of [[Point at infinity\|infinity]] = exterior of filled-in Julia set = set of escaping points for <math>f</math><br />
			<br />
			<math>A_{f}(\infty) \ \overset{\underset{\mathrm{def}}{}}{=} \ \{ z \in \mathbb{C} : f^{(k)} (z) \to \infty\ as\ k \to \infty \}. </math>

			If the filled-in Julia set has no [[Interior (topology)\|interior]] then the [[Julia set]] coincides with the filled-in Julia set. This happens when all the critical points of <math>f</math> are pre-periodic. Such critical points are often called [[Misiurewicz point]]s.

			==Spine ==

			The most studied polynomials are probably those of the form <math>f(z)=z^2 + c</math>, which are often denoted by <math>f_c</math>, where <math>c</math> is any complex number. In this case, the spine <math>S_c\,</math> of the filled Julia set <math>\ K \,</math> is defined as [[Arc (projective geometry)\|arc]] between <math>\beta\,</math>[[Periodic points of complex quadratic mappings\|-fixed point]] and <!-- its preimage --> <math>-\beta\,</math>,

			<math>S_c = \left [ - \beta , \beta \right ]\,</math>

			with such properties:
			*spine lies inside <math>\ K \,</math>.<ref>[http://www.math.rochester.edu/u/faculty/doug/oldcourses/215s98/lecture10.html Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester]</ref> This makes sense when <math>K\,</math> is connected and full <ref>[http://www.emis.de/journals/EM/expmath/volumes/13/13.1/Milnor.pdf John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)]</ref>
			* spine is invariant under 180 degree rotation,
			* spine is a finite topological tree,
			*[[Complex quadratic polynomial\|Critical point]] <math> z_{cr} = 0 \,</math> always belongs to the spine.<ref>[http://arxiv.org/abs/math/9801148 Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case]</ref>
			*<math>\beta\,</math>[[Periodic points of complex quadratic mappings\|-fixed point]] is a landing point of [[external ray]] of angle zero <math>\mathcal{R}^K _0</math>,
			*<math>-\beta\,</math> is landing point of [[external ray]] <math>\mathcal{R}^K _{1/2}</math>.

			Algorithms for constructing the spine:
			*[[b:Fractals/Iterations in the complex plane/Julia set\|detailed version]] is described by A. Douady<ref>A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986.</ref>

			*Simplified version of algorithm:
			**connect <math>- \beta\,</math> and <math> \beta\,</math> within <math>K\,</math> by an arc,
			**when <math>K\,</math> has empty interior then arc is unique,
			**otherwise take the shortest way that contains <math>0</math>.<ref>[http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521547666 K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257]</ref>

			Curve <math>R\,</math> :

			<math>R\ \overset{\underset{\mathrm{def}}{}}{=} \ R_{1/2}\ \cup\ S_c\ \cup \ R_0 \,</math>

			divides dynamical plane into two components.

			==Images==
			<gallery>
			Image:Time escape Julia set from coordinate (phi-2, 0).jpg\|Filled Julia set for f<sub>c</sub>, c=φ−2=-0.38..., where φ means [[Golden ratio]]
			Image:Julia_IIM_1.jpg\| Filled Julia with no interior = Julia set. It is for c=i.
			Image:Filled.jpg\| Filled Julia set for c=-1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
			Image:ColorDouadyRabbit1.jpg\|[[Douady rabbit]]
			</gallery>

			==Notes==
			{{Reflist}}

			==References==
			# Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
			# Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, [http://www2.mat.dtu.dk/publications/uk?id=122 MAT-Report no. 1996-42].

			{{DEFAULTSORT:Filled Julia Set}}
			[[Category:Fractals]]
			[[Category:Limit sets]]
			[[Category:Complex dynamics]]

193.171.77.1 at 11:35, 9 August 2012

2012-08-09T11:35:00Z

New page

The author is known by the name of Figures Lint. Years ago we moved to North Dakota. His wife doesn't like it the way he does but what he really likes performing is to do aerobics and he's been performing it for fairly a while. Bookkeeping is her day job now.<br><br>Also visit my web page ... [http://nationlinked.com/index.php?do=/profile-35688/info/ std home test]