Spatial descriptive statistics - Revision history

en>Rjwilmsi: /* Ripley's K and L functions */Journal cites, Added 1 doi to a journal cite using AWB (10365)

2014-08-16T14:56:40Z

Ripley's K and L functions: Journal cites, Added 1 doi to a journal cite using AWB (10365)

← Older revision		Revision as of 16:56, 16 August 2014
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	~~[[Image:Biholomorphism illustration.svg\|right\|thumb\|The complex [[exponential function]] mapping biholomorphically a rectangle~~ to ~~a quarter-[[annulus (mathematics)\|annulus]].]]~~		Nice to satisfy you, my title is Numbers Held though I don't really like becoming called like that. Bookkeeping is what I do. It's not a typical factor but what she likes performing is base jumping and now she is attempting to make cash with it. Years ago we moved to North Dakota.<br><br>my web blog :: [http://Www.gaysphere.net/blog/167593 at home std testing]
	~~In the [[mathematics\|mathematical theory]] of functions of [[complex analysis\|one]] or [[several complex variables\|more complex variables]]~~, ~~and also in [[complex algebraic geometry]], a '''biholomorphism''' or '''biholomorphic function''' is a [[bijective]] [[holomorphic function]] whose [[inverse function\|inverse]]~~ is ~~also [[holomorphic function\| holomorphic]].~~

	Formally, a ''biholomorphic function'' is a function <math>\phi</math> defined on an [[open subset]] ''U'' of the <math>n</math>-dimensional complex space '''C'''<sup>''n''</sup> with values in '''C'''<sup>''n'~~'</sup> which is [[holomorphic function\| holomorphic]] and [[injective function\|one-to-one]], such~~ that ~~its [[image (mathematics)\|image]] is an open set <math>V</math> in '''C'''<sup>''n''</sup> and the inverse <math>\phi^{-1}:V\to U</math>~~ is ~~also [[holomorphic function\| holomorphic]]~~. ~~More generally,~~ '~~'U'' and ''V'' can be [[complex manifold]]~~s~~. By applying the chain rule, one may prove that it~~ is ~~enough for <math>\phi</math> to be holomorphic~~ and ~~one-~~to~~-one in order for~~ it ~~to be biholomorphic onto its image~~.

	~~If there exists a biholomorphism <math>\phi \colon U \~~to ~~V</math>, we say that ''U'' and ''V'' are '''biholomorphically equivalent''' or that they are '''biholomorphic'''~~.

	If <~~math~~>~~n=1,~~<~~/math~~> ~~every~~ [[simply connected]] open set other than the whole complex plane is biholomorphic to the [[unit disc]] (this is the [[Riemann mapping theorem]]). The situation is very different in higher dimensions. For example, open [[unit ball]]s and open unit [[polydisc]]s are not biholomorphically equivalent for <math>n>1.</~~math> In fact, there does not exist even a [[Proper map\|proper]] holomorphic function from one to the other.~~

	~~In the case of the complex plane '''C''', some authors (e.g. Freitag 2009, Definition IV~~.~~4.1) define [[conformal map]] as a synonym for ''biholomorphism''~~. ~~The usual conditions for a function <math>\phi<~~/~~math> to be conformal on '''C''' - namely, that <math>\phi<~~/~~math> is holomorphic and that its derivative is nowhere zero - are equivalent to biholomorphism.~~

	~~==References==~~
	* {{cite book \| author=Steven G. Krantz \| title=Function Theory of Several Complex Variables \| publisher=American Mathematical Society \| year=2002 \| isbn= 0-8218-2724-3}}
	* {{cite book \| author=John P. D'Angelo \| title=Several Complex Variables and the Geometry of Real Hypersurfaces \| publisher=CRC Press \| year=1993 \| isbn= 0-8493-8272-6}}
	* {{cite book \| author=Eberhard Freitag and Rolf Busam \| title=Complex Analysis \| publisher=Springer-Verlag \| year=2009 \| isbn= 978-3-540-93982-5}}

	~~{{PlanetMath attribution\|id=6032\|title=biholomorphically equivalent}}~~

	~~[[Category:Several complex variables]]~~
	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Complex manifolds]]~~
	~~[[Category:Functions and mappings]~~]

85.180.109.254: added to math, added reference Ripley 1976

2014-02-02T18:20:30Z

added to math, added reference Ripley 1976

← Older revision		Revision as of 20:20, 2 February 2014
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	The ~~name~~ of the ~~writer~~ is ~~Numbers but it~~'s ~~not~~ the ~~most masucline title out~~ there. ~~South Dakota~~ is ~~exactly where I've always been residing. Playing baseball~~ is the ~~hobby he will by no means quit performing~~. ~~Hiring~~ is ~~my occupation~~.<br><br>~~Also visit my blog:~~ [~~http:~~//~~facehack~~.~~ir/index~~.~~php?do~~=~~/blog/20/what~~-~~you~~-~~have~~-to-do-~~facing~~-~~candida/ facehack.ir~~]		[[Image:Biholomorphism illustration.svg\|right\|thumb\|The complex [[exponential function]] mapping biholomorphically a rectangle to a quarter-[[annulus (mathematics)\|annulus]].]]
			In the [[mathematics\|mathematical theory]] of functions of [[complex analysis\|one]] or [[several complex variables\|more complex variables]], and also in [[complex algebraic geometry]], a '''biholomorphism''' or '''biholomorphic function''' is a [[bijective]] [[holomorphic function]] whose [[inverse function\|inverse]] is also [[holomorphic function\| holomorphic]].

			Formally, a ''biholomorphic function'' is a function <math>\phi</math> defined on an [[open subset]] ''U'' of the <math>n</math>-dimensional complex space '''C'''<sup>''n''</sup> with values in '''C'''<sup>''n''</sup> which is [[holomorphic function\| holomorphic]] and [[injective function\|one-to-one]], such that its [[image (mathematics)\|image]] is an open set <math>V</math> in '''C'''<sup>''n''</sup> and the inverse <math>\phi^{-1}:V\to U</math> is also [[holomorphic function\| holomorphic]]. More generally, ''U'' and ''V'' can be [[complex manifold]]s. By applying the chain rule, one may prove that it is enough for <math>\phi</math> to be holomorphic and one-to-one in order for it to be biholomorphic onto its image.

			If there exists a biholomorphism <math>\phi \colon U \to V</math>, we say that ''U'' and ''V'' are '''biholomorphically equivalent''' or that they are '''biholomorphic'''.

			If <math>n=1,</math> every [[simply connected]] open set other than the whole complex plane is biholomorphic to the [[unit disc]] (this is the [[Riemann mapping theorem]]). The situation is very different in higher dimensions. For example, open [[unit ball]]s and open unit [[polydisc]]s are not biholomorphically equivalent for <math>n>1.</math> In fact, there does not exist even a [[Proper map\|proper]] holomorphic function from one to the other.

			In the case of the complex plane '''C''', some authors (e.g. Freitag 2009, Definition IV.4.1) define [[conformal map]] as a synonym for ''biholomorphism''. The usual conditions for a function <math>\phi</math> to be conformal on '''C''' - namely, that <math>\phi</math> is holomorphic and that its derivative is nowhere zero - are equivalent to biholomorphism.

			==References==
			* {{cite book \| author=Steven G. Krantz \| title=Function Theory of Several Complex Variables \| publisher=American Mathematical Society \| year=2002 \| isbn= 0-8218-2724-3}}
			* {{cite book \| author=John P. D'Angelo \| title=Several Complex Variables and the Geometry of Real Hypersurfaces \| publisher=CRC Press \| year=1993 \| isbn= 0-8493-8272-6}}
			* {{cite book \| author=Eberhard Freitag and Rolf Busam \| title=Complex Analysis \| publisher=Springer-Verlag \| year=2009 \| isbn= 978-3-540-93982-5}}

			{{PlanetMath attribution\|id=6032\|title=biholomorphically equivalent}}

			[[Category:Several complex variables]]
			[[Category:Algebraic geometry]]
			[[Category:Complex manifolds]]
			[[Category:Functions and mappings]]

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2012-01-08T06:57:33Z

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The name of the writer is Numbers but it's not the most masucline title out there. South Dakota is exactly where I've always been residing. Playing baseball is the hobby he will by no means quit performing. Hiring is my occupation.<br><br>Also visit my blog: [http://facehack.ir/index.php?do=/blog/20/what-you-have-to-do-facing-candida/ facehack.ir]