Rho Andromedae - Revision history

en>Rjwilmsi: /* References */Journal cites (minus instead of endash in pages), using AWB (10484)

2014-11-10T16:37:06Z

References: Journal cites (minus instead of endash in pages), using AWB (10484)

← Older revision		Revision as of 18:37, 10 November 2014
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	~~In [[mathematics]]~~, ~~more precisely in [[complex analysis]], the~~ '~~''holomorphically convex hull''' of~~ a ~~given [[compact set]] in the ''n''-[[dimension]]al [[complex number\|complex]] space '''C'''<sup>''n''</sup> is defined as follows.~~		Hi there. Allow me begin by introducing the writer, her title is Sophia Boon but she by no means truly liked that name. It's not a typical factor but what she likes doing psychic love readings ([http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 galab-work.cs.pusan.ac.kr]) is to play domino but she doesn't have the time recently. He is an info officer. Alaska is the only location I've been residing in but now I'm [http://kpupf.com/xe/talk/735373 psychic chat online] contemplating other choices.<br><br>Here is my web page - psychic love readings; [http://www.indosfriends.com/profile-253/info/ sneak a peek at this web-site],

	~~Let~~ ~~<math>G \subset {\mathbb{C}}^n</math> be a domain~~ (an [~~[open set\|open]] and [[connected set\|connected]] [[Set (mathematics)\|set]]), or alternatively for a more general definition, let <math>G<~~/~~math> be an <math>n<~~/~~math> dimensional [[complex analytic manifold]]~~. ~~Further let <math>{\mathcal{O}}(G)</math> stand for the set of [[holomorphic function]]s on <math>G~~.~~</math> For a compact set <math>K \subset G<~~/~~math>, the '''holomorphically convex hull''' of <math>K<~~/~~math> is~~

	~~:<math> \hat{K}_G :~~= ~~\{ z \in G \big\| \left\| f(z~~) ~~\right\| \leq \sup_{w \in K} \left\| f(w) \right\| \mbox{ for all } f \in {\mathcal{O}}(G) \} .</math>~~

	~~(One obtains a narrower concept of~~ '~~''polynomially convex hull''' by requiring in~~ the ~~above definition that ''f'' be a [[polynomial]]~~.)

	~~The domain <math>G</math>~~ is ~~called~~ ''~~'holomorphically convex''' if for every~~ ~~<math>K \subset G<~~/~~math> compact in <math>G<~~/~~math>, <math>\hat{K}_G<~~/~~math> is also compact in~~ <~~math~~>G<~~/math~~>~~. Sometimes this~~ is ~~just abbreviated as ''holomorph~~-~~convex''.~~

	~~When <math>n=1<~~/~~math>, any domain <math>G<~~/~~math> is holomorphically convex since then <math>\hat{K}_G<~~/~~math> is the union of <math>K<~~/~~math> with the relatively compact components of <math>G \setminus K \subset G<~~/~~math>. Also note that being holomorphically convex is the same as being~~ a ~~[[domain of holomorphy]] (The Cartan–Thullen theorem). These concepts are more important in the case ''n'' > 1 of [[several complex variables~~]].

	~~==See also==~~
	* [[Stein manifold]]
	* [[Pseudoconvexity]]

	~~== References ==~~
	* [[Lars Hörmander]]. ''An Introduction to Complex Analysis in Several Variables'', ~~North-Holland Publishing Company, New York, New York, 1973.~~
	* Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.

	~~{{PlanetMath attribution\|id=6798\|title=Holomorphically convex}}~~

	~~[[Category:Several complex variables]]~~

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-29T08:57:13Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 10:57, 29 January 2014
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	~~I would like to introduce myself to you~~, ~~I am Andrew~~ and ~~my spouse doesn~~'~~t like it at~~ all. ~~Invoicing is what I do~~. ~~What me and my family members love~~ is ~~bungee jumping but I~~'~~ve been taking on new issues recently. Her family members lives~~ in ~~Ohio.~~<br><br>~~Review my web site real psychic readings ([http:~~//~~www~~.~~zavodpm~~.ru/~~blogs~~/~~glennmusserrvji~~/~~14565~~-~~great-hobby-advice-assist-allow-you-get-going zavodpm~~.ru])		In [[mathematics]], more precisely in [[complex analysis]], the '''holomorphically convex hull''' of a given [[compact set]] in the ''n''-[[dimension]]al [[complex number\|complex]] space '''C'''<sup>''n''</sup> is defined as follows.

			Let <math>G \subset {\mathbb{C}}^n</math> be a domain (an [[open set\|open]] and [[connected set\|connected]] [[Set (mathematics)\|set]]), or alternatively for a more general definition, let <math>G</math> be an <math>n</math> dimensional [[complex analytic manifold]]. Further let <math>{\mathcal{O}}(G)</math> stand for the set of [[holomorphic function]]s on <math>G.</math> For a compact set <math>K \subset G</math>, the '''holomorphically convex hull''' of <math>K</math> is

			:<math> \hat{K}_G := \{ z \in G \big\| \left\| f(z) \right\| \leq \sup_{w \in K} \left\| f(w) \right\| \mbox{ for all } f \in {\mathcal{O}}(G) \} .</math>

			(One obtains a narrower concept of '''polynomially convex hull''' by requiring in the above definition that ''f'' be a [[polynomial]].)

			The domain <math>G</math> is called '''holomorphically convex''' if for every <math>K \subset G</math> compact in <math>G</math>, <math>\hat{K}_G</math> is also compact in <math>G</math>. Sometimes this is just abbreviated as ''holomorph-convex''.

			When <math>n=1</math>, any domain <math>G</math> is holomorphically convex since then <math>\hat{K}_G</math> is the union of <math>K</math> with the relatively compact components of <math>G \setminus K \subset G</math>. Also note that being holomorphically convex is the same as being a [[domain of holomorphy]] (The Cartan–Thullen theorem). These concepts are more important in the case ''n'' > 1 of [[several complex variables]].

			==See also==
			* [[Stein manifold]]
			* [[Pseudoconvexity]]

			== References ==
			* [[Lars Hörmander]]. ''An Introduction to Complex Analysis in Several Variables'', North-Holland Publishing Company, New York, New York, 1973.
			* Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.

			{{PlanetMath attribution\|id=6798\|title=Holomorphically convex}}

			[[Category:Several complex variables]]

en>Ilvon at 07:26, 27 August 2012

2012-08-27T07:26:04Z

New page

I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Invoicing is what I do. What me and my family members love is bungee jumping but I've been taking on new issues recently. Her family members lives in Ohio.<br><br>Review my web site real psychic readings ([http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going zavodpm.ru])