Air permeability specific surface - Revision history

en>Daniele Pugliesi: removed Category:Particulates; added Category:Particle technology using HotCat

2014-05-01T15:02:31Z

removed Category:Particulates; added Category:Particle technology using HotCat

← Older revision		Revision as of 17:02, 1 May 2014
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	~~In [[complex analysis]], a field within [[mathematics]]~~, ~~'''Bloch's theorem''' gives a lower bound on the size of a disc in which an inverse~~ to ~~a [[holomorphic function]] exists. It is named after [[André Bloch (mathematician)\|André Bloch]].~~		Binder and Finisher King from Acton Vale, likes to spend time pyrotechnics, como [http://ganhedinheiro.comoganhardinheiro101.com ganhar dinheiro] na internet and swimming. Finds immense encouragement from life by touring locales like Historic Centre of Camagüey.

	~~==Statement==~~
	Let ''f'' be a [[holomorphic function]] in the [[unit disk]] \|''z''\| ≤ 1. Suppose that \|''f′''(0)\| = 1. Then there exists a disc of radius ''b'' and an analytic function φ in this disc, such that ''f''(φ(''z'')) = ''z'' for all ''z'' in this disc. Here ''b'' > 1/72 is an absolute constant.

	~~==Landau's theorem==~~
	~~If ''f'' is a holomorphic function in the unit disc with the property \|''f′''(0)\| = 1~~, ~~then the image of ''f'' contains a disc of radius ''l'', where ''l'' ≥ ''b'' is an absolute constant.~~

	~~This theorem is named after [[Edmund Landau]].~~

	~~==Valiron's theorem==~~
	~~Bloch's theorem was inspired by the following theorem of [~~[~~Georges Valiron]]~~:

	'''Theorem.''' If ''f'' is a non-constant entire function then there exist discs ''D'' of arbitrarily large radius and analytic functions φ in ''D'' such that ''f''(φ(''z'')) = ''z'' for ''z'' in ''D''.

	~~Bloch's theorem corresponds to Valiron's theorem via the so-called [[Bloch's Principle]].~~

	~~== Bloch's and Landau's constants ==~~
	~~The lower bound 1~~/72 in Bloch's theorem is not the best possible. The number ''B'' defined as the [[supremum]] of all ''b'' for which this theorem holds, is called the '''Bloch's constant'''. Bloch's theorem tells us ''B'' ≥ 1/~~72, but the exact value of ''B'' is still unknown~~.

	~~The similarly defined optimal constant ''L'' in Landau's theorem is called the '''Landau's constant'''. Its exact value is also unknown.~~

	~~The best known bounds for ''B'' at present are~~
	~~:<math>\frac{\sqrt{3}}{4}+2\times10^{-4}\leq B\leq \sqrt{\frac{\sqrt{3}-1}{2}} \cdot \frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})},</math>~~
	~~where Γ is the [[Gamma function]]~~. ~~The lower bound was proved by Chen and Gauthier, and the upper bound dates back to [[Lars Ahlfors\|Ahlfors]~~] and ~~Grunsky~~. ~~They also gave an upper bound for the Landau constant.~~

	~~In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values~~ of ~~''B'' and ''L''.~~

	~~== References ==~~
	* {{cite journal\|first=Lars Valerian\|last=Ahlfors\|authorlink=Lars Ahlfors\|last2=Grunsky\|first2=Helmut\|author2-link=Helmut Grunsky\|title=Über die Blochsche Konstante\|journal=[[Mathematische Zeitschrift]]\|year=1937\|volume=42\|number=1\|pages=671–673\|doi=10.1007/BF01160101}}
	* {{cite conference\|first=Albert II\|last=Baernstein\|coauthors=Vinson, Jade P.\|title=Local minimality results related to the Bloch and Landau constants\|booktitle=Quasiconformal mappings and analysis\|place=Ann Arbor\|publisher=Springer, New York\|year=1998\|pages=55–89}}
	* {{cite journal\|first=André\|last=Bloch\|title=Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation\|journal=Annales de la faculté des sciences de l'Université de Toulouse\|number=3\|volume=17\|year=1925\|pages=1–22\|issn=0240-2963}}
	* {{ cite journal \| first=Huaihui \| last=Chen \| coauthors=Gauthier, Paul M. \| title=On Bloch's constant\|journal=Journal d'Analyse Mathématique\|year=1996\|volume=69\|number=1\|pages=275–291\|doi=10.~~1007/BF02787110}}~~

	~~[[Category:Unsolved problems in mathematics]]~~
	~~[[Category:Theorems in complex analysis]]~~

en>Scray: Undid revision 563609600 by Gold APP Instruments (talk) rv promotional edit

2013-12-04T07:28:35Z

Undid revision 563609600 by Gold APP Instruments (talk) rv promotional edit

← Older revision		Revision as of 09:28, 4 December 2013
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	Hi there, ~~I am Felicidad Oquendo~~. ~~Interviewing~~ is ~~what I do for~~ a ~~residing but I plan on altering it~~. ~~Kansas~~ is ~~our beginning location~~ and ~~my parents live nearby~~. The ~~factor I adore most flower arranging and now I have time to take on new things~~.<br><br>~~My web page~~ - [~~http:~~/~~/vadim~~.~~amherstburg~~.ca/~~ActivityFeed/MyProfile/tabid/56/userId/15736/Default.aspx aftermarket car warranty~~]		In [[complex analysis]], a field within [[mathematics]], '''Bloch's theorem''' gives a lower bound on the size of a disc in which an inverse to a [[holomorphic function]] exists. It is named after [[André Bloch (mathematician)\|André Bloch]].

			==Statement==
			Let ''f'' be a [[holomorphic function]] in the [[unit disk]] \|''z''\| ≤ 1. Suppose that \|''f′''(0)\| = 1. Then there exists a disc of radius ''b'' and an analytic function φ in this disc, such that ''f''(φ(''z'')) = ''z'' for all ''z'' in this disc. Here ''b'' > 1/72 is an absolute constant.

			==Landau's theorem==
			If ''f'' is a holomorphic function in the unit disc with the property \|''f′''(0)\| = 1, then the image of ''f'' contains a disc of radius ''l'', where ''l'' ≥ ''b'' is an absolute constant.

			This theorem is named after [[Edmund Landau]].

			==Valiron's theorem==
			Bloch's theorem was inspired by the following theorem of [[Georges Valiron]]:

			'''Theorem.''' If ''f'' is a non-constant entire function then there exist discs ''D'' of arbitrarily large radius and analytic functions φ in ''D'' such that ''f''(φ(''z'')) = ''z'' for ''z'' in ''D''.

			Bloch's theorem corresponds to Valiron's theorem via the so-called [[Bloch's Principle]].

			== Bloch's and Landau's constants ==
			The lower bound 1/72 in Bloch's theorem is not the best possible. The number ''B'' defined as the [[supremum]] of all ''b'' for which this theorem holds, is called the '''Bloch's constant'''. Bloch's theorem tells us ''B'' ≥ 1/72, but the exact value of ''B'' is still unknown.

			The similarly defined optimal constant ''L'' in Landau's theorem is called the '''Landau's constant'''. Its exact value is also unknown.

			The best known bounds for ''B'' at present are
			:<math>\frac{\sqrt{3}}{4}+2\times10^{-4}\leq B\leq \sqrt{\frac{\sqrt{3}-1}{2}} \cdot \frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})},</math>
			where Γ is the [[Gamma function]]. The lower bound was proved by Chen and Gauthier, and the upper bound dates back to [[Lars Ahlfors\|Ahlfors]] and Grunsky. They also gave an upper bound for the Landau constant.

			In their paper, Ahlfors and Grunsky conjectured that their upper bounds are actually the true values of ''B'' and ''L''.

			== References ==
			* {{cite journal\|first=Lars Valerian\|last=Ahlfors\|authorlink=Lars Ahlfors\|last2=Grunsky\|first2=Helmut\|author2-link=Helmut Grunsky\|title=Über die Blochsche Konstante\|journal=[[Mathematische Zeitschrift]]\|year=1937\|volume=42\|number=1\|pages=671–673\|doi=10.1007/BF01160101}}
			* {{cite conference\|first=Albert II\|last=Baernstein\|coauthors=Vinson, Jade P.\|title=Local minimality results related to the Bloch and Landau constants\|booktitle=Quasiconformal mappings and analysis\|place=Ann Arbor\|publisher=Springer, New York\|year=1998\|pages=55–89}}
			* {{cite journal\|first=André\|last=Bloch\|title=Les théorèmes de M.Valiron sur les fonctions entières et la théorie de l'uniformisation\|journal=Annales de la faculté des sciences de l'Université de Toulouse\|number=3\|volume=17\|year=1925\|pages=1–22\|issn=0240-2963}}
			* {{ cite journal \| first=Huaihui \| last=Chen \| coauthors=Gauthier, Paul M. \| title=On Bloch's constant\|journal=Journal d'Analyse Mathématique\|year=1996\|volume=69\|number=1\|pages=275–291\|doi=10.1007/BF02787110}}

			[[Category:Unsolved problems in mathematics]]
			[[Category:Theorems in complex analysis]]

en>Yobot: WP:CHECKWIKI error 61 fixes + general fixes using AWB (8032)

2012-03-20T23:56:51Z

WP:CHECKWIKI error 61 fixes + general fixes using AWB (8032)

New page

Hi there, I am Felicidad Oquendo. Interviewing is what I do for a residing but I plan on altering it. Kansas is our beginning location and my parents live nearby. The factor I adore most flower arranging and now I have time to take on new things.<br><br>My web page - [http://vadim.amherstburg.ca/ActivityFeed/MyProfile/tabid/56/userId/15736/Default.aspx aftermarket car warranty]