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2015-01-12T08:59:01Z

Industrial safety problems

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	~~In [[mathematics]], a '''translation plane''~~' ~~is a particular kind of [[projective plane]], as considered as a combinatorial object.<ref>Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes]</ref>~~		Golda is what's written on my birth certification even though it is not the title on my birth certificate. Credit authorising is how he tends to make money. For a whilst I've been in Alaska but I will have to move in a year or two. Doing ballet is some thing she would never give up.<br><br>Review my web site :: [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 love psychics]

	In a projective plane, <math>\scriptstyle p</math> represents a point, and <math>\scriptstyle L</math> represents a line. A central [[collineation]] with center <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> is a collineation fixing every point on <math>\scriptstyle L</math> and every line through <math>\scriptstyle p</math>. It is called an "elation" if <math>\scriptstyle p</math> is on ~~<math>\scriptstyle L</math>, otherwise~~ it is called a "homology". The central collineations with centre <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> form a group.<ref>Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007</ref>

	~~A projective plane <math>\scriptstyle \Pi</math> is called a translation plane if there exists a line <math>\scriptstyle L</math> such that~~ the ~~group of elations with axis <math>\scriptstyle L</math> is transitive~~ on ~~the affine plane Π<sub>l</sub> (the [[Affine geometry\|affine]] derivative of Π)~~.

	~~== Relationship~~ to ~~spreads ==~~
	~~Translation planes are related~~ to ~~spreads~~ in finite projective spaces by the André/Bruck-Bose construction.<ref>{{cite web\|url=http://www-ma4.upc.es/~simeon/bblpsympspread.pdf\|title=Symplectice Spreads\|last=Ball\|first=Simeon\|coauthors=John Bamberg, Michel Lavrauw, Tim Penttila\|date=2003-09-15\|publisher=[[Polytechnic University of Catalonia]]\|accessdate=2008-10-08}}</ref> A spread of <math>\scriptstyle PG(3, q) </math> is a ~~set of ''q''<sup>2</sup> + 1 lines, with no~~ two ~~intersecting~~. ~~Equivalently, it~~ is ~~a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines~~.

	~~Given a spread~~ <~~math~~>~~\scriptstyle S~~<~~/math~~> ~~of <math>\scriptstyle PG(3, q) </math>, the André/Bruck-Bose construction<sup>1</sup> produces a translation plane <math>\scriptstyle \pi(S)</math> of order ''q''<sup>2</sup> as follows~~: Embed <math>\scriptstyle PG(3, q) </math> as a hyperplane of <math>\scriptstyle PG(4, q) </math>. Define an incidence structure <math>\scriptstyle A(S)</math> with "points," the points of <math>\scriptstyle PG(4, q) </math> not on <math>\scriptstyle PG(3, q) </math> and "lines" the planes of <math>\scriptstyle PG(4, q) </math> meeting <math>\scriptstyle PG(3, q) </math> in a line of <math>\scriptstyle S</math>. Then <math>\scriptstyle A(S)</math> is a translation affine plane of order ''q''<sup>2</sup>. Let <math>\scriptstyle \pi(S)</math> be the projective completion of <math>\scriptstyle A(S)</math>.<ref>{{cite book
	\| last =André \| first =Johannes \| authorlink = \| coauthors = \| title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe \| publisher = \| year =1954 \| location = \| pages =156–186 \| url = \| doi = \| id = }}</ref><ref>{{cite book
	\| last =Bruck \| first = R. H. \| authorlink = Richard Bruck\| coauthors = R. C. Bose \| title = The Construction of Translation Planes from Projective Spaces \| publisher = \| year =1964 \| location = \| pages = 85–102 \| url = \| doi = \| id = }}</ref>

	~~==References==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .

	~~==External links==~~
	*[http:~~//www.library.tuiasi.ro/ipm/vol13no34/pure.html Foundations_of_Translation_Planes]~~
	*[http://www~~-math~~.~~ucdenver~~.~~edu/~wcherowi~~/~~courses/m6221/pglc3a~~.~~html Lecture Notes on Projective Geometry]~~
	*[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]

	~~{{DEFAULTSORT:Translation Plane}}~~
	~~[[Category:Projective geometry]~~]

en>Dandycustard: Removed part of recent contribution

2013-10-28T18:12:02Z

Removed part of recent contribution

← Older revision		Revision as of 20:12, 28 October 2013
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	~~Hi there~~, ~~I am Sophia~~. ~~Office supervising~~ is ~~my profession~~. The ~~preferred pastime for him~~ and ~~his kids~~ is ~~fashion and he~~'~~ll be starting some thing else alongside~~ with it. ~~Ohio~~ is ~~exactly where my house is but my spouse wants us to transfer~~.<br><br>~~My website~~ ... ~~love psychic~~ ([http://~~chorokdeul~~.co.kr/~~index~~.~~php?document_srl=324263&mid=customer21 had me going~~])		In [[mathematics]], a '''translation plane''' is a particular kind of [[projective plane]], as considered as a combinatorial object.<ref>Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes]</ref>

			In a projective plane, <math>\scriptstyle p</math> represents a point, and <math>\scriptstyle L</math> represents a line. A central [[collineation]] with center <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> is a collineation fixing every point on <math>\scriptstyle L</math> and every line through <math>\scriptstyle p</math>. It is called an "elation" if <math>\scriptstyle p</math> is on <math>\scriptstyle L</math>, otherwise it is called a "homology". The central collineations with centre <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> form a group.<ref>Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007</ref>

			A projective plane <math>\scriptstyle \Pi</math> is called a translation plane if there exists a line <math>\scriptstyle L</math> such that the group of elations with axis <math>\scriptstyle L</math> is transitive on the affine plane Π<sub>l</sub> (the [[Affine geometry\|affine]] derivative of Π).

			== Relationship to spreads ==
			Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.<ref>{{cite web\|url=http://www-ma4.upc.es/~simeon/bblpsympspread.pdf\|title=Symplectice Spreads\|last=Ball\|first=Simeon\|coauthors=John Bamberg, Michel Lavrauw, Tim Penttila\|date=2003-09-15\|publisher=[[Polytechnic University of Catalonia]]\|accessdate=2008-10-08}}</ref> A spread of <math>\scriptstyle PG(3, q) </math> is a set of ''q''<sup>2</sup> + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines.

			Given a spread <math>\scriptstyle S</math> of <math>\scriptstyle PG(3, q) </math>, the André/Bruck-Bose construction<sup>1</sup> produces a translation plane <math>\scriptstyle \pi(S)</math> of order ''q''<sup>2</sup> as follows: Embed <math>\scriptstyle PG(3, q) </math> as a hyperplane of <math>\scriptstyle PG(4, q) </math>. Define an incidence structure <math>\scriptstyle A(S)</math> with "points," the points of <math>\scriptstyle PG(4, q) </math> not on <math>\scriptstyle PG(3, q) </math> and "lines" the planes of <math>\scriptstyle PG(4, q) </math> meeting <math>\scriptstyle PG(3, q) </math> in a line of <math>\scriptstyle S</math>. Then <math>\scriptstyle A(S)</math> is a translation affine plane of order ''q''<sup>2</sup>. Let <math>\scriptstyle \pi(S)</math> be the projective completion of <math>\scriptstyle A(S)</math>.<ref>{{cite book
			\| last =André \| first =Johannes \| authorlink = \| coauthors = \| title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe \| publisher = \| year =1954 \| location = \| pages =156–186 \| url = \| doi = \| id = }}</ref><ref>{{cite book
			\| last =Bruck \| first = R. H. \| authorlink = Richard Bruck\| coauthors = R. C. Bose \| title = The Construction of Translation Planes from Projective Spaces \| publisher = \| year =1964 \| location = \| pages = 85–102 \| url = \| doi = \| id = }}</ref>

			==References==
			{{Reflist}}

			==Further reading==
			* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .

			==External links==
			*[http://www.library.tuiasi.ro/ipm/vol13no34/pure.html Foundations_of_Translation_Planes]
			*[http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
			*[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]

			{{DEFAULTSORT:Translation Plane}}
			[[Category:Projective geometry]]

108.223.132.121 at 02:56, 24 July 2012

2012-07-24T02:56:32Z

New page

Hi there, I am Sophia. Office supervising is my profession. The preferred pastime for him and his kids is fashion and he'll be starting some thing else alongside with it. Ohio is exactly where my house is but my spouse wants us to transfer.<br><br>My website ... love psychic ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 had me going])