Locally integrable function - Revision history

en>Daniele.tampieri: A Typo and a minor correction

2014-05-11T12:14:17Z

A Typo and a minor correction

← Older revision		Revision as of 14:14, 11 May 2014
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	~~In the [[projective space]] ''PG(3,q)'', with ''q'' a prime power greater than 2, an '''ovoid'''~~ is ~~a set of <math>q^2+1</math> points, no three of which are collinear (the maximum size of such a set).<ref>more properly the term should be~~ '~~'ovaloid'' and ovoid has a different definition which extends to projective spaces of higher dimension~~. ~~However, in dimension 3 the two concepts are equivalent~~ and ~~the ovoid terminology~~ is ~~almost universally used, except most notably, in Hirschfeld~~.~~</ref> When <math>q = 2</math> the largest set of non-collinear points has size eight and is the complement of a plane.<ref>{{harvnb\|Hirschfeld\|1985\|loc=pg.33, Theorem 16.1.3}}</ref>~~		Ed is what individuals call me and my spouse doesn't like it at all. What me and my family members adore is bungee jumping but I've been using on new issues lately. I've usually loved living in Mississippi. Since he was eighteen he's been working as an info officer but he ideas on altering it.<br><br>My homepage ... email psychic readings; [http://checkmates.co.za/index.php?do=/profile-56347/info/ mouse click the following internet site],

	~~An important example of an ovoid~~ in ~~any finite projective three-dimensional space are the <math>q^2+1</math> points of an elliptic quadric (all of which are projectively equivalent)~~.

	~~When '~~'~~q'' is odd or <math>q = 4</math>, no ovoids exist other than the elliptic [[quadric]]~~s~~.<ref>{{harvnb\|Barlotti\|1955}} and {{harvnb\|Panella\|1955}}</ref>~~

	~~When <math>q=2^{2 h+1}</math> another type of ovoid can be constructed : the [[Jacques Tits\|Tits]] ovoid, also known~~ as ~~the [[Michio Suzuki\|Suzuki]] ovoid~~. ~~It is conjectured that no other ovoids exist in ''PG(3,q)''.~~

	~~Through every point ''P'' on the ovoid, there are exactly~~ <~~math~~>~~q+1~~<~~/math~~> ~~tangents, and it can be proven that these lines are exactly the lines through ''P'' in one specific plane through ''P''~~. ~~This means that through every point ''P'' in the ovoid, there is a unique plane intersecting the ovoid in exactly one point~~.~~<ref>{{harvnb\|Hirschfeld\|1985\|loc=pg~~. ~~34, Lemma 16.1.6}}<~~/~~ref> Also, if ''q'' is odd or <math>q = 4<~~/~~math> every plane which is not a tangent plane meets the ovoid in a [[Conic section\|conic]]~~.~~<ref>{{harvnb\|Hirschfeld\|1985\|loc=pg~~.~~35, Corollary}}<~~/~~ref>~~

	~~==See also==~~
	*[[Ovoid (polar space)]]
	*[[Oval (projective plane)]]
	*[[Inversive plane]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{citation\|last=Barlotti\|first=A.~~\|title~~=~~Un' estensione del teorema di Segre~~-~~Kustaanheimo\|journal=Boll. Un. Mat. Ital.\|year=1955\|volume=10\|pages=96–98}}~~
	*{{citation\|last=Hirschfeld\|first=J.W.P.\|title=Finite Projective Spaces of Three Dimensions\|year=1985\|publisher=Oxford University Press\|location=New York\|isbn=0-19-853536-8}}
	*{{citation\|last=Panella\|first=G.\|title=Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito\|journal=Boll. Un. Mat. Ital.\|year=1955\|volume=10\|pages=507–513}}


	~~{{DEFAULTSORT:Ovoid (Projective Geometry)}}~~
	~~[[Category:Projective geometry]]~~
	~~[[Category:Incidence geometry]~~]

en>Myasuda: /* An alternative definition */ sp

2013-12-18T14:21:45Z

An alternative definition: sp

← Older revision		Revision as of 16:21, 18 December 2013
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	~~The author~~'~~s name~~ is ~~Andera~~ and ~~she believes it sounds fairly great~~. ~~Her family members lives~~ in ~~Alaska but her husband wants them to transfer~~. ~~Distributing manufacturing~~ is ~~how he makes~~ a ~~living~~. ~~As a woman what she really likes~~ is ~~fashion~~ and ~~she~~'~~s been doing~~ it ~~for fairly~~ a ~~while~~.<br><br>~~Feel free to visit my blog; phone psychic~~ ([~~http://www~~.~~chk~~.~~woobi~~.co.~~kr/xe/?document_srl~~=~~346069 click through the next page~~])		In the [[projective space]] ''PG(3,q)'', with ''q'' a prime power greater than 2, an '''ovoid''' is a set of <math>q^2+1</math> points, no three of which are collinear (the maximum size of such a set).<ref>more properly the term should be ''ovaloid'' and ovoid has a different definition which extends to projective spaces of higher dimension. However, in dimension 3 the two concepts are equivalent and the ovoid terminology is almost universally used, except most notably, in Hirschfeld.</ref> When <math>q = 2</math> the largest set of non-collinear points has size eight and is the complement of a plane.<ref>{{harvnb\|Hirschfeld\|1985\|loc=pg.33, Theorem 16.1.3}}</ref>

			An important example of an ovoid in any finite projective three-dimensional space are the <math>q^2+1</math> points of an elliptic quadric (all of which are projectively equivalent).

			When ''q'' is odd or <math>q = 4</math>, no ovoids exist other than the elliptic [[quadric]]s.<ref>{{harvnb\|Barlotti\|1955}} and {{harvnb\|Panella\|1955}}</ref>

			When <math>q=2^{2 h+1}</math> another type of ovoid can be constructed : the [[Jacques Tits\|Tits]] ovoid, also known as the [[Michio Suzuki\|Suzuki]] ovoid. It is conjectured that no other ovoids exist in ''PG(3,q)''.

			Through every point ''P'' on the ovoid, there are exactly <math>q+1</math> tangents, and it can be proven that these lines are exactly the lines through ''P'' in one specific plane through ''P''. This means that through every point ''P'' in the ovoid, there is a unique plane intersecting the ovoid in exactly one point.<ref>{{harvnb\|Hirschfeld\|1985\|loc=pg. 34, Lemma 16.1.6}}</ref> Also, if ''q'' is odd or <math>q = 4</math> every plane which is not a tangent plane meets the ovoid in a [[Conic section\|conic]].<ref>{{harvnb\|Hirschfeld\|1985\|loc=pg.35, Corollary}}</ref>

			==See also==
			*[[Ovoid (polar space)]]
			*[[Oval (projective plane)]]
			*[[Inversive plane]]

			==Notes==
			{{reflist}}

			==References==
			*{{citation\|last=Barlotti\|first=A.\|title=Un' estensione del teorema di Segre-Kustaanheimo\|journal=Boll. Un. Mat. Ital.\|year=1955\|volume=10\|pages=96–98}}
			*{{citation\|last=Hirschfeld\|first=J.W.P.\|title=Finite Projective Spaces of Three Dimensions\|year=1985\|publisher=Oxford University Press\|location=New York\|isbn=0-19-853536-8}}
			*{{citation\|last=Panella\|first=G.\|title=Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito\|journal=Boll. Un. Mat. Ital.\|year=1955\|volume=10\|pages=507–513}}


			{{DEFAULTSORT:Ovoid (Projective Geometry)}}
			[[Category:Projective geometry]]
			[[Category:Incidence geometry]]

en>Daniele.tampieri: /* Examples */ Typo

2012-09-01T09:19:34Z

Examples: Typo

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The author's name is Andera and she believes it sounds fairly great. Her family members lives in Alaska but her husband wants them to transfer. Distributing manufacturing is how he makes a living. As a woman what she really likes is fashion and she's been doing it for fairly a while.<br><br>Feel free to visit my blog; phone psychic ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 click through the next page])