Normal-gamma distribution - Revision history

en>Mingyuanzhou: should be $\alpha-1/2$ instead of $\alpha$

2015-01-03T21:54:49Z

should be $\alpha-1/2$ instead of $\alpha$

← Older revision		Revision as of 23:54, 3 January 2015
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en>Gxhrid at 12:13, 10 February 2014

2014-02-10T12:13:42Z

← Older revision		Revision as of 14:13, 10 February 2014
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	~~In mathematics, a~~ [~~[quadratic form]] over a [[field (mathematics)\|field]] ''F'' is said to be '''isotropic''' if there is a non~~-~~zero vector on which the form evaluates to zero~~. ~~Otherwise the quadratic form is '''anisotropic'''. More precisely, if ''q'' is a quadratic form on a [[vector space]] ''V'' over ''F'', then a non~~-~~zero vector ''v'' in ''V'' is said to be '''isotropic''' if {{nowrap\|1=''q''(''v'') = 0}}. A quadratic form is isotropic if and only if there exists a non~~-~~zero isotropic vector~~ for ~~that quadratic form.~~		[http://alles-herunterladen.de/excellent-advice-for-picking-the-ideal-hobby/ love psychics] I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my complete title. Alaska is the only place I've been residing in but now I'm considering other options. As a woman what she really likes is style and she's been doing it for fairly a whilst. Credit authorising is exactly where my primary earnings arrives from.<br><br>Look into my web site: free [http://black7.mireene.com/aqw/5741 online psychic readings] ([http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ www.edmposts.com])

	Suppose that (''V'',''q'') is [[quadratic space]] and ''W'' is a [[linear subspace\|subspace]]. Then ''W'' is called an '''isotropic subspace''' of ''V'' if ''some'' vector in it is isotropic, a '''totally isotropic subspace''' if ''all'' vectors in it are isotropic, and an '''anisotropic subspace''' if it does not contain ''any'' (non-~~zero) isotropic vectors. The '''{{visible anchor\|isotropy index}}''' of a quadratic space is the maximum of~~ the ~~dimensions of the totally isotropic subspaces.<ref name=MH57/>~~

	~~A quadratic form ''q'' on a finite~~-~~dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a [[definite bilinear form\|definite form]]:~~
	~~:* either ''q'' is ''positive definite'', i.e. {{nowrap\|1=''q''(''v'') > 0}} for all non~~-~~zero ''v'' in ''V'' ;~~
	~~:* or ''q'' is ''negative definite'', i.e. {{nowrap\|1=''q''(''v'') < 0}} for all non-zero ''v'' in ''V''.~~

	~~More generally, if the quadratic form is non-degenerate and has the [[signature (quadratic form)\|signature]] (''a'',''b''), then its isotropy index is the minimum of ''a'' and ''b''.~~

	~~==Hyperbolic plane==~~
	~~Let ''V'' = ''F''<sup>2</sup> with elements (''x,y''). Then the quadratic forms ''q = xy'' and ''r'' = ''x''<sup>2</sup> − ''y''<sup>2<~~/~~sup>~~
	~~are equivalent since there is a [[linear transformation~~]~~] on ''V'' that makes ''q'' look~~ like ~~''r''~~, ~~and vice versa. Evidently (~~'~~'V,q'') and (''V,r'') are isotropic~~. ~~This example~~ is ~~called~~ the '~~''hyperbolic plane'''~~ in ~~the theory of [[quadratic form]]s. A common instance has ''F~~'~~' = [[real number]]s in which case~~
	~~<math>\scriptstyle \lbrace x \isin V : q(x)\ =\ \text{nonzero constant} \rbrace </math> and~~
	~~<math>\scriptstyle \lbrace x \isin V : r(x)\ =\ \text{nonzero constant} \rbrace </math> are [[hyperbola]]s~~. ~~In particular, <math>\scriptstyle \lbrace x \isin V : r(x) = 1 \rbrace </math> is the [[unit hyperbola]]. The notation~~
	<math>\scriptstyle \langle 1 \rangle \oplus \langle -1 \rangle </math> has been used by Milnor and Huseman<ref>Milnor & Husemoller (1973) page 9</ref> for the hyperbolic plane as the signs of the terms of the [[polynomial\|bivariate polynomial]] ''r'' are exhibited.

	~~==Split quadratic space==~~
	~~A space with quadratic form is '''split''' (or '''metabolic''') if there is~~ a ~~subspace which is equal to its own [[orthogonal complement]]: equivalently, the index of isotropy~~ is ~~equal to half the dimension.<ref name=MH57>Milnor & Husemoller (1973) p.57</ref> The hyperbolic plane is an example,~~ and ~~over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.<ref>Milnor & Husemoller (1973) pp.12–13</ref>~~

	~~== Relation with classification of quadratic forms ==~~

	From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field ''F'', classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By [[Witt's ~~decomposition theorem]], every inner product space over a field is an [[orthogonal direct sum]] of a split space and an anisotropic space.<ref>Milnor & Husemoller (1973) p.56</ref>~~

	~~==Field theory==~~
	* If ''F'' is an [[algebraically closed]] field, for ~~example, the field of [[complex numbers]], and (''V'',''q'') is~~ a ~~quadratic space of dimension at least two, then it is isotropic~~.
	* If ''F'' is ~~a [[finite field]] and (''V'',''q'') is a quadratic space of dimension at least three, then it is isotropic~~.
	* If ''F'' is the field ''Q''<~~sub~~>~~''p''~~<~~/sub~~> ~~of [[p-adic number]]s and (''V'',''q'') is a quadratic space of dimension at least five, then it is isotropic.~~
	~~== See also ==~~
	*[[Null vector]]
	*[[Witt group]]
	*[[Witt ring (forms)]]
	*[[Witt's theorem]]
	*[[Symmetric bilinear form]]
	*[[Universal quadratic form]]

	~~== References ==~~
	~~{{reflist}}~~
	* Pete L. Clark, [http://~~www~~.~~math~~.~~miami.edu~~/~~~armstrong~~/~~685fa12/pete_clark.pdf Quadratic forms chapter I: Witts theory] from [[University of Miami]] in [[Coral Gables, Florida]].~~
	* [[Tsit Yuen Lam]] (~~1973) ''Algebraic Theory of Quadratic Forms'', §1.3 Hyperbolic plane and hyperbolic spaces,~~ [[W. A. ~~Benjamin]].~~
	* Tsit Yuen Lam (2005) ''Introduction to Quadratic Forms over Fields'', [[American Mathematical Society]] ISBN 0-~~8218~~-~~1095-2 .~~
	* {{cite book \| first1=J. \| last1=Milnor \| author1-~~link=John Milnor\| first2=D. \| last2=Husemoller \| title=Symmetric Bilinear Forms \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] \| volume=73 \| publisher=[[Springer~~-~~Verlag]] \| year=1973 \| isbn=3~~-~~540~~-~~06009~~-~~X \| zbl=0292.10016 }}~~
	* {{cite book \| first=O.~~T \| last=O'Meara \| authorlink=O~~. ~~Timothy O'Meara \| year=1963 \| title=Introduction to Quadratic Forms \| page=94 §42D Isotropy \| publisher=[[Springer-Verlag]] \| isbn=3-540-66564-1 }}~~
	* {{cite book \| first=Jean-Pierre \| last=Serre \| authorlink=Jean-Pierre Serre \| title=A Course in Arithmetic \| series=[[Graduate Texts in Mathematics]] \| volume=7 \| publisher=[[Springer-Verlag]] \| year=2000 \| origyear=1973 \| edition=reprint of 3rd \| series=Classics in mathematics \| isbn=0-387-90040-3 \| zbl=1034.11003 }}

	~~[[Category:Quadratic forms]]~~
	~~[[Category:Bilinear forms]~~]

en>BeyondNormality: /* References */

2013-12-10T04:05:04Z

References

← Older revision		Revision as of 06:05, 10 December 2013
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	Hi there. ~~Let clairvoyants~~ ([~~http~~:~~//conniecolin~~.~~com/xe/community/24580 conniecolin~~.~~com~~]) ~~me start by introducing~~ the ~~writer~~, ~~her name~~ is ~~Sophia~~. ~~I am really fond of to go to karaoke but I~~'~~ve been using on new things recently~~. ~~She works as a travel agent but quickly she~~'~~ll be on her personal~~. ~~For years he~~'s ~~been residing~~ in [~~http~~://~~www~~.~~taehyuna~~.~~net~~/xe/~~?document_srl~~=~~78721~~ are ~~psychics real~~] ~~Alaska~~ and ~~he doesn~~'~~t plan on changing~~ it.<br><br>~~my site free tarot readings~~ ([http://~~myoceancounty~~.~~net~~/~~groups~~/~~apply~~-~~these~~-~~guidelines~~-~~when~~-~~gardening~~-~~and~~-~~grow/ http://myoceancounty~~.~~net/groups/apply~~-~~these~~-~~guidelines~~-~~when~~-~~gardening~~-~~and~~-~~grow~~])		In mathematics, a [[quadratic form]] over a [[field (mathematics)\|field]] ''F'' is said to be '''isotropic''' if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is '''anisotropic'''. More precisely, if ''q'' is a quadratic form on a [[vector space]] ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be '''isotropic''' if {{nowrap\|1=''q''(''v'') = 0}}. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector for that quadratic form.

			Suppose that (''V'',''q'') is [[quadratic space]] and ''W'' is a [[linear subspace\|subspace]]. Then ''W'' is called an '''isotropic subspace''' of ''V'' if ''some'' vector in it is isotropic, a '''totally isotropic subspace''' if ''all'' vectors in it are isotropic, and an '''anisotropic subspace''' if it does not contain ''any'' (non-zero) isotropic vectors. The '''{{visible anchor\|isotropy index}}''' of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.<ref name=MH57/>

			A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a [[definite bilinear form\|definite form]]:
			:* either ''q'' is ''positive definite'', i.e. {{nowrap\|1=''q''(''v'') > 0}} for all non-zero ''v'' in ''V'' ;
			:* or ''q'' is ''negative definite'', i.e. {{nowrap\|1=''q''(''v'') < 0}} for all non-zero ''v'' in ''V''.

			More generally, if the quadratic form is non-degenerate and has the [[signature (quadratic form)\|signature]] (''a'',''b''), then its isotropy index is the minimum of ''a'' and ''b''.

			==Hyperbolic plane==
			Let ''V'' = ''F''<sup>2</sup> with elements (''x,y''). Then the quadratic forms ''q = xy'' and ''r'' = ''x''<sup>2</sup> − ''y''<sup>2</sup>
			are equivalent since there is a [[linear transformation]] on ''V'' that makes ''q'' look like ''r'', and vice versa. Evidently (''V,q'') and (''V,r'') are isotropic. This example is called the '''hyperbolic plane''' in the theory of [[quadratic form]]s. A common instance has ''F'' = [[real number]]s in which case
			<math>\scriptstyle \lbrace x \isin V : q(x)\ =\ \text{nonzero constant} \rbrace </math> and
			<math>\scriptstyle \lbrace x \isin V : r(x)\ =\ \text{nonzero constant} \rbrace </math> are [[hyperbola]]s. In particular, <math>\scriptstyle \lbrace x \isin V : r(x) = 1 \rbrace </math> is the [[unit hyperbola]]. The notation
			<math>\scriptstyle \langle 1 \rangle \oplus \langle -1 \rangle </math> has been used by Milnor and Huseman<ref>Milnor & Husemoller (1973) page 9</ref> for the hyperbolic plane as the signs of the terms of the [[polynomial\|bivariate polynomial]] ''r'' are exhibited.

			==Split quadratic space==
			A space with quadratic form is '''split''' (or '''metabolic''') if there is a subspace which is equal to its own [[orthogonal complement]]: equivalently, the index of isotropy is equal to half the dimension.<ref name=MH57>Milnor & Husemoller (1973) p.57</ref> The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.<ref>Milnor & Husemoller (1973) pp.12–13</ref>

			== Relation with classification of quadratic forms ==

			From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field ''F'', classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By [[Witt's decomposition theorem]], every inner product space over a field is an [[orthogonal direct sum]] of a split space and an anisotropic space.<ref>Milnor & Husemoller (1973) p.56</ref>

			==Field theory==
			* If ''F'' is an [[algebraically closed]] field, for example, the field of [[complex numbers]], and (''V'',''q'') is a quadratic space of dimension at least two, then it is isotropic.
			* If ''F'' is a [[finite field]] and (''V'',''q'') is a quadratic space of dimension at least three, then it is isotropic.
			* If ''F'' is the field ''Q''<sub>''p''</sub> of [[p-adic number]]s and (''V'',''q'') is a quadratic space of dimension at least five, then it is isotropic.
			== See also ==
			*[[Null vector]]
			*[[Witt group]]
			*[[Witt ring (forms)]]
			*[[Witt's theorem]]
			*[[Symmetric bilinear form]]
			*[[Universal quadratic form]]

			== References ==
			{{reflist}}
			* Pete L. Clark, [http://www.math.miami.edu/~armstrong/685fa12/pete_clark.pdf Quadratic forms chapter I: Witts theory] from [[University of Miami]] in [[Coral Gables, Florida]].
			* [[Tsit Yuen Lam]] (1973) ''Algebraic Theory of Quadratic Forms'', §1.3 Hyperbolic plane and hyperbolic spaces, [[W. A. Benjamin]].
			* Tsit Yuen Lam (2005) ''Introduction to Quadratic Forms over Fields'', [[American Mathematical Society]] ISBN 0-8218-1095-2 .
			* {{cite book \| first1=J. \| last1=Milnor \| author1-link=John Milnor\| first2=D. \| last2=Husemoller \| title=Symmetric Bilinear Forms \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] \| volume=73 \| publisher=[[Springer-Verlag]] \| year=1973 \| isbn=3-540-06009-X \| zbl=0292.10016 }}
			* {{cite book \| first=O.T \| last=O'Meara \| authorlink=O. Timothy O'Meara \| year=1963 \| title=Introduction to Quadratic Forms \| page=94 §42D Isotropy \| publisher=[[Springer-Verlag]] \| isbn=3-540-66564-1 }}
			* {{cite book \| first=Jean-Pierre \| last=Serre \| authorlink=Jean-Pierre Serre \| title=A Course in Arithmetic \| series=[[Graduate Texts in Mathematics]] \| volume=7 \| publisher=[[Springer-Verlag]] \| year=2000 \| origyear=1973 \| edition=reprint of 3rd \| series=Classics in mathematics \| isbn=0-387-90040-3 \| zbl=1034.11003 }}

			[[Category:Quadratic forms]]
			[[Category:Bilinear forms]]

en>Howeman: /* Interpretation of parameters */

2012-08-29T17:06:10Z

Interpretation of parameters

New page

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