Volatility arbitrage - Revision history

81.157.54.88 at 02:44, 6 June 2014

2014-06-06T02:44:17Z

← Older revision		Revision as of 04:44, 6 June 2014
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	~~Alyson Meagher~~ is ~~the title her parents gave her~~ but ~~she doesn~~'~~t like when people use her full name~~. ~~Mississippi is where her home~~ is but ~~her husband desires them to move~~. To perform lacross is ~~the factor~~ I ~~love most of all. Credit authorising is how he makes cash~~.<br><br>Also visit my ~~blog post~~ [http://~~medialab~~.~~zendesk~~.com/~~entries~~/~~54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- authentic psychic readings~~]		The writer's free online tarot card readings - [http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies such a good point] - name is Andera and she believes it seems quite good. My day occupation is an invoicing officer psychic chat online ([http://modenpeople.co.kr/modn/qna/292291 such a good point]) but I've currently utilized for another 1. Alaska is the only place I've been residing in but now I'm considering other choices. To perform lacross is something I really enjoy doing.<br><br>Also visit my weblog ... are psychics real ([http://Youronlinepublishers.com/authWiki/AdolphvhBladenqq http://Youronlinepublishers.com/authWiki/AdolphvhBladenqq])

en>Colonycat: Fix link to Black swan theory rather than animal

2014-02-14T08:00:17Z

Fix link to Black swan theory rather than animal

← Older revision		Revision as of 10:00, 14 February 2014
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	~~In [[algebraic geometry]], a '''Weil cohomology''' or '''Weil cohomology theory'''~~ is ~~a [[cohomology]] satisfying certain axioms concerning~~ the ~~interplay of [[algebraic cycles]] and cohomology groups. The~~ name ~~is in honor of [[André Weil]]~~. ~~Weil cohomology theories play an important role in the theory of [[motive (algebraic geometry)\|motives]], insofar as the [[category (mathematics)\|category]] of [[Chow motive]]s~~ is a universal Weil cohomology theory in the sense that any Weil cohomology function factors through Chow motives. Note that, however, the category of Chow motives does not give a Weil cohomology theory since it is ~~not [[abelian category\|abelian]].~~		Alyson Meagher is the title her parents gave her but she doesn't like when people use her full name. Mississippi is where her home is but her husband desires them to move. To perform lacross is the factor I love most of all. Credit authorising is how he makes cash.<br><br>Also visit my blog post [http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- authentic psychic readings]

	~~==Definition==~~
	~~A ''Weil cohomology theory'' is a [[functor#Covariance and contravariance\|contravariant functor]]:~~

	~~::::''H<sup>*</sup>'': {smooth projective [[algebraic variety\|varieties]] over a field ''k''} → {graded ''K''-algebras}~~

	~~subject~~ to ~~the axioms below~~. ~~Note that the field ''K''~~ is ~~not to be confused with ''k'';~~ the ~~former is a field~~ of ~~characteristic zero, called the ''coefficient field'', whereas the base field ''k'' can be arbitrary~~. ~~Suppose ''X'' is a smooth [[projective variety\|projective algebraic variety]] of dimension ''n'', then the [[graded algebra\|graded]] ''[[K-algebra]]'' ''H<sup>*</sup>(X)'' = ⊕''H<sup>i</sup>(X)''~~ is ~~subject to the following:~~

	~~#''H<sup>i</sup>(X)'' are finite-dimensional ''K''-[[vector space]]s~~.
	~~#''H~~<~~sup~~>i<~~/sup~~>~~(X)'' vanish for ''i < 0'' or ''i > 2n''.~~
	~~#''H<sup>2n</sup>(X)'' is isomorphic to ''K'' (so-called orientation map).~~
	~~#There is a [~~[~~Poincaré duality]], i.e. a non-degenerate pairing~~: ~~''H<sup>i</sup>(X)'' × ''H<sup>2n−i<~~/~~sup>(X) → H<sup>2n<~~/~~sup>(X) ≅ K''~~.
	~~#There is a canonical [[Künneth theorem\|Künneth]] isomorphism: ''H<sup></sup>(X)'' ⊗ ''H<sup></sup>(Y)'' → ''H<sup>*</sup>(X × Y)''~~.
	~~#There is a ''cycle-map'': γ<sub>''X''</sub>: ''Z<sup>i<~~/~~sup>(X)'' → ''H<sup>2i<~~/sup>(X)'', where the former group means algebraic cycles of codimension ''i'', satisfying certain compatibility conditions with respect to functionality of ''H'', the Künneth isomorphism and such that for ''X'' a point, the cycle map is the inclusion '''Z''' ⊂ ''K''.
	#''Weak Lefschetz axiom'': For any smooth [[hyperplane section]] ''j: W ⊂ X'' (i.e. ''W = X ∩ H'', ''H'' some hyperplane in the ambient projective space), the maps ''j<sup>*</sup>: H<sup>i</sup>(X)'' → ''H<sup>i</sup>(W)'' are isomorphisms for ''i ≤ n-~~2'' and a monomorphism for ''i ≤ n~~-~~1''.~~
	#''Hard Lefschetz axiom'': Again let ''W'' be a hyperplane section and ''w'' = γ<sub>''X''</sub>(''W'') ∈ ''H''<sup>2</sup>''(X)''be its image under the cycle class map. The ''Lefschetz operator'' ''L: H<sup>i</sup>(X)'' → ''H<sup>i+2</sup>(X)'' maps ''x'' to ''x·w'' (the dot denotes the product in the algebra ''H<sup>*</sup>(X)''). The axiom states that ''L<sup>i</sup>: H<sup>n−i</sup>(X) → H<sup>n+i</sup>(X)'' is an isomorphism for ''i=1, ..., n''.

	~~==Examples==~~
	~~There are four so-called classical Weil cohomology theories:~~
	*[[Betti cohomology\|singular (=Betti) cohomology]], regarding varieties over '''C''' as topological spaces using their [[analytic topology]] (see [[GAGA]])
	*[[de Rham cohomology]] over a base field of [[characteristic (algebra)\|characteristic]] zero: over '''C''' defined by [[differential forms]] and in general by means of the complex of Kähler differentials (see [[algebraic de Rham cohomology]])
	*[[etale cohomology\|l-adic cohomology]] for varieties over fields of characteristic different from ''l''
	*[[crystalline cohomology]]

	The proofs of the axioms in the case of Betti and de Rham cohomology are comparatively easy and classical, whereas for ''l''-adic cohomology, for example, most of the above properties are deep theorems.

	The vanishing of Betti cohomology groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension ''n'' has real dimension ''2n'', so these higher cohomology groups vanish (for example by comparing them to [[simplicial homology\|simplicial (co)homology]]). The cycle map also has a down-to-earth explanation: given any (complex-)''i''-dimensional sub-variety of (the compact manifold) ''X'' of complex dimension ''n'', one can integrate a differential (''2n−i'')-form along this sub-variety. The classical statement of [[Poincaré duality]] is, that this gives a non-degenerate pairing:
	~~::<math>H_i (X) \otimes H_{\text{dR}}^{2n-i}(X) \rightarrow \mathbf{C}</math>,~~
	~~thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism:~~
	~~::<math>H_i (X) \cong H_{\text{dR}}^{2n-i}(X)^{\vee} \cong H^i(X).</math>~~

	~~==References==~~
	* {{Citation \| last1=Griffiths \| first1=Phillip \| last2=Harris \| first2=Joseph \| title=Principles of algebraic geometry \| publisher=Wiley \| location=New York \| series=Wiley Classics Library \| isbn=978-0-~~471~~-~~05059~~-~~9 \| id={{MathSciNet \| id = 1288523}} \| year=1994}} (contains proofs of all of the axioms for Betti and de~~-~~Rham cohomology)~~
	* {{Citation \| last1=Milne \| first1=James S. \| title=Étale cohomology \| publisher=[[Princeton University Press]] \| location=Princeton, NJ \| isbn=978-0-~~691~~-~~08238~~-~~7 \| year=1980}} (idem for ''l''~~-~~adic cohomology)~~
	* {{Citation \| last1=Kleiman \| first1=S. L. \| title=Dix exposés sur la cohomologie des schémas \| publisher=North-~~Holland \| location=Amsterdam \| id={{MathSciNet \| id = 0292838}} \| year=1968 \| chapter=Algebraic cycles and the Weil conjectures \| pages=359–386}}~~

	~~[[Category:Topological methods of algebraic geometry]]~~
	~~[[Category:Cohomology theories]~~]

en>Cydebot: Robot - Speedily moving category Options to :Category:Options (finance) per CFDS.

2013-11-09T08:06:13Z

Robot - Speedily moving category Options to Category:Options (finance) per CFDS.

← Older revision		Revision as of 10:06, 9 November 2013
Line 1:		Line 1:
	The ~~writer~~ is called ~~Irwin~~. ~~Supervising~~ is ~~my occupation~~. ~~Playing baseball~~ is ~~the hobby he will never quit doing~~. ~~Puerto Rico~~ is ~~exactly~~ where he and ~~his wife reside~~.<br><br>~~My web~~-~~site~~ ... ~~std home test~~ ([~~http~~:~~//www~~.~~siccus~~.~~net~~/~~blog~~/~~15356 click~~ the ~~up coming website~~])		In [[algebraic geometry]], a '''Weil cohomology''' or '''Weil cohomology theory''' is a [[cohomology]] satisfying certain axioms concerning the interplay of [[algebraic cycles]] and cohomology groups. The name is in honor of [[André Weil]]. Weil cohomology theories play an important role in the theory of [[motive (algebraic geometry)\|motives]], insofar as the [[category (mathematics)\|category]] of [[Chow motive]]s is a universal Weil cohomology theory in the sense that any Weil cohomology function factors through Chow motives. Note that, however, the category of Chow motives does not give a Weil cohomology theory since it is not [[abelian category\|abelian]].

			==Definition==
			A ''Weil cohomology theory'' is a [[functor#Covariance and contravariance\|contravariant functor]]:

			::::''H<sup>*</sup>'': {smooth projective [[algebraic variety\|varieties]] over a field ''k''} → {graded ''K''-algebras}

			subject to the axioms below. Note that the field ''K'' is not to be confused with ''k''; the former is a field of characteristic zero, called the ''coefficient field'', whereas the base field ''k'' can be arbitrary. Suppose ''X'' is a smooth [[projective variety\|projective algebraic variety]] of dimension ''n'', then the [[graded algebra\|graded]] ''[[K-algebra]]'' ''H<sup>*</sup>(X)'' = ⊕''H<sup>i</sup>(X)'' is subject to the following:

			#''H<sup>i</sup>(X)'' are finite-dimensional ''K''-[[vector space]]s.
			#''H<sup>i</sup>(X)'' vanish for ''i < 0'' or ''i > 2n''.
			#''H<sup>2n</sup>(X)'' is isomorphic to ''K'' (so-called orientation map).
			#There is a [[Poincaré duality]], i.e. a non-degenerate pairing: ''H<sup>i</sup>(X)'' × ''H<sup>2n−i</sup>(X) → H<sup>2n</sup>(X) ≅ K''.
			#There is a canonical [[Künneth theorem\|Künneth]] isomorphism: ''H<sup></sup>(X)'' ⊗ ''H<sup></sup>(Y)'' → ''H<sup>*</sup>(X × Y)''.
			#There is a ''cycle-map'': γ<sub>''X''</sub>: ''Z<sup>i</sup>(X)'' → ''H<sup>2i</sup>(X)'', where the former group means algebraic cycles of codimension ''i'', satisfying certain compatibility conditions with respect to functionality of ''H'', the Künneth isomorphism and such that for ''X'' a point, the cycle map is the inclusion '''Z''' ⊂ ''K''.
			#''Weak Lefschetz axiom'': For any smooth [[hyperplane section]] ''j: W ⊂ X'' (i.e. ''W = X ∩ H'', ''H'' some hyperplane in the ambient projective space), the maps ''j<sup>*</sup>: H<sup>i</sup>(X)'' → ''H<sup>i</sup>(W)'' are isomorphisms for ''i ≤ n-2'' and a monomorphism for ''i ≤ n-1''.
			#''Hard Lefschetz axiom'': Again let ''W'' be a hyperplane section and ''w'' = γ<sub>''X''</sub>(''W'') ∈ ''H''<sup>2</sup>''(X)''be its image under the cycle class map. The ''Lefschetz operator'' ''L: H<sup>i</sup>(X)'' → ''H<sup>i+2</sup>(X)'' maps ''x'' to ''x·w'' (the dot denotes the product in the algebra ''H<sup>*</sup>(X)''). The axiom states that ''L<sup>i</sup>: H<sup>n−i</sup>(X) → H<sup>n+i</sup>(X)'' is an isomorphism for ''i=1, ..., n''.

			==Examples==
			There are four so-called classical Weil cohomology theories:
			*[[Betti cohomology\|singular (=Betti) cohomology]], regarding varieties over '''C''' as topological spaces using their [[analytic topology]] (see [[GAGA]])
			*[[de Rham cohomology]] over a base field of [[characteristic (algebra)\|characteristic]] zero: over '''C''' defined by [[differential forms]] and in general by means of the complex of Kähler differentials (see [[algebraic de Rham cohomology]])
			*[[etale cohomology\|l-adic cohomology]] for varieties over fields of characteristic different from ''l''
			*[[crystalline cohomology]]

			The proofs of the axioms in the case of Betti and de Rham cohomology are comparatively easy and classical, whereas for ''l''-adic cohomology, for example, most of the above properties are deep theorems.

			The vanishing of Betti cohomology groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension ''n'' has real dimension ''2n'', so these higher cohomology groups vanish (for example by comparing them to [[simplicial homology\|simplicial (co)homology]]). The cycle map also has a down-to-earth explanation: given any (complex-)''i''-dimensional sub-variety of (the compact manifold) ''X'' of complex dimension ''n'', one can integrate a differential (''2n−i'')-form along this sub-variety. The classical statement of [[Poincaré duality]] is, that this gives a non-degenerate pairing:
			::<math>H_i (X) \otimes H_{\text{dR}}^{2n-i}(X) \rightarrow \mathbf{C}</math>,
			thus (via the comparison of de Rham cohomology and Betti cohomology) an isomorphism:
			::<math>H_i (X) \cong H_{\text{dR}}^{2n-i}(X)^{\vee} \cong H^i(X).</math>

			==References==
			* {{Citation \| last1=Griffiths \| first1=Phillip \| last2=Harris \| first2=Joseph \| title=Principles of algebraic geometry \| publisher=Wiley \| location=New York \| series=Wiley Classics Library \| isbn=978-0-471-05059-9 \| id={{MathSciNet \| id = 1288523}} \| year=1994}} (contains proofs of all of the axioms for Betti and de-Rham cohomology)
			* {{Citation \| last1=Milne \| first1=James S. \| title=Étale cohomology \| publisher=[[Princeton University Press]] \| location=Princeton, NJ \| isbn=978-0-691-08238-7 \| year=1980}} (idem for ''l''-adic cohomology)
			* {{Citation \| last1=Kleiman \| first1=S. L. \| title=Dix exposés sur la cohomologie des schémas \| publisher=North-Holland \| location=Amsterdam \| id={{MathSciNet \| id = 0292838}} \| year=1968 \| chapter=Algebraic cycles and the Weil conjectures \| pages=359–386}}

			[[Category:Topological methods of algebraic geometry]]
			[[Category:Cohomology theories]]

199.96.107.58: /* Overview */

2012-08-06T18:49:44Z

Overview

New page

The writer is called Irwin. Supervising is my occupation. Playing baseball is the hobby he will never quit doing. Puerto Rico is exactly where he and his wife reside.<br><br>My web-site ... std home test ([http://www.siccus.net/blog/15356 click the up coming website])