Parallel tempering - Revision history

en>P99am: /* Implementations */

2014-11-30T08:49:08Z

Implementations

← Older revision		Revision as of 10:49, 30 November 2014
Line 1:		Line 1:
	In [[noncommutative geometry]] and related branches of mathematics, '''cyclic homology''' and '''cyclic cohomology''' are certain (co)homology theories for [[associative algebra]]s which generalize the [[de Rham cohomology\|de Rham (co)homology]] of manifolds. These notions were independently introduced by [[Alain Connes]] (cohomology)<ref>Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.		The author is known by the title of Figures Wunder. My day job is a librarian. South Dakota is exactly where me and my spouse live. One of the things she loves most is to do aerobics and now she is attempting to earn money with it.<br><br>My blog post :: [http://Ref.pw/healthyfooddelivery98296 weight loss food delivery]
	</ref> and Boris Tsygan (homology)<ref>Boris L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983. Translation in Russ. Math. Survey 38(2) (1983), 198–199.
	</ref> around 1980. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the [[K-theory]]. The principal contributors to the development of theory include [[Max Karoubi]], Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, {{Link-interwiki\|en=Jean-Louis Loday\|lang=fr}}, Victor Nistor, [[Daniel Quillen]], Joachim Cuntz, Ryszard Nest, Ralf Meyer, Michael Puschnigg, and many others.

	~~== Hints about definition ==~~

	The ~~first definition of the cyclic homology of a ring ''A'' over a field of [[characteristic (algebra)\|characteristic]] zero, denoted~~

	~~:''HC''<sub>''n''</sub>(''A'') or ''H''<sub>''n''</sub><sup>λ</sup>(''A''),~~

	proceeded by the means of an explicit [[chain complex]] related to the [[Hochschild homology\|Hochschild homology complex]] of ''A''. Connes later found a more categorical approach to cyclic homology using a notion of '''cyclic object''' in an [[abelian category]], which is ~~analogous to the notion of [[simplicial object]]. In this way, cyclic homology (and cohomology) may be interpreted as a [[derived functor]], which can be explicitly computed~~ by the ~~means~~ of ~~the (''b'', ''B'')-bicomplex~~.

	~~One of the striking features of cyclic homology~~ is ~~the existence of~~ a ~~long exact sequence connecting~~
	~~Hochschild and cyclic homology~~. ~~This long exact sequence~~ is ~~referred to as the periodicity sequence.~~

	~~== Case of commutative rings ==~~

	Cyclic cohomology of the commutative algebra ''A'' of regular functions on an [[affine algebraic variety]] over a field ''k'' of characteristic zero can be computed in terms of [[Grothendieck]]'s [[crystalline cohomology\|algebraic de Rham complex]].<ref>Boris L. Fegin and Boris L. Tsygan. Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen., 19(2):52–62, 96, 1985.</ref> In particular, if the variety ''V''=Spec ''A'' is smooth, cyclic cohomology of ''A'' are expressed in terms of the [[de Rham cohomology]] of ''V'' as follows:
	~~:<math> HC_n(A)\simeq \Omega^n\!A/d\Omega^{n-1}\!A\oplus \bigoplus_{i\geq 1}H^{n-2i}_{DR}(V)~~.~~</math>~~

	~~This formula suggests a way to define de Rham cohomology for a 'noncommutative spectrum' of a noncommutative algebra ''A'', which was extensively developed by Connes.~~

	~~== Variants of cyclic homology ==~~

	One ~~motivation~~ of ~~cyclic homology was~~ the ~~need for an approximation of [[K-theory]] that be defined, unlike K-theory, as the homology of a [[chain complex]]. Cyclic cohomology~~ is ~~in fact endowed with a pairing with K-theory,~~ and ~~one hopes this pairing to be non-degenerate.~~

	~~There has been defined a number of variants whose purpose~~ is to ~~fit better~~ with algebras with topology, such as [[Fréchet algebra]]s, <math>C^</math>-algebras, etc. The reason is that K-theory behaves much better on topological algebras such as [[Banach algebra]]s or [[C-algebras]] than on algebras without additional structure. Since, on the other hand, cyclic homology degenerates on C*-algebras, there came up the need to define modified theories. Among them are entire cyclic homology due to [[Alain Connes]], analytic cyclic homology due to Ralf Meyer<ref>
	Ralf Meyer. Analytic cyclic cohomology. PhD thesis, Universität Münster, 1999</ref> or asymptotic and local cyclic homology due to Michael Puschnigg.<ref>Michael Puschnigg. Diffeotopy functors of ind-algebras and local cyclic cohomology. Doc.
	~~Math., 8:143–245 (electronic), 2003.</ref> The last one is very near to [[K-theory]] as~~ it ~~is endowed with a bivariant [[Chern character]] from [[KK-theory]]~~.

	~~==Applications==~~

	~~One of the applications of cyclic homology is to find new proofs and generalizations of the [[Atiyah-Singer index theorem]]. Among these generalizations are index theorems based on spectral triples~~<~~ref~~>~~Alain Connes and Henri Moscovici. The local index formula in noncommutative geometry. Geom. Funct. Anal., 5(2):174–243, 1995.~~<~~/ref~~> ~~and [[deformation quantization]] of Poisson structures.<ref>Ryszard Nest and Boris Tsygan. Algebraic index theorem. Comm. Math. Phys., 172(2)~~:~~223–262, 1995.</ref><!-- needs to be a good representative of the theory, with enough context and relevance. The index theorem for quantum tori is linked to the [[quantum Hall effect]],<ref>http~~://citeseer.ist.psu.edu/old/404503.html</ref> and the index theorem for deformation quantization to the study of band energy redistribution in the [[Born-Oppenheimer approximation]] in molecular physics.<ref>http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.3618</ref> -->

	~~==See also==~~

	*[[Hochschild homology]]
	*[[Noncommutative geometry]]
	*[[Homology (mathematics)\|Homology]]
	*[[Homology theory]]

	~~== References ==~~
	~~<references/>~~

	* Jean-Louis Loday, ''Cyclic Homology'', Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0

	~~==External links==~~
	*[http://~~mathsci~~.~~kaist.ac.kr/~jinhyun/note~~/~~cyclic/cyclic.pdf A personal note on Hochschild and Cyclic homology]~~

	~~{{DEFAULTSORT:Cyclic Homology}}~~
	~~[[Category:Homological algebra]~~]

en>Mjharvey2: /* Implementations */

2013-11-20T15:04:10Z

Implementations

← Older revision		Revision as of 17:04, 20 November 2013
Line 1:		Line 1:
	~~Greetings~~. ~~The~~ [~~http~~://~~calorielab~~.~~com~~/~~news~~/~~diet-delivery~~/ ~~writer~~'s] ~~name~~ is ~~Phebe~~ and ~~she feels healthy food delivery comfortable when people use~~ the ~~full name~~. ~~In her professional life she~~ is a ~~payroll clerk but she~~'s ~~always needed her own business~~. ~~diet meal delivery For years he~~'~~s meal delivery service been residing~~ in [~~http~~://~~healthyeater~~.~~com~~/~~diet~~-~~food~~-~~delivery~~-~~services North Dakota~~] and ~~his family enjoys~~ it. ~~Doing ceramics~~ is ~~what my family members~~ and ~~I appreciate~~.<br><br>~~Stop by my weblog~~ - [http://~~www~~.~~Myprgenie~~.~~com~~/~~view-publication~~/~~how-~~to-be-~~successful~~-~~with~~-~~your~~-~~weight~~-~~loss~~-~~goals~~ http://~~www~~.~~Myprgenie~~.~~com~~/]		In [[noncommutative geometry]] and related branches of mathematics, '''cyclic homology''' and '''cyclic cohomology''' are certain (co)homology theories for [[associative algebra]]s which generalize the [[de Rham cohomology\|de Rham (co)homology]] of manifolds. These notions were independently introduced by [[Alain Connes]] (cohomology)<ref>Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.
			</ref> and Boris Tsygan (homology)<ref>Boris L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983. Translation in Russ. Math. Survey 38(2) (1983), 198–199.
			</ref> around 1980. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the [[K-theory]]. The principal contributors to the development of theory include [[Max Karoubi]], Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, {{Link-interwiki\|en=Jean-Louis Loday\|lang=fr}}, Victor Nistor, [[Daniel Quillen]], Joachim Cuntz, Ryszard Nest, Ralf Meyer, Michael Puschnigg, and many others.

			== Hints about definition ==

			The first definition of the cyclic homology of a ring ''A'' over a field of [[characteristic (algebra)\|characteristic]] zero, denoted

			:''HC''<sub>''n''</sub>(''A'') or ''H''<sub>''n''</sub><sup>λ</sup>(''A''),

			proceeded by the means of an explicit [[chain complex]] related to the [[Hochschild homology\|Hochschild homology complex]] of ''A''. Connes later found a more categorical approach to cyclic homology using a notion of '''cyclic object''' in an [[abelian category]], which is analogous to the notion of [[simplicial object]]. In this way, cyclic homology (and cohomology) may be interpreted as a [[derived functor]], which can be explicitly computed by the means of the (''b'', ''B'')-bicomplex.

			One of the striking features of cyclic homology is the existence of a long exact sequence connecting
			Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence.

			== Case of commutative rings ==

			Cyclic cohomology of the commutative algebra ''A'' of regular functions on an [[affine algebraic variety]] over a field ''k'' of characteristic zero can be computed in terms of [[Grothendieck]]'s [[crystalline cohomology\|algebraic de Rham complex]].<ref>Boris L. Fegin and Boris L. Tsygan. Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen., 19(2):52–62, 96, 1985.</ref> In particular, if the variety ''V''=Spec ''A'' is smooth, cyclic cohomology of ''A'' are expressed in terms of the [[de Rham cohomology]] of ''V'' as follows:
			:<math> HC_n(A)\simeq \Omega^n\!A/d\Omega^{n-1}\!A\oplus \bigoplus_{i\geq 1}H^{n-2i}_{DR}(V).</math>

			This formula suggests a way to define de Rham cohomology for a 'noncommutative spectrum' of a noncommutative algebra ''A'', which was extensively developed by Connes.

			== Variants of cyclic homology ==

			One motivation of cyclic homology was the need for an approximation of [[K-theory]] that be defined, unlike K-theory, as the homology of a [[chain complex]]. Cyclic cohomology is in fact endowed with a pairing with K-theory, and one hopes this pairing to be non-degenerate.

			There has been defined a number of variants whose purpose is to fit better with algebras with topology, such as [[Fréchet algebra]]s, <math>C^</math>-algebras, etc. The reason is that K-theory behaves much better on topological algebras such as [[Banach algebra]]s or [[C-algebras]] than on algebras without additional structure. Since, on the other hand, cyclic homology degenerates on C*-algebras, there came up the need to define modified theories. Among them are entire cyclic homology due to [[Alain Connes]], analytic cyclic homology due to Ralf Meyer<ref>
			Ralf Meyer. Analytic cyclic cohomology. PhD thesis, Universität Münster, 1999</ref> or asymptotic and local cyclic homology due to Michael Puschnigg.<ref>Michael Puschnigg. Diffeotopy functors of ind-algebras and local cyclic cohomology. Doc.
			Math., 8:143–245 (electronic), 2003.</ref> The last one is very near to [[K-theory]] as it is endowed with a bivariant [[Chern character]] from [[KK-theory]].

			==Applications==

			One of the applications of cyclic homology is to find new proofs and generalizations of the [[Atiyah-Singer index theorem]]. Among these generalizations are index theorems based on spectral triples<ref>Alain Connes and Henri Moscovici. The local index formula in noncommutative geometry. Geom. Funct. Anal., 5(2):174–243, 1995.</ref> and [[deformation quantization]] of Poisson structures.<ref>Ryszard Nest and Boris Tsygan. Algebraic index theorem. Comm. Math. Phys., 172(2):223–262, 1995.</ref><!-- needs to be a good representative of the theory, with enough context and relevance. The index theorem for quantum tori is linked to the [[quantum Hall effect]],<ref>http://citeseer.ist.psu.edu/old/404503.html</ref> and the index theorem for deformation quantization to the study of band energy redistribution in the [[Born-Oppenheimer approximation]] in molecular physics.<ref>http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.3618</ref> -->

			==See also==

			*[[Hochschild homology]]
			*[[Noncommutative geometry]]
			*[[Homology (mathematics)\|Homology]]
			*[[Homology theory]]

			== References ==
			<references/>

			* Jean-Louis Loday, ''Cyclic Homology'', Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0

			==External links==
			*[http://mathsci.kaist.ac.kr/~jinhyun/note/cyclic/cyclic.pdf A personal note on Hochschild and Cyclic homology]

			{{DEFAULTSORT:Cyclic Homology}}
			[[Category:Homological algebra]]

en>Stevenj: rv Karimi paper spam

2012-08-08T18:29:30Z

rv Karimi paper spam

New page

Greetings. The [http://calorielab.com/news/diet-delivery/ writer's] name is Phebe and she feels healthy food delivery comfortable when people use the full name. In her professional life she is a payroll clerk but she's always needed her own business. diet meal delivery For years he's meal delivery service been residing in [http://healthyeater.com/diet-food-delivery-services North Dakota] and his family enjoys it. Doing ceramics is what my family members and I appreciate.<br><br>Stop by my weblog - [http://www.Myprgenie.com/view-publication/how-to-be-successful-with-your-weight-loss-goals http://www.Myprgenie.com/]