141.70.93.9: /* Recursion */

2013-11-12T17:20:56Z

Recursion

← Older revision		Revision as of 19:20, 12 November 2013
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	I~~'m Malinda~~ and ~~I live in~~ a ~~seaside city in northern Canada~~, ~~Calgary. I'm 28~~ and I'~~m will soon finish my study~~ at ~~Continuing Education and Summer Sessions~~.<br><br>~~Visit my webpage [http~~://~~www~~.~~3cheers2u~~.~~com~~/~~LinkExchange~~.~~html cheapdomains~~.~~com~~]		In mathematics, the '''eta invariant''' of a self-adjoint [[elliptic differential operator\|elliptic]] [[differential operator]] on a [[compact manifold]] is formally the number of positive [[eigenvalue]]s minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using [[zeta function regularization]]. It was introduced by {{harvs\|txt\|last1=Atiyah\|author1-link=Michael Atiyah\|last2=Patodi\|author2-link=Vijay Kumar Patodi\|last3=Singer\|author3-link=Isadore Singer\|year1=1973\|year2=1975}} who used it to extend the [[Hirzebruch signature theorem]] to manifolds with boundary.

			They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd dimensional smooth manifold.

			{{harvs\|txt \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last2=Donnelly \| first2=H. \| last3=Singer \| first3=I. M. \| title=Eta invariants, signature defects of cusps, and values of L-functions \| doi=10.2307/2006957 \| doi=10.2307/2006957 \| mr=707164 \| year=1983 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=118 \| issue=1 \| pages=131–177}}
			defined the [[signature defect]] of the boundary of a manifold as the eta invariant, and used this to show that Hirzebruch's signature defect of a cusp of a [[Hilbert modular surface]] can be expressed in terms of the value at ''s''=0 or 1 of a [[Shimizu L-function]].

			==Definition==

			The eta invariant of self-adjoint operator ''A'' is given by ''η''<sub>''A''</sub>(0), where ''η'' is the analytic continuation of

			:<math>\eta(s)=\sum_{\lambda\ne 0} \frac{\operatorname{sign}(\lambda)}{\|\lambda\|^s}</math>

			and the sum is over the nonzero eigenvalues λ of ''A''.

			==References==

			*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last2=Patodi \| first2=V. K. \| last3=Singer \| first3=I. M. \| title=Spectral asymmetry and Riemannian geometry \| doi=10.1112/blms/5.2.229 \| mr=0331443 \| year=1973 \| journal=The Bulletin of the London Mathematical Society \| issn=0024-6093 \| volume=5 \| pages=229–234}}
			*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last2=Patodi \| first2=V. K. \| last3=Singer \| first3=I. M. \| title=Spectral asymmetry and Riemannian geometry. I \| doi=10.1017/S0305004100049410 \| mr=0397797 \| year=1975 \| journal=Mathematical Proceedings of the Cambridge Philosophical Society \| issn=0305-0041 \| volume=77 \| pages=43–69}}
			*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last2=Donnelly \| first2=H. \| last3=Singer \| first3=I. M. \| title=Eta invariants, signature defects of cusps, and values of L-functions \| doi=10.2307/2006957 \| mr=707164 \| year=1983 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=118 \| issue=1 \| pages=131–177}}

			[[Category:Differential operators]]

en>Katharineamy: added Category:Sequence alignment algorithms; removed {{uncategorized}} using HotCat

2012-03-24T01:01:14Z

added Category:Sequence alignment algorithms; removed {{uncategorized}} using HotCat

New page

I'm Malinda and I live in a seaside city in northern Canada, Calgary. I'm 28 and I'm will soon finish my study at Continuing Education and Summer Sessions.<br><br>Visit my webpage [http://www.3cheers2u.com/LinkExchange.html cheapdomains.com]

Probalign - Revision history

141.70.93.9: /* Recursion */

en>Katharineamy: added Category:Sequence alignment algorithms; removed {{uncategorized}} using HotCat