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	~~In [[mathematics]]~~, ~~the '''signature operator''' is~~ an ~~[[elliptic differential operator]] defined~~ on ~~a certain subspace of the space of [[differential~~ form~~]]s on an even-dimensional [[closed manifold\|compact]] [[Riemannian manifold]]~~, ~~whose [[Atiyah-Singer_index_theorem#The_analytical_index\|analytic index]] is~~ the ~~same as~~ the ~~[[Signature (topology)\|topological signature]] of~~ the ~~manifold if~~ the ~~dimension~~ of the ~~manifold is~~ a ~~multiple of four~~.<~~ref~~>~~{{Harvnb\|Atiyah\|Bott\|1967}}~~<~~/ref~~> ~~It is an instance of a Dirac-type operator.~~		If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. * A community forum for debate of the product together with some other customers in the comments spot. They found out all the possible information about bringing up your baby and save money at the same time. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br>

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	~~Let <math>M</math>~~ be a ~~compact [[Riemannian manifold]] of even dimension <math>2l</math>~~. ~~Let~~

	~~:<math> d : \Omega^p(M)\rightarrow \Omega^{p+1}(M) </math>~~

	be the ~~[[exterior derivative]] on <math>i</math>-th order [[differential forms]] on <math>M</math>. The Riemannian metric on <math>M</math> allows us~~ to ~~define~~ the ~~[[Hodge star operator]] <math>\star</math>~~ and ~~with~~ it ~~the inner product~~

	~~:<math>\langle\omega~~,~~\eta\rangle=\int_M\omega\wedge\star\eta</math>~~

	~~on forms. Denote by~~

	~~:<math> d^*: \Omega^{p+1}(M)\rightarrow \Omega^p(M) </math>~~

	the ~~[[Hermitian adjoint\|adjoint operator]] of~~ the ~~exterior differential~~ <~~math~~>d<~~/math~~>. ~~This operator can be expressed purely in terms~~ of the ~~Hodge star operator as follows:~~

	~~:<math>d^*= (-1)^{2l(p+1) + 2l + 1} \star d \star= - \star d \star</math>~~

	~~Now consider <math>d + d^*</math> acting on the space of all forms <math>\Omega(M)=\bigoplus_{p=0}^{2l}\Omega^{p}(M)</math>.~~
	~~One way~~ to ~~consider~~ this ~~as a graded operator~~ is ~~the following~~: ~~Let <math>\tau</math> be~~ an ~~[[Involution (mathematics)#Linear algebra\|involution]]~~ on ~~the space of ''all'' forms defined by:~~

	~~:<math> \tau(\omega)=i^{p(p-1)+l}\star \omega\quad~~,~~\quad\omega \in \Omega^p(M) </math>~~

	~~It is verified that <math>d + d^*</math> anti-commutes with <math>\tau</math>~~ and, ~~consequently~~, ~~switches the <math>(\pm 1) </math>~~-~~[[eigenspace]]s~~ ~~<math>\Omega_{\pm}(M)</math> of <math>\tau</math>~~

	~~Consequently,~~

	~~:<math> d + d^* = \begin{pmatrix} 0 & D \\ D^* & 0 \end{pmatrix}</math>~~

	~~'''Definition:''' The operator <math> d + d^*</math> with~~ the ~~above grading respectively the above operator <math>D~~: ~~\Omega_+(M) \rightarrow \Omega_-(M) <~~/~~math> is called the '''signature operator''' of <math>M<~~/~~math>~~.~~<ref>{{Harvnb\|Atiyah\|Bott\|1967}}<~~/~~ref>~~

	~~==Definition in~~ the ~~odd-dimensional case==~~

	~~In the odd dimensional case one defines the signature operator~~ to ~~be <math>i(d+d^*)\tau</math> acting~~
	~~on the~~ even~~-dimensional forms of <math>M</math>~~.

	~~==Hirzebruch Signature Theorem==~~

	If <~~math~~> ~~l = 2k~~ <~~/math~~>~~, so that~~ the ~~dimension~~ of ~~<math>M</math> is a multiple of four, then [[Hodge theory]] implies that:~~

	~~:<math>\mathrm{index}(D) = \mathrm{sign}(M) </math>~~

	~~where~~ the ~~right hand side is the [[Signature (topology)\|topological signature]] (''i~~.~~e.''~~ the ~~[[signature~~ of ~~a quadratic form]] on <math>H^{2k}(M)\ </math> defined by the [[cup product]])~~.

	~~The ''Heat Equation'' approach to~~ the ~~[[Atiyah-Singer index theorem]] can then be used~~ to ~~show that~~:

	~~:<math>\mathrm{sign}(M) = \int_M L(p_1~~,~~\ldots~~,~~p_l)~~ <~~/math~~>

	~~where~~ <~~math~~>~~L</math> is the [[Hirzebruch_L-polynomial#L_genus_and_the_Hirzebruch_signature_theorem\|Hirzebruch L-Polynomial]]~~,~~<ref>{{Harvnb\|Hirzebruch\|1995}}</ref>~~ and ~~the <math>p_i\ </math> the [[Pontryagin class\|Pontrjagin form]]s on <math>M</math>~~.~~<ref>{{Harvnb\|Gilkey\|1973}}~~, ~~{{Harvnb\|Atiyah\|Bott\|Patodi\|1973}}</ref>~~

	~~==Homotopy invariance of the higher indices==~~

	~~Kaminker~~ and ~~Miller proved that the higher indices of the signature operator are homotopy-invariant~~.~~<ref>{{Harvnb\|Kaminker\|Miller\|1985}}</ref>~~

	~~==See also==~~
	*[[Hirzebruch signature theorem]]
	*[[Pontryagin class]]
	*[[Friedrich Hirzebruch]]
	*[[Michael Atiyah]]
	*[[Isadore Singer]]
	~~==Notes==~~
	~~{{reflist\|2}}~~
	~~==References==~~
	*{{Citation \| last1 = Atiyah \| first1 = M.F. \| last2 = Bott \| first2 = R. \| title = A Lefschetz fixed-point formula for elliptic complexes I \| journal = Annals of Mathematics \| volume = 86 \| year = 1967 \| pages = 374–407}}
	*{{Citation \| last1 = Atiyah \| first1 = M.F. \| last2 = Bott \|first2= R. \| last3 = Patodi \|first3 = V.K.\| title = On the ~~heat equation~~ and the ~~index theorem \|journal = Inventiones Math~~. ~~\| volume = 19 \| year = 1973 \| pages = 279–330}}~~
	*{{Citation \|last1 = Gilkey \| first1 = P.B. \| title = Curvature and the eigenvalues of ~~the Laplacian for elliptic complexes \| journal = Advances~~ in ~~Mathematics \| year = 1973 \| volume = 10 \| pages = 344–382}}~~
	*{{Citation \| last = Hirzebruch \| first = Friedrich \| title = Topological Methods in Algebraic Geometry, 4th edition\|year = 1995 \| publisher = Berlin and Heidelberg: Springer-Verlag. ~~Pp. 234\|isbn= 3-540-58663-6}}~~
	{{Citation \| last1 = Kaminker \| first1 = Jerome \| last2 = Miller \| first2 = John G. \| title = Homotopy Invariance of the ~~Analytic Index of Signature Operators over C-Algebras \| journal = Journal of Operator Theory \| year = 1985 \| volume = 14 \| pages = 113–127 \| url = http://jot~~.~~theta.ro/jot/archive/1985-014-001/1985-014-001-006.pdf}}~~

	~~[[Category:Elliptic partial differential equations]]~~

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