en>NOrbeck: Removed redundant reference

2013-07-25T23:40:42Z

Removed redundant reference

← Older revision		Revision as of 01:40, 26 July 2013
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	~~Emilia Shryock~~ is ~~my name but you can contact me anything you like~~. ~~Years ago we moved to Puerto Rico and my family loves it~~. ~~Since she~~ was ~~eighteen she~~'s ~~been working~~ as a ~~meter reader but she~~'s ~~usually needed her personal business~~. It's ~~not a [http:~~//~~Www~~.~~Stopthinkbesafe~~.~~org~~/~~youngpeople~~/~~hotline~~.~~asp typical factor] but what she likes~~ ~~home std test performing~~ is ~~foundation jumping and now she~~ is ~~trying~~ to ~~make money with it~~.<br><br>~~Feel free to visit~~ [~~http~~://~~www~~.~~fanproof~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~115513~~/~~info~~/ ~~std home test~~] ~~my webpage: at home~~ [~~http~~://~~3bbc~~.~~com~~/~~index~~.~~php?do=~~/~~profile-548128~~/~~info~~/ ~~over the counter std test~~] ~~test~~ [[http://~~www~~.~~pponline~~.co.uk/~~user~~/~~miriamlinswkucrd a knockout post~~]]		The '''Minakshisundaram–Pleijel zeta function''' is a [[Zeta function (operator)\|zeta function]] encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced
			by {{harvs\|txt\|author1-link=Subbaramiah Minakshisundaram \|first1=Subbaramiah \|last1=Minakshisundaram \|author2-link=Åke Pleijel\|first2=Åke \|last2=Pleijel\|year=1949}}. The case of a compact region of the plane was treated earlier by {{harvtxt\|Carleman\|1935}}.

			==Definition==

			For a compact Riemannian manifold ''M'' of dimension ''N'' with eigenvalues
			<math>\lambda_1, \lambda_2, \ldots </math> of the [[Laplace–Beltrami operator]] Δ the zeta function
			is given for <math>\operatorname{Re}(s) </math> sufficiently large by

			:<math> Z(s) = \operatorname{Tr}(\Delta^{-s}) = \sum_{n=1}^{\infty} \vert \lambda_{n} \vert^{-s}.</math>

			(where if an eigenvalue is zero it is omitted in the sum). The manifold may have a boundary, in which case one has to prescribe suitable boundary conditions, such as Dirichlet or Neumann boundary conditions.

			More generally one can define

			:<math> Z(P, Q, s) = \sum_{n=1}^{\infty} \frac{f_n(P)f_n(Q)}{ \lambda_{n}^s}</math>

			for ''P'' and ''Q'' on the manifold, where the ''f''<sub>''n''</sub> are normalized eigenfunctions. This can be analytically continued to a meromorphic function of ''s'' for all complex ''s'', and is holomorphic for ''P''≠''Q''.

			The only possible poles are simple poles at the points ''s'' = ''N''/2, ''N''/2−1, ''N''/2−2,..., 1/2,−1/2, −3/2,... for ''N'' odd, and at the points ''s'' = ''N''/2, ''N''/2−1, ''N''/2−2, ...,2, 1 for ''N'' even. If ''N'' is odd then ''Z''(''P'',''P'',''s'') vanishes at ''s'' = 0, −1, −2,... If ''N'' is even its values can be explicitly by [[Wiener-Ikehara theorem]] as a corollary the relation

			:<math>\displaystyle Z(P,P,s)\sim\frac{T^{N/2}}{(2\sqrt{\pi})^N\Gamma(N/2+1)} </math>

			where the sign ~ indicates that the quatient of both the sides tend to 1 when T tends to +∞.

			The function ''Z''(''s'') can be recovered from this by integrating ''Z''(''P'', ''P'', ''s'') over the whole manifold ''M'':
			:<math>\displaystyle Z(s) = \int_M Z(P,P,s)dP</math>

			==Heat kernel==

			The analytic continuation of the zeta function can be found by expressing it in terms of the [[heat kernel]]
			:<math> K(P,Q,t) = \sum_{n=1}^{\infty} f_n(P)f_n(Q) e^{- \lambda_{n}t} </math>

			as the [[Mellin transform]]
			:<math> Z(P,Q,s) = \frac{1}{\Gamma(s)} \int_0^\infty K(P,Q,t) t^{s-1} dt </math>

			In case of heat kernel, for given a Riemannian manifold (M,g) we can take the orthonormal basis of eigenfunctions and obtain the partition function
			:<math> Z(s)=\sum^\infty_{i=1}e^{-\lambda_i s}. </math>

			The poles of the zeta function can be found from the asymptotic behavior of the heat kernel as ''t''→0.

			==Example==

			If the manifold is a circle of dimension ''N''=1, then the eigenvalues of the Laplacian are ''n''<sup>2</sup> for integers ''n''. The zeta function
			:<math>Z(s) = \sum_{n\ne 0}\frac{1}{(n^2)^s} = \zeta(2s)</math>
			where ζ is the [[Riemann zeta function]].

			==Applications==

			Apply the method of heat kernel to asymptotic expansion for Riemannian manifold (M,g) we obtain the two following theorems. Both are the resolutions of the inverse problem in which we get the geometric properties or quantities from spectra of the operators.

			1,Minakshisundaram-Pleijel Asymptotic Expansion

			Let (M,g) be an ''n''-dimensional Riemannian manifold. The following asymptotic expansions hold as ''t''→0+:
			:<math> Z(s)\sim(4\pi s)^{-n/2}\sum^\infty_{m=0}a_ms^m. </math>
			In dim=2, this means that the integral of [[scalar curvature]] tells us the [[Euler characteristic]] of M, i.e. [[Gauss-Bonnet Theorem]].

			In particular,
			:<math> a_0=Vol(M,g),\ \ \ \ a_1=\frac{1}{6}\int_MS(x)dV </math>
			where S(x) is scalar curvature, the trace of the [[Ricci curvature]], on M.

			2,Weyl Asymptotic Formula
			Let M be a compact Riemannian manifold, with eigenvalues
			<math> 0=\lambda_0\le\lambda_1\le\lambda_2\cdots, </math>
			with each distinct eigenvalue repeated with its multiplicity. Define N(λ) to be the number of eigen values less than or equal to <math>\lambda</math>, and let <math>\omega_n</math> denote the volume of the unit disk in <math>\mathbb{R}^n</math>. Then
			:<math>N(\lambda)\sim\frac{\omega_n Vol(M)\lambda^{n/2}}{(2\pi)^n},</math>
			as λ→∞. Additionally, as k→∞,
			:<math> (\lambda_k)^{n/2}\sim\frac{(2\pi)^nk}{\omega_n Vol(M)}.</math>
			This is also called [[Weyl's law]], refined from Minakshisundaram-Pleijel asymptotic expansion.

			==References==

			*{{Citation \| last1=Berger \| first1=Marcel \| author1-link=Marcel Berger \| last2=Gauduchon \| first2=Paul \| last3=Mazet \| first3=Edmond \| title=Le spectre d'une variété riemannienne \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics \| doi=10.1007/BFb0064643 \| mr=0282313 \| year=1971 \| volume=194}}
			*{{Citation \| last1=Carleman \| first1=Torsten \| title=Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes. \| url=http://books.google.com/books?id=ZdHGSgAACAAJ \| language=French \| zbl=0012.07001 \| year=1935 \| journal=8. Skand. Mat.-Kongr. \| pages=34–44}}
			*{{Citation \| last1=Minakshisundaram \| first1=S. \| last2=Pleijel \| first2=Å. \| title=Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds \| url=http://math.ca/10.4153/CJM-1949-021-5 \| doi=10.4153/CJM-1949-021-5 \| mr=0031145 \| year=1949 \| journal=[[Canadian Journal of Mathematics]] \| issn=0008-414X \| volume=1 \| pages=242–256}}

			{{DEFAULTSORT:Minakshisundaram-Pleijel zeta function}}
			[[Category:Harmonic analysis]]
			[[Category:Differential geometry]]
			[[Category:Zeta and L-functions]]

en>BD2412: minor fixes, mostly disambig links, replaced: half-space → half-space using AWB

2012-06-14T12:20:31Z

minor fixes, mostly disambig links, replaced: half-space → half-space using AWB

New page

Emilia Shryock is my name but you can contact me anything you like. Years ago we moved to Puerto Rico and my family loves it. Since she was eighteen she's been working as a meter reader but she's usually needed her personal business. It's not a [http://Www.Stopthinkbesafe.org/youngpeople/hotline.asp typical factor] but what she likes home std test performing is foundation jumping and now she is trying to make money with it.<br><br>Feel free to visit [http://www.fanproof.com/index.php?do=/profile-115513/info/ std home test] my webpage: at home [http://3bbc.com/index.php?do=/profile-548128/info/ over the counter std test] test [[http://www.pponline.co.uk/user/miriamlinswkucrd a knockout post]]

Phase boundary - Revision history

en>NOrbeck: Removed redundant reference

en>BD2412: minor fixes, mostly disambig links, replaced: half-space → half-space using AWB