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They also later used the eta invariant of a self-adjoint operator to define the eta invariant of a compact odd-dimensional smooth manifold.
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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
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357:; Donnelly, H.; Singer, I. M. (1983), "Eta invariants, signature defects of cusps, and values of L-functions",
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minus the number of negative eigenvalues. In practice both numbers are often infinite so are defined using
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226:{\displaystyle \eta (s)=\sum _{\lambda \neq 0}{\frac {\operatorname {sign} (\lambda )}{|\lambda |^{s}}}}
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to manifolds with boundary. The name comes from the fact that it is a generalization of the
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Mathematical
Proceedings of the Cambridge Philosophical Society
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and the sum is over the nonzero eigenvalues λ of
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94:, H. Donnelly, and I. M. Singer (
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255:The Bulletin of the London Mathematical Society
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122:The eta invariant of self-adjoint operator
106:can be expressed in terms of the value at
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49:is formally the number of positive
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18:Atiyah–Patodi–Singer eta invariant
139:is the analytic continuation of
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1:
79:Hirzebruch signature theorem
77:) who used it to extend the
55:zeta function regularization
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92:Michael Francis Atiyah
324:10.1017/S0305004100049410
355:Atiyah, Michael Francis
300:Atiyah, Michael Francis
251:Atiyah, Michael Francis
104:Hilbert modular surface
57:. It was introduced by
413:Differential operators
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83:Dirichlet eta function
360:Annals of Mathematics
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43:differential operator
27:Differential operator
277:10.1112/blms/5.2.229
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316:1975MPCPS..77...43A
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112:Shimizu L-function
38:of a self-adjoint
363:, Second Series,
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16:(Redirected from
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367:(1): 131–177,
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110:=0 or 1 of a
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36:eta invariant
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126:is given by
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135:(0), where
51:eigenvalues
32:mathematics
244:References
118:Definition
381:0003-486X
332:0305-0041
285:0024-6093
263:CiteSeerX
206:λ
193:λ
187:
173:≠
170:λ
166:∑
150:η
407:Category
348:17638224
40:elliptic
397:0707164
389:2006957
340:0397797
312:Bibcode
293:0331443
69: (
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67:Singer
63:Patodi
61:,
59:Atiyah
34:, the
385:JSTOR
344:S2CID
45:on a
377:ISSN
328:ISSN
281:ISSN
184:sign
96:1983
75:1975
71:1973
369:doi
365:118
320:doi
273:doi
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