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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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346:; 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
215:{\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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291:; Patodi, V. K.; Singer, I. M. (1975), "Spectral asymmetry and Riemannian geometry. I",
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242:; Patodi, V. K.; Singer, I. M. (1973), "Spectral asymmetry and Riemannian geometry",
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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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83:, H. Donnelly, and I. M. Singer (
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244:The Bulletin of the London Mathematical Society
111:The eta invariant of self-adjoint operator
95:can be expressed in terms of the value at
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38:is formally the number of positive
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128:is the analytic continuation of
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1:
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68:Hirzebruch signature theorem
66:) who used it to extend the
44:zeta function regularization
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81:Michael Francis Atiyah
313:10.1017/S0305004100049410
344:Atiyah, Michael Francis
289:Atiyah, Michael Francis
240:Atiyah, Michael Francis
93:Hilbert modular surface
46:. It was introduced by
402:Differential operators
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72:Dirichlet eta function
349:Annals of Mathematics
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32:differential operator
16:Differential operator
266:10.1112/blms/5.2.229
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305:1975MPCPS..77...43A
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101:Shimizu L-function
27:of a self-adjoint
352:, Second Series,
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89:signature defect
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356:(1): 131–177,
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87:) defined the
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99:=0 or 1 of a
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25:eta invariant
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299:(1): 43–69,
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115:is given by
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124:(0), where
40:eigenvalues
21:mathematics
233:References
107:Definition
370:0003-486X
321:0305-0041
274:0024-6093
252:CiteSeerX
195:λ
182:λ
176:
162:≠
159:λ
155:∑
139:η
396:Category
337:17638224
29:elliptic
386:0707164
378:2006957
329:0397797
301:Bibcode
282:0331443
58: (
384:
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368:
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254:
56:Singer
52:Patodi
50:,
48:Atiyah
23:, the
374:JSTOR
333:S2CID
34:on a
366:ISSN
317:ISSN
270:ISSN
173:sign
85:1983
64:1975
60:1973
358:doi
354:118
309:doi
262:doi
19:In
398::
382:MR
380:,
372:,
364:,
331:,
325:MR
323:,
315:,
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297:77
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278:MR
276:,
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248:5
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179:(
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151:=
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145:s
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126:η
121:A
117:η
113:A
97:s
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