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Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `subscript' on input line 466. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref) removing `math shift' on input line 466. Package hyperref Warning: Composite letter `\textacute+3' (hyperref) not defined in PD1 encoding, (hyperref) removing `\textacute' on input line 466. Package hyperref Warning: Composite letter `\textdotaccent+"' (hyperref) not defined in PD1 encoding, (hyperref) removing `\textdotaccent' on input line 466. Overfull \hbox (3.99167pt too wide) in paragraph at lines 490--491 []\T1/LinuxLibertineT-TLF/m/n/12 The group $\OT1/cmr/m/n/12 ^^@$ \T1/LinuxLiber tineT-TLF/m/n/12 acts dis-cretely on $[][]$ by Möbius trans-for-ma-tions. The t he-o-rem of Cerednik-- [] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [8] Overfull \vbox (6.183pt too high) has occurred while \output is active [] [9] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [10pdfTeX warning (ext4): destination with the same identifier (name{equation.2 .16}) has been already used, duplicate ignored \@EveryShipout@Output ...@Org@Shipout \box \@cclv l.652 \end{equation} ] Overfull \hbox (15.49968pt too wide) in paragraph at lines 690--692 [][]\T1/LinuxLibertineT-TLF/b/n/12 Proposition 7. []\T1/LinuxLibertineT-TLF/m/i t/12 The $[]$-valued pair-ing $[][]\OT1/cmr/m/n/12 (\OMS/cmsy/m/n/12 h \OML/cm m/m/it/12 ; \OMS/cmsy/m/n/12 i[]\OT1/cmr/m/n/12 )$ \T1/LinuxLibertineT-TLF/m/i t/12 is pos-i-tive def-i-nite and hence non-degenerate. 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Tak-ing its ho-mol-ogy and in-vok-ing Shapiro's lemma shows th at $[][]\OT1/cmr/m/n/12 (^^@\OML/cmm/m/it/12 ; []\OT1/cmr/m/n/12 )$ [] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [18] Overfull \hbox (1.8348pt too wide) in paragraph at lines 1155--1160 []\T1/LinuxLibertineT-TLF/m/n/12 The iden-ti-fi-ca-tion of $[][]\OT1/cmr/m/n/12 (^^@\OML/cmm/m/it/12 ; [][]\OT1/cmr/m/n/12 )$ \T1/LinuxLibertineT-TLF/m/n/12 w ith $[][]\OT1/cmr/m/n/12 (^^@[]\OML/cmm/m/it/12 ; []\OT1/cmr/m/n/12 )[]$ \T1/Li nuxLibertineT-TLF/m/n/12 fol-lows from Shapiro's lemma, [] Overfull \vbox (6.183pt too high) has occurred while \output is active [] [19] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [20] Overfull \vbox (6.183pt too high) has occurred while \output is active [] [21] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [22] Overfull \vbox (6.183pt too high) has occurred while \output is active [] [23] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [24] Overfull \vbox (6.183pt too high) has occurred while \output is active [] [25] Overfull \hbox (1.61398pt too wide) in paragraph at lines 1601--1607 []\T1/LinuxLibertineT-TLF/m/n/12 The ra-tio-nal func-tion on the right hand sid e of [] has di-vi-sor equal to $[][]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 ^^M\OT1/c mr/m/n/12 )$ [] Overfull \vbox (5.90399pt too high) has occurred while \output is active [] [26] Overfull \hbox (13.26575pt too wide) detected at line 1655 [] [] (./paper.brf) \tf@brf=\write8 \openout8 = `paper.brf'. 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