This is TeX, Version 3.14159 (MiKTeX 1.20e) (preloaded format=latex 2001.7.15) 22 JAN 2002 19:44 **Book.tex (Book.tex LaTeX2e <2000/06/01> Babel and hyphenation patterns for american, french, german, ngerman, n ohyphenation, loaded. (C:\texmf\tex\latex\amscls\amsbook.cls Document Class: amsbook 1999/11/29 v2.0 \linespacing=\dimen102 \normalparindent=\dimen103 \normaltopskip=\skip41 (C:\texmf\tex\latex\amslatex\amsgen.sty File: amsgen 1996/10/29 v1.2b \@emptytoks=\toks14 \ex@=\dimen104 ) (C:\texmf\tex\latex\amslatex\amsmath.sty Package: amsmath 1997/03/20 v1.2d AMS math features \@mathmargin=\skip42 (C:\texmf\tex\latex\amslatex\amstext.sty Package: amstext 1996/10/28 v1.2b (C:\texmf\tex\latex\amslatex\amsgen.sty File: amsgen 1996/10/29 v1.2b \@emptytoks=\toks15 \ex@=\dimen105 )) (C:\texmf\tex\latex\amslatex\amsbsy.sty Package: amsbsy 1996/10/28 v1.2b \pmbraise@=\dimen106 ) (C:\texmf\tex\latex\amslatex\amsopn.sty Package: amsopn 1996/10/28 v1.2b operator names ) LaTeX Info: Redefining \frac on input line 188. \uproot@=\count79 \leftroot@=\count80 \classnum@=\count81 \DOTSCASE@=\count82 LaTeX Info: Redefining \dots on input line 335. LaTeX Info: Redefining \ldots on input line 418. 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(C:\texmf\tex\latex\amsfonts\umsa.fd File: umsa.fd 1995/01/05 v2.2e AMS font definitions ) (C:\texmf\tex\latex\amsfonts\amsfonts.sty Package: amsfonts 1997/09/17 v2.2e \symAMSa=\mathgroup4 \symAMSb=\mathgroup5 LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' (Font) U/euf/m/n --> U/euf/b/n on input line 133. ) \copyins=\insert233 \abstractbox=\box28 \listisep=\skip45 \c@part=\count93 \c@chapter=\count94 \c@section=\count95 \c@subsection=\count96 \c@subsubsection=\count97 \c@paragraph=\count98 \c@subparagraph=\count99 \c@figure=\count100 \c@table=\count101 \abovecaptionskip=\skip46 \belowcaptionskip=\skip47 \captionindent=\dimen116 \thm@style=\toks19 \thm@bodyfont=\toks20 \thm@headfont=\toks21 \thm@notefont=\toks22 \thm@headpunct=\toks23 \thm@preskip=\skip48 \thm@postskip=\skip49 \thm@headsep=\skip50 \dth@everypar=\toks24 ) (C:\texmf\tex\generic\dvips\epsf.sty This is `epsf.tex' v2.7 <25 October 1996> \epsffilein=\read1 \epsfframemargin=\dimen117 \epsfframethickness=\dimen118 \epsfrsize=\dimen119 \epsftmp=\dimen120 \epsftsize=\dimen121 \epsfxsize=\dimen122 \epsfysize=\dimen123 \pspoints=\dimen124 ) (C:\texmf\tex\latex\amsfonts\amssymb.sty Package: amssymb 1996/11/03 v2.2b ) (C:\texmf\tex\latex\graphics\graphicx.sty Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR) (C:\texmf\tex\latex\graphics\keyval.sty Package: keyval 1999/03/16 v1.13 key=value parser (DPC) \KV@toks@=\toks25 ) (C:\texmf\tex\latex\graphics\graphics.sty Package: graphics 1999/02/16 v1.0l Standard LaTeX Graphics (DPC,SPQR) (C:\texmf\tex\latex\graphics\trig.sty Package: trig 1999/03/16 v1.09 sin cos tan (DPC) ) (C:\texmf\tex\latex\config\graphics.cfg) Package graphics Info: Driver file: dvips.def on input line 80. (C:\texmf\tex\latex\graphics\dvips.def File: dvips.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR) )) \Gin@req@height=\dimen125 \Gin@req@width=\dimen126 ) (C:\texmf\tex\latex\tools\delarray.sty Package: delarray 1994/03/14 v1.01 array delimiter package (DPC) (C:\texmf\tex\latex\tools\array.sty Package: array 1998/05/13 v2.3m Tabular extension package (FMi) \col@sep=\dimen127 \extrarowheight=\dimen128 \NC@list=\toks26 \extratabsurround=\skip51 \backup@length=\skip52 )) (C:\texmf\tex\latex\tools\enumerate.sty Package: enumerate 1999/03/05 v3.00 enumerate extensions (DPC) \@enLab=\toks27 ) (amsmidx.sty) \book-a@idxfile=\write3 Writing index file book-a.idx \book-b@idxfile=\write4 Writing index file book-b.idx \book-c@idxfile=\write5 Writing index file book-c.idx (bibmods.sty) \c@theorem=\count102 \c@lemma=\count103 \c@example=\count104 \c@proposition=\count105 \c@corollary=\count106 \c@problem=\count107 \c@exercise=\count108 \c@definition=\count109 \c@remark=\count110 (Book.aux (Preface.aux) (intro.aux) (ch-0.aux) (ch-1.aux) (ch-2.aux) (ch-3.aux) (ch-4.aux) (ch-5.aux) (ch-6.aux) (ch-7.aux) (ch-8.aux) (ch-9.aux) (app-A.aux)) LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 143. 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(C:\texmf\tex\latex\amsfonts\umsb.fd File: umsb.fd 1995/01/05 v2.2e AMS font definitions ) [1 ] LaTeX Font Info: Try loading font information for OMS+cmr on input line 151. (C:\texmf\tex\latex\base\omscmr.fd File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions ) LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10.95> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 151. [1] (Preface.tex) [2 ] Contents (Book.toc [3 ] [4]) \tf@toc=\write6 [5] (book-c.ind [6 ] Glossary of notation [7 ] [8]) (intro.tex Introduction [1 ] LaTeX Font Warning: Font shape `OT1/cmr/bx/sc' undefined (Font) using `OT1/cmr/bx/n' instead on input line 112. [2] [3]) [4] (ch-0.tex Chapter 0 [5 ] LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <10> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 201. [6] [7] [8] [9] [10] [11] [12] LaTeX Warning: Command \L invalid in math mode on input line 884. [13] [14] [15] [16] [17] [18] Overfull \hbox (13.6395pt too wide) in paragraph at lines 1614--1617 [][]\OT1/cmr/m/n/10.95 (b) with any two per-mu-ta-tions $\OML/cmm/m/it/10.95 [ ]; []$\OT1/cmr/m/n/10.95 , their \OT1/cmr/m/it/10.95 prod-uct $\OML/cmm/m/it/1 0.95 [][]$\OT1/cmr/m/n/10.95 , de-fined as $\OML/cmm/m/it/10.95 []$ \OT1/cmr /m/n/10.95 ap-plied to $\OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 []\OT1/cmr/m/ n/10.95 (1)\OML/cmm/m/it/10.95 ; []\OT1/cmr/m/n/10.95 (2)\OML/cmm/m/it/10.95 ; [] ; []\OT1/cmr/m/n/10.95 (\OML/cmm/m/it/10.95 n\OT1/cmr/m/n/10.95 )\OMS/cms y/m/n/10.95 g$\OT1/cmr/m/n/10.95 , | [] [19] [20] [21] LaTeX Font Warning: Command \small invalid in math mode on input line 2024. LaTeX Font Warning: Command \small invalid in math mode on input line 2024. LaTeX Font Warning: Command \small invalid in math mode on input line 2024. LaTeX Font Warning: Command \small invalid in math mode on input line 2024. 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[28] [29] [30] LaTeX Font Warning: Command \tiny invalid in math mode on input line 2875. LaTeX Font Warning: Command \tiny invalid in math mode on input line 2875. LaTeX Font Warning: Command \tiny invalid in math mode on input line 2878. LaTeX Font Warning: Command \tiny invalid in math mode on input line 2884. [31] LaTeX Font Warning: Command \tiny invalid in math mode on input line 2894. [32] [33] LaTeX Font Info: Font shape `OMS/cmr/bx/n' in size <10> not available (Font) Font shape `OMS/cmsy/b/n' tried instead on input line 3096. ) [34] (ch-1.tex Chapter 1 [35 ] [36] [37] [38] [39] [40] [41] [42] [43]) [44] (ch-2.tex Chapter 2 [45 ] [46] [47] LaTeX Font Warning: Command \small invalid in math mode on input line 350. LaTeX Font Warning: Command \small invalid in math mode on input line 350. LaTeX Font Warning: Command \small invalid in math mode on input line 372. LaTeX Font Warning: Command \small invalid in math mode on input line 372. 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[88]) [89] (ch-3.tex [90 ] Chapter 3 [91] [92] [93] [94] [95] LaTeX Font Warning: Command \small invalid in math mode on input line 504. LaTeX Font Warning: Command \small invalid in math mode on input line 504. LaTeX Font Warning: Command \small invalid in math mode on input line 513. LaTeX Font Warning: Command \small invalid in math mode on input line 513. LaTeX Font Warning: Command \small invalid in math mode on input line 518. LaTeX Font Warning: Command \small invalid in math mode on input line 518. [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] LaTeX Font Info: Font shape `OMS/cmr/m/n' in size <9> not available (Font) Font shape `OMS/cmsy/m/n' tried instead on input line 1659. [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] LaTeX Font Warning: Command \small invalid in math mode on input line 3201. LaTeX Font Warning: Command \small invalid in math mode on input line 3201. [125] [126] [127] [128] [129] [130] LaTeX Font Warning: Command \small invalid in math mode on input line 3761. LaTeX Font Warning: Command \small invalid in math mode on input line 3761. [131] LaTeX Font Warning: Command \small invalid in math mode on input line 3809. LaTeX Font Warning: Command \small invalid in math mode on input line 3809. [132] LaTeX Font Warning: Command \small invalid in math mode on input line 3876. LaTeX Font Warning: Command \small invalid in math mode on input line 3876. LaTeX Font Warning: Command \small invalid in math mode on input line 3887. LaTeX Font Warning: Command \small invalid in math mode on input line 3887. LaTeX Font Warning: Command \small invalid in math mode on input line 3928. LaTeX Font Warning: Command \small invalid in math mode on input line 3928. [133]) [134] (ch-4.tex Chapter 4 [135 ] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1070. LaTeX Font Warning: Command \small invalid in math mode on input line 1106. LaTeX Font Warning: Command \small invalid in math mode on input line 1106. LaTeX Font Warning: Command \small invalid in math mode on input line 1106. LaTeX Font Warning: Command \small invalid in math mode on input line 1106. [146] [147] Overfull \hbox (25.03271pt too wide) in paragraph at lines 1260--1270 []\OMS/cmsy/m/n/10 f[] \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 AB\OT1/cmr/m/n/10 )\OM L/cmm/m/it/10 ; [] \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 BA\OT1/cmr/m/n/10 )\OMS/cm sy/m/n/10 g$\OT1/cmr/m/n/10 . Then $(\OML/cmm/m/it/10 AB\OT1/cmr/m/n/10 )[]\OML /cmm/m/it/10 Y \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 AB\OT1/cmr/m/n/10 )[]\OML/cm m/m/it/10 A\OT1/cmr/m/n/10 [(\OML/cmm/m/it/10 BA\OT1/cmr/m/n/10 )[]][]\OML/cmm/ m/it/10 B \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 AB\OT1/cmr/m/n/10 )[]\OML/cmm/m/i t/10 ABA\OT1/cmr/m/n/10 [(\OML/cmm/m/it/10 BA\OT1/cmr/m/n/10 )[]][]\OML/cmm/m/i t/10 B \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 AB\OT1/cmr/m/n/10 )[]\OML/cmm/m/it/1 0 A\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 BA\OT1/cmr/m/n/10 )[]\OML/cmm/m/it/10 B \O T1/cmr/m/n/10 =| [] [148] [149] [150] [151] [152] [153]) [154] (ch-5.tex Chapter 5 [155 ] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] LaTeX Font Warning: Command \small invalid in math mode on input line 1413. LaTeX Font Warning: Command \small invalid in math mode on input line 1413. LaTeX Font Warning: Command \small invalid in math mode on input line 1413. 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[200] [201] [202] [203] [204] [205] [206] [207] Overfull \hbox (12.8145pt too wide) in paragraph at lines 2734--2738 [][]\OT1/cmr/m/n/10.95 The geo-met-ric mean-ing of $[] (\OML/cmm/m/it/10.95 L; M\OT1/cmr/m/n/10.95 )$ is given in the next lemma, which fol-lows from the CS-- decomposition| [] [208] [209] LaTeX Font Warning: Command \tiny invalid in math mode on input line 3015. [210] LaTeX Font Warning: Command \small invalid in math mode on input line 3061. LaTeX Font Warning: Command \small invalid in math mode on input line 3061. [211] LaTeX Font Warning: Command \small invalid in math mode on input line 3170. LaTeX Font Warning: Command \small invalid in math mode on input line 3170. [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227]) [228] (ch-7.tex Chapter 7 [229 ] [230] [231] [232] [233] [234] [235] [236] [237] LaTeX Font Warning: Command \small invalid in math mode on input line 934. 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[324] [325] [326] [327] [328]) [329] (app-A.tex [330 ] Appendix A LaTeX Font Info: Try loading font information for U+euf on input line 57. 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