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[8] \openout2 = `frontmatter/preface.aux'. (./frontmatter/preface.tex Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \hbox (badness 10000) in paragraph at lines 3--4 [] Underfull \hbox (badness 10000) in paragraph at lines 5--6 [] LaTeX Warning: Reference `ch6' on page ix undefined on input line 65. LaTeX Warning: Reference `ch7' on page ix undefined on input line 75. LaTeX Warning: Reference `ch8' on page ix undefined on input line 82. LaTeX Warning: Reference `ch9' on page ix undefined on input line 97. LaTeX Warning: Reference `ch10' on page ix undefined on input line 114. LaTeX Warning: Reference `ch11' on page ix undefined on input line 126. LaTeX Warning: Reference `ch12' on page ix undefined on input line 138. ) Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [9 ] [10] [11] \openout2 = `frontmatter/contributor.aux'. (./frontmatter/contributor.tex Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] ) Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [12 ] \openout2 = `chapters/chapter1/ch1.aux'. (./chapters/chapter1/ch1.tex Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Chapter 1. LaTeX Font Info: Calculating math sizes for size <8.5> on input line 6. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <8.5> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 6. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <5.94997> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 6. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <4.25> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 6. Overfull \hbox (9.50653pt too wide) in paragraph at lines 4--13 \OT1/ptm/m/it/10.5 Piergiorgio Vitello,$[]\OML/ztmcm/m/it/10.5 ;$\OT1/ptm/m/it/ 10.5 * An-drea Capponi,$[]$ Clau-dio Fiandrino$[]$ \OT1/ptm/m/n/10.5 and \OT1/p tm/m/it/10.5 Guido Cantelmo$[]$ [] Underfull \hbox (badness 10000) in paragraph at lines 25--26 [] Underfull \hbox (badness 10000) in paragraph at lines 28--29 [] Underfull \hbox (badness 10000) in paragraph at lines 30--31 [] Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [1 ] Underfull \hbox (badness 10000) in paragraph at lines 32--33 [] LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <10> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 40. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <7.4> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 40. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <6> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 40. [2] [3] [4] Underfull \hbox (badness 10000) in paragraph at lines 82--83 [] Underfull \hbox (badness 10000) in paragraph at lines 84--85 [] [5] LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <9> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 125. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <7> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 125. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <5> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 125. [6] [7] Underfull \hbox (badness 10000) in paragraph at lines 186--195 [] [8] Underfull \hbox (badness 10000) in paragraph at lines 215--217 [] [9] Underfull \hbox (badness 10000) in paragraph at lines 220--221 [] Underfull \hbox (badness 10000) in paragraph at lines 222--223 [] Underfull \hbox (badness 10000) in paragraph at lines 225--226 [] Underfull \hbox (badness 10000) in paragraph at lines 227--228 [] Overfull \hbox (14.40472pt too wide) in paragraph at lines 229--233 []\OT1/ptm/m/n/10.5 It is in-ter-est-ing to con-sider now the vec-tor spaces $\ U/msb/m/n/10.5 R[] \OT1/ztmcm/m/n/10.5 = \OMS/zmtt/m/n/10.5 ff\OML/ztmcm/m/it/1 0.5 x[]\OMS/zmtt/m/n/10.5 g[] \OT1/ztmcm/m/n/10.5 : \OML/ztmcm/m/it/10.5 x[] \O MS/zmtt/m/n/10.5 2 \U/msb/m/n/10.5 R []\OML/ztmcm/m/it/10.5 for []all []n\OMS/z mtt/m/n/10.5 g$\OT1/ptm/m/n/10.5 , [] [10] Underfull \hbox (badness 10000) in paragraph at lines 237--239 [] [11] Overfull \hbox (23.70163pt too wide) detected at line 246 \OML/ztmcm/m/it/10.5 P[] \OT1/ztmcm/m/n/10.5 = \OMS/zmtt/m/n/10.5 f\OML/ztmcm/m /it/10.5 f \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/10.5 V[] \OT1/ztmcm/m/n/10.5 : []\OML/ztmcm/m/it/10.5 f\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 t\OT1/ztmcm/ m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^U \OT1/ztmcm/m/n/10.5 0 []\OML/ztmcm/m/it/10.5 for []all []t \OMS/zmtt/m/n/10.5 2 \OT1/ztmcm/m/n/10.5 [0\OML/ztmcm/m/it/10.5 ; []\OT1/ztmcm/m/n/10.5 ) []\OML/ztmcm/m/it/10.5 and []f\OT1/ztmcm/m/n/10.5 ([]) = 0 []\OML/ztmcm/m/it/10.5 and []f\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 t\ OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OT1/ztmcm/m/n/10.5 0 []\OML/ztmcm/ m/it/10.5 for []all []t \OMS/zmtt/m/n/10.5 2 \OT1/ztmcm/m/n/10.5 ([]\OML/ztmcm/ m/it/10.5 ; \OT1/ztmcm/m/n/10.5 1]\OMS/zmtt/m/n/10.5 g\OML/ztmcm/m/it/10.5 : [] Underfull \hbox (badness 10000) in paragraph at lines 246--248 [] Underfull \hbox (badness 10000) in paragraph at lines 249--250 [] Underfull \hbox (badness 10000) in paragraph at lines 251--252 [] Underfull \hbox (badness 10000) in paragraph at lines 253--254 [] Underfull \hbox (badness 10000) in paragraph at lines 257--259 [] Underfull \hbox (badness 10000) in paragraph at lines 260--261 [] [12]) [13] \openout2 = `chapters/chapter2/ch2.aux'. (./chapters/chapter2/ch2.tex Underfull \vbox (badness 10000) detected at line 2 [] Overfull \hbox (336.25pt too wide) detected at line 2 [] [] Underfull \vbox (badness 10000) detected at line 2 [] Overfull \hbox (336.25pt too wide) detected at line 2 [] [] Underfull \vbox (badness 10000) detected at line 2 [] Overfull \hbox (336.25pt too wide) detected at line 2 [] [] Underfull \vbox (badness 10000) detected at line 2 [] Overfull \hbox (336.25pt too wide) detected at line 2 [] [] Chapter 2. Underfull \hbox (badness 10000) in paragraph at lines 22--23 [] Underfull \hbox (badness 10000) in paragraph at lines 24--25 [] Underfull \hbox (badness 10000) in paragraph at lines 26--27 [] Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [14 ] LaTeX Font Info: Font shape `U/lasy/b/n' in size <7.4> not available (Font) Font shape `U/lasy/m/n' tried instead on input line 49. LaTeX Font Info: Font shape `U/lasy/b/n' in size <6> not available (Font) Font shape `U/lasy/m/n' tried instead on input line 49. [15] Underfull \hbox (badness 10000) in paragraph at lines 83--94 [] Underfull \hbox (badness 10000) in paragraph at lines 95--96 [] Underfull \hbox (badness 10000) in paragraph at lines 107--112 [] [16] Underfull \hbox (badness 10000) in paragraph at lines 168--172 [] [17] Underfull \hbox (badness 10000) in paragraph at lines 184--188 [] [18] Underfull \hbox (badness 10000) in paragraph at lines 249--252 [] [19] Overfull \hbox (36.67903pt too wide) in paragraph at lines 287--289 \OT1/ptm/m/n/10.5 duced by the met-ric. In fact, $\OML/ztmcm/m/it/10.5 ^^\[] \O T1/ztmcm/m/n/10.5 = \OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 E \OMS/zmtt/m/n/10 .5 ^^R \OML/ztmcm/m/it/10.5 X \OT1/ztmcm/m/n/10.5 : \OML/ztmcm/m/it/10.5 E []is []an []open []set []in []the []sense []of []metric []spaces\OMS/zmtt/m/n/10.5 g$\OT1/ptm/m/n/10.5 . [] Underfull \hbox (badness 10000) in paragraph at lines 287--289 [] Underfull \hbox (badness 10000) in paragraph at lines 290--291 [] LaTeX Warning: Citation `R' on page 20 undefined on input line 299. LaTeX Warning: Citation `J1' on page 20 undefined on input line 299. LaTeX Warning: Citation `J2' on page 20 undefined on input line 299. LaTeX Warning: Citation `J3' on page 20 undefined on input line 299. LaTeX Warning: Citation `ST' on page 20 undefined on input line 299. LaTeX Warning: Citation `P' on page 20 undefined on input line 299. LaTeX Warning: Citation `Kran' on page 20 undefined on input line 299. LaTeX Warning: Citation `Van' on page 20 undefined on input line 299. LaTeX Warning: Citation `Zab' on page 20 undefined on input line 299. LaTeX Warning: Citation `HZ' on page 20 undefined on input line 299. [20] Underfull \hbox (badness 10000) in paragraph at lines 301--302 [] Underfull \hbox (badness 10000) in paragraph at lines 303--304 [] Underfull \hbox (badness 10000) in paragraph at lines 305--306 [] Underfull \hbox (badness 10000) in paragraph at lines 307--308 [] Underfull \hbox (badness 10000) in paragraph at lines 309--310 [] Underfull \hbox (badness 10000) in paragraph at lines 311--312 [] Underfull \hbox (badness 10000) in paragraph at lines 313--317 [] LaTeX Warning: Citation `HZ' on page 21 undefined on input line 321. Underfull \hbox (badness 10000) in paragraph at lines 320--322 [] Underfull \hbox (badness 10000) in paragraph at lines 323--324 [] [21] Underfull \hbox (badness 10000) in paragraph at lines 325--332 [] LaTeX Warning: Citation `HZ' on page 22 undefined on input line 333. Underfull \hbox (badness 10000) in paragraph at lines 335--340 [] LaTeX Warning: Citation `HZ' on page 22 undefined on input line 349. [22] LaTeX Warning: Citation `HZ' on page 23 undefined on input line 372. Underfull \hbox (badness 10000) in paragraph at lines 372--373 [] [23] [24] [25] [26] [27] LaTeX Warning: Citation `HZ' on page 28 undefined on input line 615. LaTeX Warning: Citation `HZ' on page 28 undefined on input line 617. [28] [29] LaTeX Warning: Citation `HZ' on page 30 undefined on input line 721. [30] LaTeX Warning: Citation `Mat92' on page 31 undefined on input line 762. LaTeX Warning: Citation `Mat92' on page 31 undefined on input line 776. LaTeX Warning: Citation `A2011' on page 31 undefined on input line 777. LaTeX Warning: Citation `Mat92' on page 31 undefined on input line 777. LaTeX Warning: Citation `IPR' on page 31 undefined on input line 784. LaTeX Warning: Citation `A2011' on page 31 undefined on input line 789. [31] LaTeX Warning: Citation `CZ' on page 32 undefined on input line 801. LaTeX Warning: Citation `CZ' on page 32 undefined on input line 815. LaTeX Warning: Citation `TS' on page 32 undefined on input line 818. LaTeX Warning: Citation `TS' on page 32 undefined on input line 818. LaTeX Warning: Citation `TS' on page 32 undefined on input line 818. LaTeX Warning: Citation `TS' on page 32 undefined on input line 819. LaTeX Warning: Citation `TS' on page 32 undefined on input line 819. LaTeX Warning: Citation `TS' on page 32 undefined on input line 820. [32] LaTeX Warning: Citation `TS' on page 33 undefined on input line 831. Underfull \hbox (badness 10000) in paragraph at lines 831--832 [] LaTeX Warning: Citation `Kh' on page 33 undefined on input line 833. Underfull \hbox (badness 10000) in paragraph at lines 833--834 [] Underfull \hbox (badness 10000) in paragraph at lines 835--836 [] ) [33] \openout2 = `chapters/chapter3/ch3.aux'. (./chapters/chapter3/ch3.tex Underfull \vbox (badness 10000) detected at line 8 [] Overfull \hbox (336.25pt too wide) detected at line 8 [] [] Underfull \vbox (badness 10000) detected at line 8 [] Overfull \hbox (336.25pt too wide) detected at line 8 [] [] Underfull \vbox (badness 10000) detected at line 8 [] Overfull \hbox (336.25pt too wide) detected at line 8 [] [] Underfull \vbox (badness 10000) detected at line 8 [] Overfull \hbox (336.25pt too wide) detected at line 8 [] [] Chapter 3. Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [34 ] [35] Overfull \hbox (3.63309pt too wide) detected at line 90 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; x[]\OT1/z tmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 Tx[]; Tx[]\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/ zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 k\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10 .5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; x[]\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; x[]\OT1/z tmcm/m/n/10.5 )) + \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/1 0.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 : [] Overfull \hbox (10.21779pt too wide) in paragraph at lines 90--94 \OT1/ptm/m/n/10.5 Thus, $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 Tx[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T []\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 kd\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x []\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm /m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) \OML/ztmcm/m/it/10.5 < [] \OT1/ztmcm/ m/n/10.5 + [] = \OML/ztmcm/m/it/10.5 "$\OT1/ptm/m/n/10.5 . Thus, $\OML/ztmcm/m/ it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; x[]\OT1/ztmcm/m/n/10.5 ) = [] [36] [37] Underfull \hbox (badness 10000) in paragraph at lines 182--183 [] Underfull \hbox (badness 10000) in paragraph at lines 184--185 [] [38] Underfull \hbox (badness 10000) in paragraph at lines 188--191 [] Underfull \hbox (badness 10000) in paragraph at lines 192--193 [] Underfull \hbox (badness 10000) in paragraph at lines 194--195 [] [39] Overfull \hbox (17.1369pt too wide) detected at line 218 []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OMS/ zmtt/m/n/10.5 g ^^T \OML/ztmcm/m/it/10.5 ^^U []d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm /m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 : [] Underfull \hbox (badness 10000) in paragraph at lines 218--220 [] Underfull \hbox (badness 10000) in paragraph at lines 221--222 [] Overfull \hbox (40.16678pt too wide) detected at line 225 []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OMS/ zmtt/m/n/10.5 g ^^T \OML/ztmcm/m/it/10.5 ^^U []d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm /m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 : [] Overfull \hbox (13.25757pt too wide) detected at line 231 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 v; Tv\OT1/ztmcm /m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T []v; T[]v\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 ^^U []d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 ) \O MS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 ^^U[] []d\OT1/ztmcm/m/n/10.5 (\OML/zt mcm/m/it/10.5 T[]v; T[]v\OT1/ztmcm/m/n/10.5 ) [] \OMS/zmtt/m/n/10.5 ^^T \OML/zt mcm/m/it/10.5 ^^U[] []d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 v; Tv\OT1/ztmc m/m/n/10.5 ) [] [40] Underfull \hbox (badness 10000) in paragraph at lines 231--233 [] Underfull \hbox (badness 10000) in paragraph at lines 234--235 [] [41] [42] [43] Underfull \hbox (badness 10000) in paragraph at lines 311--316 [] Underfull \hbox (badness 10000) in paragraph at lines 320--321 [] Underfull \hbox (badness 10000) in paragraph at lines 322--324 [] Underfull \hbox (badness 10000) in paragraph at lines 325--326 [] Underfull \hbox (badness 10000) in paragraph at lines 330--332 [] [44] Underfull \hbox (badness 10000) in paragraph at lines 333--334 [] Underfull \hbox (badness 10000) in paragraph at lines 341--342 [] Underfull \hbox (badness 10000) in paragraph at lines 343--344 [] [45] [46] [47] Overfull \hbox (18.626pt too wide) detected at line 458 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]x[]; T[]x[]\ OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 a[] []d\OT1/zt mcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/ m/it/10.5 a[] []d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; T[]x[]\OT1/ztm cm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 b[] []d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 Tx[]; T[]x[]\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 b[] []d\OT1/ztm cm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]x[]; T[]x[]\OT1/ztmcm/m/n/10.5 ) [] [48] Overfull \hbox (14.89017pt too wide) detected at line 477 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Ty; y\OT1/ztmcm /m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T []Ty; T[]y\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 a[] d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Ty; T[]y\OT1/ztmcm/m/n/10.5 ) + \OML /ztmcm/m/it/10.5 a[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]y; T[]y\OT1/z tmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it /10.5 y; Ty\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Ty; T[]y\OT1/ztmcm/m/n/10.5 ) [] [49] Overfull \hbox (4.13068pt too wide) detected at line 492 \OT1/ztmcm/m/n/10.5 2 = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T[]\OT1/ztmcm/m/n/10.5 1\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 3) \OML/ztmcm/m/it/10.5 > a[]d\OT1/ztmcm/m/n/10.5 (1\OML/ztmcm/m/it/10.5 ; T\OT 1/ztmcm/m/n/10.5 1) + \OML/ztmcm/m/it/10.5 a[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 T\OT1/ztmcm/m/n/10.5 1\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 1 ) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (3\OML/ztmcm/m/it/10.5 ; T\OT1 /ztmcm/m/n/10.5 3) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T\OT1/ztmcm/m/n/10.5 3\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 3) = 0\OML/ztmcm/m/it/10.5 :\OT1/ztmcm/m/n/10.5 8\OML/ztmcm/m/it/10.5 : [] LaTeX Font Info: Calculating math sizes for size <14> on input line 509. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <14> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 509. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <9.79996> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 509. Underfull \hbox (badness 10000) in paragraph at lines 510--511 [] [50] Underfull \hbox (badness 10000) in paragraph at lines 512--513 [] Underfull \hbox (badness 10000) in paragraph at lines 516--517 [] Underfull \hbox (badness 10000) in paragraph at lines 518--519 [] [51] Overfull \hbox (10.12529pt too wide) in paragraph at lines 523--527 \OT1/ptm/m/it/10 contractive and $\OML/ztmcm/m/it/10 $\OT1/ptm/m/it/10 -admiss ible self-map on $\OML/ztmcm/m/it/10 X$\OT1/ptm/m/it/10 . If there ex-ists $\OM L/ztmcm/m/it/10 x[] \OMS/zmtt/m/n/10 2 \OML/ztmcm/m/it/10 X$ \OT1/ptm/m/it/10 s uch that $\OML/ztmcm/m/it/10 \OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; Tx[]\O T1/ztmcm/m/n/10 ) \OMS/zmtt/m/n/10 ^^U [] Overfull \hbox (3.64766pt too wide) detected at line 537 \OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 ) = []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\O T1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )\OML/ztmc m/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/ n/10.5 )\OML/ztmcm/m/it/10.5 ; [] []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\ OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )\OML/ztm cm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m /n/10.5 )\OMS/zmtt/m/n/10.5 gg [] [52] Overfull \hbox (5.54825pt too wide) in paragraph at lines 542--545 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , then $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/z tmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1 /ztmcm/m/n/10.5 ))$ \OT1/ptm/m/n/10.5 and if $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/ m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = []\OML/ztmcm/m/i t/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) $\OT1/ptm/m/n/10.5 , [] Overfull \hbox (18.61877pt too wide) detected at line 546 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 ([]) \OMS/zmtt/m/n/10.5 ^^T [] \OML/ztmcm/m/it/10.5 < [] \OT1/ztmcm/m/n/10.5 ( \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 )) + []\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/ 10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) [] [53] Underfull \hbox (badness 10000) in paragraph at lines 584--585 [] [54] Underfull \hbox (badness 10000) in paragraph at lines 607--612 [] [55] Overfull \hbox (20.9451pt too wide) detected at line 681 [] [] \OMS/zmtt/m/n/10.5 ^^T [] [] \OT1/ztmcm/m/n/10.5 + [] [](3\OML/ztmcm/m/it /10.5 "[] \OT1/ztmcm/m/n/10.5 + (2\OML/ztmcm/m/it/10.5 N \OMS/zmtt/m/n/10.5 ^^@ \OT1/ztmcm/m/n/10.5 1)\OML/ztmcm/m/it/10.5 "[]\OT1/ztmcm/m/n/10.5 ) [] [56] [57] [58] Overfull \hbox (10.37825pt too wide) detected at line 780 \OML/ztmcm/m/it/10.5 r\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 t[]; x[]\OT1/zt mcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10. 5 t[]; T[]^^M[]\OT1/ztmcm/m/n/10.5 ) + \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 T[]^^M[]; T[]^^M[]\OT1/ztmcm/m/n/10.5 ) + [] + \OML/ztm cm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]^^M[]; x[]\OT1/ztmcm /m/n/10.5 ) [] Underfull \hbox (badness 10000) in paragraph at lines 794--795 [] [59] Underfull \hbox (badness 10000) in paragraph at lines 796--802 [] [60] [61] Underfull \hbox (badness 10000) in paragraph at lines 894--903 [] [62] [63] Underfull \hbox (badness 1221) in paragraph at lines 932--932 []\OT1/ptm/m/n/10.5 Let $\OML/ztmcm/m/it/10.5 A$ \OT1/ptm/m/n/10.5 and $\OML/zt mcm/m/it/10.5 B$ \OT1/ptm/m/n/10.5 be nonempty closed sub-sets of a met-ric spa ce $\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 X; d\OT1/ztmcm/m/n/10.5 )$ \OT1/p tm/m/n/10.5 and [] Underfull \hbox (badness 10000) in paragraph at lines 932--933 [] Underfull \hbox (badness 10000) in paragraph at lines 934--935 [] Underfull \hbox (badness 10000) in paragraph at lines 936--937 [] Underfull \hbox (badness 10000) in paragraph at lines 938--939 [] [64] [65] [66] [67] Overfull \hbox (7.7724pt too wide) detected at line 972 \OMS/zmtt/m/n/10.5 k\OML/ztmcm/m/it/10.5 x \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/ it/10.5 y\OMS/zmtt/m/n/10.5 k \OT1/ztmcm/m/n/10.5 = [] \OMS/zmtt/m/n/10.5 j\OML /ztmcm/m/it/10.5 f\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/1 0.5 ^^@ \OML/ztmcm/m/it/10.5 x[]\OT1/ztmcm/m/n/10.5 )\OMS/zmtt/m/n/10.5 j ^^T [ ] k\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 x[]\OMS /zmtt/m/n/10.5 k \OT1/ztmcm/m/n/10.5 = \OMS/zmtt/m/n/10.5 k\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 x[]\OMS/zmtt/m/n/10.5 k \OT1/z tmcm/m/n/10.5 = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 A; B\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 : [] Underfull \hbox (badness 10000) in paragraph at lines 979--981 [] Underfull \hbox (badness 10000) in paragraph at lines 984--986 [] [68] Underfull \hbox (badness 10000) in paragraph at lines 987--988 [] Underfull \hbox (badness 10000) in paragraph at lines 1001--1003 [] [69] Underfull \hbox (badness 10000) in paragraph at lines 1014--1015 [] [70] [71] [72] Underfull \hbox (badness 10000) in paragraph at lines 1053--1055 [] Overfull \hbox (15.51868pt too wide) in paragraph at lines 1056--1056 \OML/ztmcm/m/it/10.5 T[]y \OMS/zmtt/m/n/10.5 ! \OML/ztmcm/m/it/10.5 Tx[]$\OT1/p tm/m/n/10.5 . Thus, $\OML/ztmcm/m/it/10.5 E[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/ it/10.5 Tx[]\OT1/ztmcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 B$\OT1/ptm/m/n/10.5 . If $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\O T1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 6\OT1/ztmcm/m/n/10.5 = \OML/ztmcm/m/it/1 0.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 A; B\OT1/ztmcm/m/n/10.5 )$\OT1/p tm/m/n/10.5 , then $\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10 .5 (\OML/ztmcm/m/it/10.5 T[]x[]; T[]x[]\OT1/ztmcm/m/n/10.5 )\OMS/zmtt/m/n/10.5 g$ [] Overfull \hbox (26.42406pt too wide) in paragraph at lines 1056--1056 \OT1/ptm/m/n/10.5 is a de-creas-ing se-quence be-cause $\OT1/ztmcm/m/n/10.5 (\O ML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 2 \OML/ztm cm/m/it/10.5 X[]$\OT1/ptm/m/n/10.5 . By The-o-rem 3.29[], $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]x[]; T[]x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 # [] [73] Overfull \hbox (25.54877pt too wide) in paragraph at lines 1056--1056 \OT1/ptm/m/n/10.5 Put $\OML/ztmcm/m/it/10.5 x \OT1/ztmcm/m/n/10.5 = \OML/ztmcm/ m/it/10.5 T[]x[]$ \OT1/ptm/m/n/10.5 and $\OML/ztmcm/m/it/10.5 y \OT1/ztmcm/m/n/ 10.5 = \OML/ztmcm/m/it/10.5 T[]x[]$\OT1/ptm/m/n/10.5 . Since $\OT1/ztmcm/m/n/10 .5 (\OML/ztmcm/m/it/10.5 x; y\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 2 \OML/zt mcm/m/it/10.5 X[]$\OT1/ptm/m/n/10.5 , $\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10. 5 Tx; Ty\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/10.5 X[]$ \O T1/ptm/m/n/10.5 and so $\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 T[]x; T[]y\OT1/ztmcm/m/n/10.5 )\OMS/zmtt/m/n/10.5 g$ [] Overfull \hbox (11.8272pt too wide) in paragraph at lines 1056--1057 \OT1/ptm/m/n/10.5 which is a con-tra-dic-tion. There-fore, $\OML/ztmcm/m/it/10. 5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) = \ OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 A; B\OT1/ztmcm/m /n/10.5 )$\OT1/ptm/m/n/10.5 . As-sume that $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) = [] Underfull \hbox (badness 10000) in paragraph at lines 1056--1057 [] Underfull \hbox (badness 10000) in paragraph at lines 1058--1059 [] [74] [75] [76] Overfull \hbox (6.71735pt too wide) in paragraph at lines 1116--1116 \OT1/ptm/m/n/10.5 Note that $\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\OT1/ztm cm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]x; T[]y\OT1/ztmcm/m/n/10.5 )\OMS/zmtt/m/n/ 10.5 g$ \OT1/ptm/m/n/10.5 is a de-creas-ing se-quence and $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]x; T[]y\OT1/ztmcm/m/n/10.5 ) \OM S/zmtt/m/n/10.5 # \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10 .5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 . [] [77] Underfull \hbox (badness 10000) in paragraph at lines 1139--1140 [] [78] [79] [80] [81] Overfull \hbox (4.13068pt too wide) detected at line 1298 \OT1/ztmcm/m/n/10.5 2 = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T[]\OT1/ztmcm/m/n/10.5 1\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 3) \OML/ztmcm/m/it/10.5 > a[]d\OT1/ztmcm/m/n/10.5 (1\OML/ztmcm/m/it/10.5 ; T\OT 1/ztmcm/m/n/10.5 1) + \OML/ztmcm/m/it/10.5 a[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 T\OT1/ztmcm/m/n/10.5 1\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 1 ) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (3\OML/ztmcm/m/it/10.5 ; T\OT1 /ztmcm/m/n/10.5 3) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T\OT1/ztmcm/m/n/10.5 3\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 3) = 0\OML/ztmcm/m/it/10.5 :\OT1/ztmcm/m/n/10.5 8\OML/ztmcm/m/it/10.5 : [] Overfull \hbox (4.13068pt too wide) detected at line 1310 \OT1/ztmcm/m/n/10.5 2 = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T[]\OT1/ztmcm/m/n/10.5 1\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 3) \OML/ztmcm/m/it/10.5 > a[]d\OT1/ztmcm/m/n/10.5 (1\OML/ztmcm/m/it/10.5 ; T\OT 1/ztmcm/m/n/10.5 1) + \OML/ztmcm/m/it/10.5 a[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 T\OT1/ztmcm/m/n/10.5 1\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 1 ) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (3\OML/ztmcm/m/it/10.5 ; T\OT1 /ztmcm/m/n/10.5 3) + \OML/ztmcm/m/it/10.5 b[]d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m /it/10.5 T\OT1/ztmcm/m/n/10.5 3\OML/ztmcm/m/it/10.5 ; T[]\OT1/ztmcm/m/n/10.5 3) = 0\OML/ztmcm/m/it/10.5 :\OT1/ztmcm/m/n/10.5 8\OML/ztmcm/m/it/10.5 : [] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] \openout4 = `bu.aux'. (./document.bbl [94] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [95 ])) [96] \openout2 = `chapters/chapter4/ch4.aux'. (./chapters/chapter4/ch4.tex Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Chapter 4. Underfull \hbox (badness 10000) in paragraph at lines 22--24 [] Underfull \hbox (badness 10000) in paragraph at lines 25--26 [] Underfull \hbox (badness 10000) in paragraph at lines 27--28 [] Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [97 ] LaTeX Warning: Citation `NAD' on page 98 undefined on input line 34. Underfull \hbox (badness 10000) in paragraph at lines 34--35 [] Underfull \hbox (badness 10000) in paragraph at lines 39--41 [] Overfull \hbox (9.2403pt too wide) in paragraph at lines 49--60 \OT1/ptm/m/n/10.5 com-plete-ness of $\OML/ztmcm/m/it/10.5 X$\OT1/ptm/m/n/10.5 , there ex-ists $\OML/ztmcm/m/it/10.5 v \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/10. 5 X$ \OT1/ptm/m/n/10.5 such that $\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 ! \OML/ztmcm/m/it/10.5 v$\OT1/ptm/m/n/10.5 . Thus, $[][] \OML/ztmcm/m/it/10.5 H\ OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; Tv\OT1/ztmcm/m/n/10.5 ) = [] [98] Underfull \hbox (badness 10000) in paragraph at lines 72--73 [] [99] LaTeX Warning: Citation `NAD' on page 100 undefined on input line 89. Underfull \hbox (badness 10000) in paragraph at lines 92--94 [] Underfull \hbox (badness 10000) in paragraph at lines 97--99 [] [100] Underfull \hbox (badness 10000) in paragraph at lines 127--129 [] Underfull \hbox (badness 10000) in paragraph at lines 132--134 [] Underfull \hbox (badness 10000) in paragraph at lines 142--144 [] Underfull \hbox (badness 10000) in paragraph at lines 147--149 [] [101] Underfull \hbox (badness 10000) in paragraph at lines 167--171 [] Underfull \hbox (badness 10000) in paragraph at lines 175--178 [] Overfull \hbox (37.77158pt too wide) in paragraph at lines 181--191 \OT1/ztmcm/m/n/10.5 [\OML/ztmcm/m/it/10.5 m\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it /10.5 "\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; n[]\OT1/ztmcm/m/n/10.5 )$ \O T1/ptm/m/n/10.5 and $\OML/ztmcm/m/it/10.5 j \OMS/zmtt/m/n/10.5 2 \OT1/ztmcm/m/n /10.5 [\OML/ztmcm/m/it/10.5 m\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 "\OT1/zt mcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; n[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 . Then, we have $\OML/ztmcm/m/it/10.5 d[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/ 10.5 T[]x[]; T[]x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it /10.5 d[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]x[]; T[]x[]\OT1/ztmcm/m/n /10.5 ) \OMS/zmtt/m/n/10.5 ^^T [] Underfull \hbox (badness 10000) in paragraph at lines 195--196 [] Overfull \hbox (1.10686pt too wide) in paragraph at lines 197--209 []\OT1/ptm/m/n/10.5 Now, we com-plete the proof of the the-o-rem. Con-sider the se-quences $\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 T[]x[]\OMS/zmtt/m/n/10.5 g[]$ [] LaTeX Font Warning: Command \tiny invalid in math mode on input line 211. LaTeX Font Warning: Command \tiny invalid in math mode on input line 214. [102] LaTeX Font Warning: Command \tiny invalid in math mode on input line 217. LaTeX Font Warning: Command \tiny invalid in math mode on input line 219. LaTeX Warning: Citation `LA' on page 103 undefined on input line 224. LaTeX Warning: Citation `RZ' on page 103 undefined on input line 240. Underfull \hbox (badness 10000) in paragraph at lines 240--241 [] [103] [104] [105] [106] LaTeX Warning: Citation `RZ' on page 107 undefined on input line 327. [107] Overfull \hbox (5.8982pt too wide) in paragraph at lines 403--408 \OT1/ptm/m/n/10.5 From (ii), we ob-tain $\OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/1 0.5 (\OML/ztmcm/m/it/10.5 Ty[]; Tx[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^ T \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; y[]\OT1/ ztmcm/m/n/10.5 )$ \OT1/ptm/m/n/10.5 and so $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 y[]; Tx[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; y[]\OT 1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 . [] [108] [109] Underfull \hbox (badness 10000) in paragraph at lines 500--501 [] [110] LaTeX Warning: Citation `KS' on page 111 undefined on input line 516. LaTeX Warning: Citation `MP' on page 111 undefined on input line 520. [111] LaTeX Warning: Citation `DY' on page 112 undefined on input line 524. Overfull \hbox (17.53804pt too wide) in paragraph at lines 527--536 \OT1/ptm/m/n/10.5 b) $\OML/ztmcm/m/it/10.5 g$ \OT1/ptm/m/n/10.5 is sub-homogene ous, that is, $\OML/ztmcm/m/it/10.5 g\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]; x[]; x[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/z tmcm/m/it/10.5 g\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]; x[]; x[]; x[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 for [] Overfull \hbox (18.78613pt too wide) in paragraph at lines 527--536 \OT1/ptm/m/n/10.5 c) If $\OML/ztmcm/m/it/10.5 x[]; y[] \OMS/zmtt/m/n/10.5 2 \OT 1/ztmcm/m/n/10.5 [0\OML/ztmcm/m/it/10.5 ; \OMS/zmtt/m/n/10.5 1\OT1/ztmcm/m/n/10 .5 )$ \OT1/ptm/m/n/10.5 and $\OML/ztmcm/m/it/10.5 x[] < y[]$ \OT1/ptm/m/n/10.5 for $\OML/ztmcm/m/it/10.5 i \OT1/ztmcm/m/n/10.5 = 1\OML/ztmcm/m/it/10.5 ; :::; \OT1/ztmcm/m/n/10.5 4$\OT1/ptm/m/n/10.5 , then $\OML/ztmcm/m/it/10.5 g\OT1/ztmc m/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]; x[]; x[]; \OT1/ztmcm/m/n/10.5 0) \OM L/ztmcm/m/it/10.5 < g\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 y[]; y[]; y[]; y []; \OT1/ztmcm/m/n/10.5 0)$ [] Underfull \hbox (badness 10000) in paragraph at lines 527--536 [] LaTeX Warning: Citation `C' on page 112 undefined on input line 538. [112] Overfull \hbox (11.32562pt too wide) in paragraph at lines 563--569 \OT1/ptm/m/n/10.5 Thus, there ex-ists $\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/1 0.5 2 \OML/ztmcm/m/it/10.5 Gx[]$ \OT1/ptm/m/n/10.5 such that $\OML/ztmcm/m/it/1 0.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) \O ML/ztmcm/m/it/10.5 < ^^V < rd\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[] \OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 :$ \OT1/ptm/m/n/10.5 Since $\OML/ztmc m/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Gx[]\OT1/ztmcm/m/n /10.5 ) \OML/ztmcm/m/it/10.5 < [] [113] Overfull \hbox (19.67795pt too wide) detected at line 595 \OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Fx[]; Gx\OT1/zt mcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 g\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x\OT1/zt mcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Fx\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/zt mcm/m/it/10.5 x[]; Gx[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/ m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Gx[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Fx\OT1/ztmcm/m/n/10.5 )) [] [114] LaTeX Warning: Citation `DY' on page 115 undefined on input line 622. [115] [116] Overfull \hbox (21.50435pt too wide) in paragraph at lines 665--672 \OML/ztmcm/m/it/10.5 x[] [] Tx[]$\OT1/ptm/m/n/10.5 . Then $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/ zmtt/m/n/10.5 6\OT1/ztmcm/m/n/10.5 = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 . Thus, $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/z tmcm/m/n/10.5 ) = []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10 .5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; [] d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )\OMS/z mtt/m/n/10.5 g$\OT1/ptm/m/n/10.5 . [] Overfull \hbox (5.94028pt too wide) in paragraph at lines 665--672 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , then $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10 .5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT 1/ztmcm/m/n/10.5 ))$\OT1/ptm/m/n/10.5 . If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = []\OML/ztmcm/m/it/ 10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )$ \OT1/ptm/m/n/10.5 , [] Overfull \hbox (10.29425pt too wide) in paragraph at lines 672--674 \OT1/ptm/m/n/10.5 because $\OML/ztmcm/m/it/10.5 $ \OT1/ptm/m/n/10.5 is sub-lin -ear. Thus there ex-ists $\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 2 \OML/zt mcm/m/it/10.5 Tx[]$ \OT1/ptm/m/n/10.5 such that $\OML/ztmcm/m/it/10.5 d\OT1/ztm cm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/1 0.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/z tmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ))$\OT1/ptm/m/n /10.5 . [] Overfull \hbox (5.94028pt too wide) in paragraph at lines 677--689 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , then $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10 .5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT 1/ztmcm/m/n/10.5 ))$\OT1/ptm/m/n/10.5 . If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = []\OML/ztmcm/m/it/ 10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )$ \OT1/ptm/m/n/10.5 , [] Overfull \hbox (10.507pt too wide) detected at line 693 \OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 ) = []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\O T1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )\OML/ztmc m/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/ n/10.5 )\OML/ztmcm/m/it/10.5 ; [] []\OMS/zmtt/m/n/10.5 f\OML/ztmcm/m/it/10.5 d\ OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )\OML/ztm cm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m /n/10.5 )\OMS/zmtt/m/n/10.5 gg\OML/ztmcm/m/it/10.5 : [] [117] Overfull \hbox (17.31366pt too wide) detected at line 728 \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; y\OT1/ztmcm/ m/n/10.5 ) = [] [] Overfull \hbox (4.2294pt too wide) in paragraph at lines 731--732 \OT1/ptm/m/n/10.5 generalized $\OML/ztmcm/m/it/10.5 $\OT1/ptm/m/n/10.5 -$\OML/ ztmcm/m/it/10.5 $\OT1/ptm/m/n/10.5 -contraction if $\OML/ztmcm/m/it/10.5 ^^R\O T1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 r\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/1 0.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Tx\OT1/ztmcm/m/n/10.5 ) \OMS/ zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10 .5 x; y\OT1/ztmcm/m/n/10.5 )$ \OT1/ptm/m/n/10.5 im-plies $\OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; y\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/ m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx; Ty\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T [] [118] Overfull \hbox (7.95166pt too wide) in paragraph at lines 744--753 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 )$ \OT1/ptm/m/n/10.5 and so $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^ T \OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; Tx[]\OT 1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 H\ OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx[]; Tx[]\OT1/ztmcm/m/n/10.5 ) \OMS/z mtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10. 5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ))$\OT 1/ptm/m/n/10.5 , [] [119] Overfull \hbox (21.27661pt too wide) in paragraph at lines 782--787 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x\OT1/ztmcm/m/n/10.5 ) = []\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x; Tx\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , then $\O ML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx\OT1/ztmcm /m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 ([ ]\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Tx\OT1/ztmc m/m/n/10.5 )) \OMS/zmtt/m/n/10.5 ^^T []\OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; x[]\OT1/z tmcm/m/n/10.5 )) + []\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it /10.5 x[]; Tx\OT1/ztmcm/m/n/10.5 )$ [] [120] Overfull \hbox (16.39813pt too wide) in paragraph at lines 814--818 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , then $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T [] \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n /10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[] \OT1/ztmcm/m/n/10.5 )) = 0$ \OT1/ptm/m/n/10.5 and so $\OML/ztmcm/m/it/10.5 d\OT 1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) = [] Overfull \hbox (4.64288pt too wide) in paragraph at lines 818--822 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = []\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/1 0.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , th en $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\O T1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T [] \OML/ztmcm/m/it/10.5 \OT1/ztmcm/ m/n/10.5 ([]\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[] ; Tx[]\OT1/ztmcm/m/n/10.5 )) \OML/ztmcm/m/it/10.5 < []d\OT1/ztmcm/m/n/10.5 (\OM L/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 )$ [] [121] Underfull \hbox (badness 10000) in paragraph at lines 830--831 [] LaTeX Warning: Citation `Y' on page 122 undefined on input line 835. LaTeX Warning: Citation `Y' on page 122 undefined on input line 835. LaTeX Warning: Citation `BP' on page 122 undefined on input line 840. LaTeX Warning: Citation `BP' on page 122 undefined on input line 840. [122] LaTeX Warning: Citation `Y' on page 123 undefined on input line 856. [123] Overfull \hbox (14.85074pt too wide) detected at line 869 \OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]\OT1/ztmcm/m /n/10.5 (\OML/ztmcm/m/it/10.5 x\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; T\OT 1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x\OT1/ztmcm/m/n/10.5 )) \OML/ztmcm/m/it/ 10.5 < ^^O; H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]\OT1/ztmcm/m/n/10.5 ( \OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; T\OT1/ztmcm/m /n/10.5 (\OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )) \OML/ztmcm/m/it/10.5 < "; H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 ^^Ux \OT1/ztmcm/m/n/10.5 + (1 \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/ 10.5 ^^U\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )\OML/zt mcm/m/it/10.5 ; T\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 ^^Ux \OT1/ztmcm/m/n/ 10.5 + (1 \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 ^^U\OT1/ztmcm/m/n/10.5 )\ OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )) \OML/ztmcm/m/it/10.5 < " [] [124] Overfull \hbox (27.74515pt too wide) detected at line 871 \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 ^^Ux \OT1/ztmcm /m/n/10.5 + (1 \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 ^^U\OT1/ztmcm/m/n/10 .5 )\OML/ztmcm/m/it/10.5 y; T[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 ^^Ux \ OT1/ztmcm/m/n/10.5 + (1 \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 ^^U\OT1/ztm cm/m/n/10.5 )\OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )) + \OML/ztmcm/m/it/10. 5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm /m/it/10.5 ^^Ux \OT1/ztmcm/m/n/10.5 + (1 \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it /10.5 ^^U\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )\OML/z tmcm/m/it/10.5 ; T\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 ^^Ux \OT1/ztmcm/m/n /10.5 + (1 \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 ^^U\OT1/ztmcm/m/n/10.5 ) \OML/ztmcm/m/it/10.5 y\OT1/ztmcm/m/n/10.5 )) \OML/ztmcm/m/it/10.5 < r \OT1/ztmc m/m/n/10.5 + 2\OML/ztmcm/m/it/10.5 ": [] [125] LaTeX Warning: Citation `ROY' on page 126 undefined on input line 896. LaTeX Warning: Citation `RU' on page 126 undefined on input line 897. LaTeX Warning: Citation `AOS' on page 126 undefined on input line 902. [126] [127] LaTeX Warning: Citation `AOS' on page 128 undefined on input line 926. Underfull \hbox (badness 10000) in paragraph at lines 933--934 [] LaTeX Warning: Citation `BGP' on page 128 undefined on input line 938. LaTeX Warning: Citation `RUS' on page 128 undefined on input line 950. [128] Underfull \hbox (badness 10000) in paragraph at lines 967--968 [] [129] [130] LaTeX Warning: Citation `RPS' on page 131 undefined on input line 1033. LaTeX Warning: Citation `RPS' on page 131 undefined on input line 1041. LaTeX Warning: Citation `RPS' on page 131 undefined on input line 1056. [131] Overfull \hbox (13.87688pt too wide) in paragraph at lines 1085--1092 []\OT1/ptm/m/n/10.5 , there ex-ists $\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10. 5 2 \OML/ztmcm/m/it/10.5 Tx[]$ \OT1/ptm/m/n/10.5 and $\OML/ztmcm/m/it/10.5 M\OT 1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/ m/n/10.5 2 []$ \OT1/ptm/m/n/10.5 such that $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^ ^T \OML/ztmcm/m/it/10.5 '\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/ m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ))$\OT1/ptm/m/n/10.5 . [] [132] Overfull \hbox (3.95374pt too wide) detected at line 1114 \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 '[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10. 5 )) + \OML/ztmcm/m/it/10.5 '[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) + [] + \OML /ztmcm/m/it/10.5 '[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/1 0.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) = [] \OML/ztmcm/m/it/1 0.5 '[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztm cm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ))\OML/ztmcm/m/it/10.5 : [] Overfull \hbox (15.4075pt too wide) in paragraph at lines 1125--1128 \OT1/ptm/m/n/10.5 Hence, $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm /m/it/10.5 u[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it /10.5 '[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/z tmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) + \OML/ztmcm/m/it/10.5 d\OT1/ztmc m/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$ \OT1/ptm/m/n/10 .5 and so $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 u[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ! \OT1/ztmcm/m/n/10.5 0$\OT1/ptm/m /n/10.5 . If $\OML/ztmcm/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x []; x[]\OT1/ztmcm/m/n/10.5 ) = [] [133] Overfull \hbox (10.65732pt too wide) in paragraph at lines 1135--1138 \OT1/ptm/m/n/10.5 Thus, $\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 u[]; x[]\OT1/ztmcm/m/n/10.5 ) \OML/ztmcm/m/it/10.5 < '\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\ OT1/ztmcm/m/n/10.5 )) + 2\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$ \OT1/ptm/m/n/10.5 and so $\OML/ztmcm/m /it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 u[]; x[]\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ! \OT1/ztmcm/m/n/10.5 0$\OT1/ptm/m/n/10.5 . If $\OML/ztmc m/m/it/10.5 M\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/1 0.5 ) = [] [134] [135] [136] [137] Overfull \hbox (12.3364pt too wide) detected at line 1282 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 I \OT1/ztmcm/m/n/10.5 + \OML/ztmcm/m/ it/10.5 A\OT1/ztmcm/m/n/10.5 )[]\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML /ztmcm/m/it/10.5 x; Tx\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OT1/ztmcm/m /n/10.5 [(\OML/ztmcm/m/it/10.5 I \OT1/ztmcm/m/n/10.5 + \OML/ztmcm/m/it/10.5 A\O T1/ztmcm/m/n/10.5 )[] + 2(\OML/ztmcm/m/it/10.5 I \OT1/ztmcm/m/n/10.5 + \OML/ztm cm/m/it/10.5 A\OT1/ztmcm/m/n/10.5 )[](\OML/ztmcm/m/it/10.5 I \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 A\OT1/ztmcm/m/n/10.5 )[]\OML/ztmcm/m/it/10.5 A\OT1/zt mcm/m/n/10.5 ]\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x ; x[]\OT1/ztmcm/m/n/10.5 ) = (\OML/ztmcm/m/it/10.5 I \OMS/zmtt/m/n/10.5 ^^@ \OM L/ztmcm/m/it/10.5 A\OT1/ztmcm/m/n/10.5 )[]\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/ 10.5 (\OML/ztmcm/m/it/10.5 x; x[]\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 : [] [138] [139] Overfull \hbox (2.30222pt too wide) in paragraph at lines 1339--1341 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 L\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/i t/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ) = \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10. 5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )$\OT1/ptm/m/n/10.5 , then $\OML/ztmcm/m/it/10.5 F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/ n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it/10.5 AF\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmc m/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) \OMS/zmtt/m/n/1 0.5 ^^T \OML/ztmcm/m/it/10.5 A[]F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT 1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 ))$\OT1/ptm/ m/n/10.5 . Thus, [] [140] [141] Overfull \hbox (23.46289pt too wide) detected at line 1400 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 I \OT1/ztmcm/m/n/10.5 + \OML/ztmcm/m/ it/10.5 A\OT1/ztmcm/m/n/10.5 )[]\OML/ztmcm/m/it/10.5 F\OT1/ztmcm/m/n/10.5 (\OML /ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Tx\OT1/ztmcm/m/n /10.5 )) \OMS/zmtt/m/n/10.5 ^^T \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 I \OT 1/ztmcm/m/n/10.5 + \OML/ztmcm/m/it/10.5 A\OT1/ztmcm/m/n/10.5 )[][\OML/ztmcm/m/i t/10.5 I \OT1/ztmcm/m/n/10.5 + 2(\OML/ztmcm/m/it/10.5 I \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 A\OT1/ztmcm/m/n/10.5 )[]\OML/ztmcm/m/it/10.5 A\OT1/ztmcm/m /n/10.5 ]\OML/ztmcm/m/it/10.5 F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; x[]\OT1/ztmcm/m/n/10.5 )) = (\OML/ztmcm /m/it/10.5 I \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 A\OT1/ztmcm/m/n/10.5 ) []\OML/ztmcm/m/it/10.5 F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m /n/10.5 (\OML/ztmcm/m/it/10.5 x; x[]\OT1/ztmcm/m/n/10.5 ))\OML/ztmcm/m/it/10.5 : [] Overfull \hbox (8.2726pt too wide) detected at line 1412 \OML/ztmcm/m/it/10.5 F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n /10.5 (\OML/ztmcm/m/it/10.5 v[]; x[]\OT1/ztmcm/m/n/10.5 )) \OMS/zmtt/m/n/10.5 ^ ^T \OML/ztmcm/m/it/10.5 F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/ m/n/10.5 (\OML/ztmcm/m/it/10.5 v[]; x[]\OT1/ztmcm/m/n/10.5 )) + \OML/ztmcm/m/it /10.5 F\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztm cm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m /it/10.5 AF\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML /ztmcm/m/it/10.5 x[]; x[]\OT1/ztmcm/m/n/10.5 )) + \OML/ztmcm/m/it/10.5 F\OT1/zt mcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x []; x[]\OT1/ztmcm/m/n/10.5 )) [] [142] Underfull \hbox (badness 10000) in paragraph at lines 1440--1443 [] [143] [144] [145] LaTeX Warning: Reference `p2.1' on page 146 undefined on input line 1583. [146] LaTeX Warning: Reference `p2.1' on page 147 undefined on input line 1643. Overfull \hbox (8.19215pt too wide) in paragraph at lines 1657--1665 \OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/10.5 X$ \OT1/ptm/ m/n/10.5 such that $[][] \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 f[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x\OT1/ztmcm/m/n/10.5 )\O ML/ztmcm/m/it/10.5 ; x[]\OT1/ztmcm/m/n/10.5 ) = 0$\OT1/ptm/m/n/10.5 . By ($\OMS /zmtt/m/n/10.5 ^^C$\OT1/ptm/m/n/10.5 ), $[][] \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/ m/n/10.5 (\OML/ztmcm/m/it/10.5 f[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x\O T1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; f[]\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/ it/10.5 x[]\OT1/ztmcm/m/n/10.5 )) = [] [147] [148] [149] Overfull \hbox (1.24933pt too wide) in paragraph at lines 1779--1784 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 x[]$ \OT1/ptm/m/n/10.5 has been give n, then choose $\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/1 0.5 Tx[]$ \OT1/ptm/m/n/10.5 such that $\OML/ztmcm/m/it/10.5 d[]\OT1/ztmcm/m/n/1 0.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) = \OMS/zmtt/m/n/10.5 k \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 )\OMS/zmtt/m/n/10.5 k$\OT1/ptm/m/n/10.5 . [] [150] Overfull \hbox (1.24933pt too wide) in paragraph at lines 1814--1819 \OT1/ptm/m/n/10.5 If $\OML/ztmcm/m/it/10.5 x[]$ \OT1/ptm/m/n/10.5 has been give n, then choose $\OML/ztmcm/m/it/10.5 x[] \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/1 0.5 Tx[]$ \OT1/ptm/m/n/10.5 such that $\OML/ztmcm/m/it/10.5 d[]\OT1/ztmcm/m/n/1 0.5 (\OML/ztmcm/m/it/10.5 x[]; Tx[]\OT1/ztmcm/m/n/10.5 ) = \OMS/zmtt/m/n/10.5 k \OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x[]; x[]\OT1/zt mcm/m/n/10.5 )\OMS/zmtt/m/n/10.5 k$\OT1/ptm/m/n/10.5 . [] [151] [152] \openout4 = `bu.aux'. (./document.bbl [153] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [154 ])) [155] \openout2 = `chapters/chapter5/ch5.aux'. (./chapters/chapter5/ch5.tex Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Underfull \vbox (badness 10000) detected at line 1 [] Overfull \hbox (336.25pt too wide) detected at line 1 [] [] Chapter 5. LaTeX Font Info: Calculating math sizes for size <30> on input line 1. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <30> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 1. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <20.99991> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 1. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <15> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 1. Underfull \hbox (badness 10000) has occurred while \output is active [][][][] [] [156 ] Package amsmath Warning: Foreign command \over; (amsmath) \frac or \genfrac should be used instead (amsmath) on input line 46. [157] Overfull \hbox (42.04506pt too wide) in paragraph at lines 49--52 \OML/ztmcm/m/it/10 Tx[]$\OT1/ptm/m/n/10 . Then, we have $\OT1/ztmcm/m/n/10 0 \O ML/ztmcm/m/it/10 < d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; Tx[]\OT1/ztmcm/m /n/10 ) \OMS/zmtt/m/n/10 ^^T \OML/ztmcm/m/it/10 []\OT1/ztmcm/m/n/10 (\OML/ztmc m/m/it/10 Tx[]; Tx[]\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 H\OT1/ztmcm/m/n/10 (\ OML/ztmcm/m/it/10 Tx[]; Tx[]\OT1/ztmcm/m/n/10 ) \OML/ztmcm/m/it/10 < q[] []\OT1 /ztmcm/m/n/10 (\OML/ztmcm/m/it/10 Tx[]; Tx[]\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/ 10 H\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 Tx[]; Tx[]\OT1/ztmcm/m/n/10 )$\OT1/pt m/m/n/10 . [] [158] Underfull \hbox (badness 10000) in paragraph at lines 83--94 [] [159] [160] [161] Overfull \hbox (2.19572pt too wide) detected at line 182 \OML/ztmcm/m/it/10.5 ^^L\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx; Ty\OT1/zt mcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 T x; Ty\OT1/ztmcm/m/n/10.5 ) = (\OML/ztmcm/m/it/10.5 a \OMS/zmtt/m/n/10.5 ^^@ \OM L/ztmcm/m/it/10.5 x\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ztmcm/m/it /10.5 \OT1/ztmcm/m/n/10.5 ((\OML/ztmcm/m/it/10.5 a \OMS/zmtt/m/n/10.5 ^^@ \OML /ztmcm/m/it/10.5 x\OT1/ztmcm/m/n/10.5 ) + (\OML/ztmcm/m/it/10.5 y \OMS/zmtt/m/n /10.5 ^^@ \OML/ztmcm/m/it/10.5 a\OT1/ztmcm/m/n/10.5 ) + (\OML/ztmcm/m/it/10.5 y \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 x\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m /n/10.5 ^^@ []f\OML/ztmcm/m/it/10.5 a \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10 .5 x; y \OMS/zmtt/m/n/10.5 ^^@ \OML/ztmcm/m/it/10.5 a\OMS/zmtt/m/n/10.5 g\OT1/z tmcm/m/n/10.5 ) [] [162] [163] Underfull \hbox (badness 10000) in paragraph at lines 219--220 [] [164] Overfull \hbox (2.24458pt too wide) detected at line 247 \OML/ztmcm/m/it/10 \OT1/ztmcm/m/n/10 ([]\OMS/zmtt/m/n/10 f\OML/ztmcm/m/it/10 d \OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; x[]\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/i t/10 ; d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; x[]\OT1/ztmcm/m/n/10 )\OML/z tmcm/m/it/10 ; d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; Tx[]\OT1/ztmcm/m/n/1 0 )\OML/ztmcm/m/it/10 ; []\OT1/ztmcm/m/n/10 [\OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n /10 (\OML/ztmcm/m/it/10 x[]; Tx[]\OT1/ztmcm/m/n/10 ) + \OML/ztmcm/m/it/10 d\OT1 /ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; x[]\OT1/ztmcm/m/n/10 )]\OMS/zmtt/m/n/10 g\OT1/ztmcm/m/n/10 ) + \OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/1 0 x[]; x[]\OT1/ztmcm/m/n/10 ) [] LaTeX Warning: Citation `CIR' on page 165 undefined on input line 251. [165] LaTeX Warning: Citation `CIR' on page 166 undefined on input line 261. LaTeX Warning: Citation `CIR' on page 166 undefined on input line 264. Underfull \hbox (badness 10000) in paragraph at lines 268--271 [] [166] Underfull \hbox (badness 10000) in paragraph at lines 277--278 [] [167] LaTeX Warning: Citation `JJ' on page 168 undefined on input line 322. [168] [169] [170] LaTeX Warning: Citation `a2010' on page 171 undefined on input line 363. LaTeX Warning: Citation `SF2011' on page 171 undefined on input line 368. [171] [172] [173] Overfull \hbox (9.88873pt too wide) in paragraph at lines 440--441 \OT1/ptm/m/n/10 If $\OML/ztmcm/m/it/10 N\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x []; x[]\OT1/ztmcm/m/n/10 ) = \OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/ m/it/10 x[]; x[]\OT1/ztmcm/m/n/10 )$ \OT1/ptm/m/n/10 or $\OML/ztmcm/m/it/10 N\O T1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; x[]\OT1/ztmcm/m/n/10 ) \OMS/zmtt/m/n/1 0 ^^T \OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x[]; x[]\OT1/zt mcm/m/n/10 )$\OT1/ptm/m/n/10 , then it is easy to find $\OML/ztmcm/m/it/10 H\OT 1/ztmcm/m/n/10 (\OMS/zmtt/m/n/10 f\OML/ztmcm/m/it/10 x[]\OMS/zmtt/m/n/10 g\OML/ ztmcm/m/it/10 ; Tx[]\OT1/ztmcm/m/n/10 ) = [] [174] [175] [176] [177] Overfull \hbox (5.91522pt too wide) in paragraph at lines 541--543 \OT1/ptm/m/n/10 But, $\OML/ztmcm/m/it/10 N\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; y\OT1/ztmcm/m/n/10 ) = \OML/ztmcm/m/it/10 max\OMS/zmtt/m/n/10 f\OML/ztmcm/m /it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; y\OT1/ztmcm/m/n/10 )\OML/ztmcm /m/it/10 ; d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; Tx\OT1/ztmcm/m/n/10 )\OML/ ztmcm/m/it/10 ; d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y; Ty\OT1/ztmcm/m/n/10 ) \OML/ztmcm/m/it/10 ; []\OMS/zmtt/m/n/10 g \OT1/ztmcm/m/n/10 = \OML/ztmcm/m/it/1 0 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; y\OT1/ztmcm/m/n/10 )$\OT1/ptm/m/n/1 0 . Thus, $\OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; y\OT1/z tmcm/m/n/10 ) = [] [178] [179] [180] Underfull \hbox (badness 10000) in paragraph at lines 612--614 [] [181] Underfull \hbox (badness 10000) in paragraph at lines 616--617 [] Overfull \hbox (44.75601pt too wide) in paragraph at lines 618--619 \OT1/ptm/m/n/10.5 ex-ists $\OML/ztmcm/m/it/10.5 g \OMS/zmtt/m/n/10.5 2 R$ \OT1/ ptm/m/n/10.5 such that $\OML/ztmcm/m/it/10.5 ^^L\OT1/ztmcm/m/n/10.5 (\OML/ztmcm /m/it/10.5 Tx; Ty\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 Tx; Ty\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^T \OML/ ztmcm/m/it/10.5 g\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; y\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/zt mcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Tx\OT1/ztmcm/m/n/10.5 )\OML/ztmcm/m/it/10 .5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 y; Ty\OT1/ztmcm/m/n/10.5 )\OML/ ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 x; Ty\OT1/ztmcm/m/ n/10.5 )\OML/ztmcm/m/it/10.5 ; d\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 y; Tx \OT1/ztmcm/m/n/10.5 ))$ [] [182] [183] [184] [185] [186] [187] [188] [189] Overfull \hbox (4.12213pt too wide) in paragraph at lines 772--773 \OT1/ptm/m/it/10.5 for all $\OML/ztmcm/m/it/10.5 x; y \OMS/zmtt/m/n/10.5 2 \OML /ztmcm/m/it/10.5 X$\OT1/ptm/m/it/10.5 . Hence, $\OML/ztmcm/m/it/10.5 T$ \OT1/pt m/m/it/10.5 is a $\OML/ztmcm/m/it/10.5 ^^L$\OT1/ptm/m/it/10.5 -quasi-contractio n while is not a quasi-contraction. [] [190] [191] [192] Underfull \hbox (badness 10000) in paragraph at lines 851--852 [] Underfull \hbox (badness 10000) in paragraph at lines 853--858 [] Underfull \hbox (badness 10000) in paragraph at lines 859--860 [] Overfull \hbox (5.38731pt too wide) in paragraph at lines 862--863 \OML/ztmcm/m/it/10.5 \OT1/ztmcm/m/n/10.5 (\OML/ztmcm/m/it/10.5 y; z\OT1/ztmcm/ m/n/10.5 ) \OMS/zmtt/m/n/10.5 ^^U \OT1/ztmcm/m/n/10.5 1$ \OT1/ptm/m/n/10.5 for all $\OML/ztmcm/m/it/10.5 z \OMS/zmtt/m/n/10.5 2 \OML/ztmcm/m/it/10.5 Fy$\OT1/p tm/m/n/10.5 . Also, re-call that $\OML/ztmcm/m/it/10.5 F$ \OT1/ptm/m/n/10.5 is con-tin-u-ous when-ever $\OML/ztmcm/m/it/10.5 H\OT1/ztmcm/m/n/10.5 (\OML/ztmcm/ m/it/10.5 Fx[]; Fx\OT1/ztmcm/m/n/10.5 ) \OMS/zmtt/m/n/10.5 ! [] [193] [194] [195] [196] Overfull \hbox (13.76953pt too wide) detected at line 977 \OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 ; F\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/1 0 ))) \OMS/zmtt/m/n/10 ^^T \OML/ztmcm/m/it/10 H\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/ it/10 F\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m /it/10 y\OT1/ztmcm/m/n/10 ))\OML/ztmcm/m/it/10 ; F\OT1/ztmcm/m/n/10 (\OML/ztmcm /m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 ))) \OMS/z mtt/m/n/10 ^^T \OML/ztmcm/m/it/10 \OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT 1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 ; '[] \OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 ))\OML/ztmcm/m/it/10 H \OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 F\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[ ]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 ))\OML/ztmcm/m/it/10 ; F\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/ 10 y\OT1/ztmcm/m/n/10 ))) [] [197] [198] Overfull \hbox (0.1193pt too wide) detected at line 1033 \OML/ztmcm/m/it/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 ; F\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/1 0 ))) \OMS/zmtt/m/n/10 ^^T \OML/ztmcm/m/it/10 hd\OT1/ztmcm/m/n/10 (\OML/ztmcm/m /it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )\OML/ztmcm/ m/it/10 ; '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )) \OML/z tmcm/m/it/10 < rd\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\O ML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 ; '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )) \OML/ztmcm/m/it/10 < rd\OT1/ztmcm/m/n /10 (\OML/ztmcm/m/it/10 '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm/m/ n/10 )\OML/ztmcm/m/it/10 ; '[]\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 y\OT1/ztmcm /m/n/10 )) + \OML/ztmcm/m/it/10 q[]: [] [199] [200] [201] [202] [203] Overfull \hbox (13.18486pt too wide) detected at line 1137 \OML/ztmcm/m/it/10 ^^Q\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 r\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 D\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; F\OT1/ztmcm/m/n/1 0 (\OML/ztmcm/m/it/10 x\OT1/ztmcm/m/n/10 )) \OMS/zmtt/m/n/10 ^^T \OML/ztmcm/m/i t/10 d\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; y\OT1/ztmcm/m/n/10 ) [][] []\OML /ztmcm/m/it/10 H\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 F\OT1/ztmcm/m/n/10 (\OML/ ztmcm/m/it/10 x\OT1/ztmcm/m/n/10 )\OML/ztmcm/m/it/10 ; F\OT1/ztmcm/m/n/10 (\OML /ztmcm/m/it/10 y\OT1/ztmcm/m/n/10 )) \OMS/zmtt/m/n/10 ^^T \OML/ztmcm/m/it/10 ^^ Ld\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; y\OT1/ztmcm/m/n/10 ) + \OML/ztmcm/m/ it/10 ^^LD\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x; F\OT1/ztmcm/m/n/10 (\OML/ztm cm/m/it/10 x\OT1/ztmcm/m/n/10 )) + \OML/ztmcm/m/it/10 ^^LD\OT1/ztmcm/m/n/10 (\O ML/ztmcm/m/it/10 y; F\OT1/ztmcm/m/n/10 (\OML/ztmcm/m/it/10 x\OT1/ztmcm/m/n/10 ) ) [] [204] Overfull \hbox (14.96515pt too wide) detected at line 1148 \OML/ztmcm/m/it/9 ^^Q\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 r\OT1/ztmcm/m/n/9 )\OM L/ztmcm/m/it/9 D\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 x; F\OT1/ztmcm/m/n/9 (\OML/ ztmcm/m/it/9 x\OT1/ztmcm/m/n/9 )) \OMS/zmtt/m/n/9 ^^T \OML/ztmcm/m/it/9 d\OT1/z tmcm/m/n/9 (\OML/ztmcm/m/it/9 x; y\OT1/ztmcm/m/n/9 ) [][] [][\OML/ztmcm/m/it/9 H\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 F\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 x\OT1 /ztmcm/m/n/9 )\OML/ztmcm/m/it/9 ; F\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 y\OT1/zt mcm/m/n/9 ))][] \OMS/zmtt/m/n/9 ^^T \OML/ztmcm/m/it/9 ^^L\OT1/ztmcm/m/n/9 [\OML /ztmcm/m/it/9 d\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 x; y\OT1/ztmcm/m/n/9 )][] + \OML/ztmcm/m/it/9 ^^L\OT1/ztmcm/m/n/9 [\OML/ztmcm/m/it/9 D\OT1/ztmcm/m/n/9 (\OM L/ztmcm/m/it/9 x; F\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 x\OT1/ztmcm/m/n/9 ))][] + \OML/ztmcm/m/it/9 ^^L\OT1/ztmcm/m/n/9 [\OML/ztmcm/m/it/9 D\OT1/ztmcm/m/n/9 (\ OML/ztmcm/m/it/9 y; F\OT1/ztmcm/m/n/9 (\OML/ztmcm/m/it/9 x\OT1/ztmcm/m/n/9 ))][ ] [] [205]) [206] (./document.ind) (./document.aux (./frontmatter/preface.aux) (./frontmatter/contributor.aux) (./chapters/chapter1/ch1.aux) (./chapters/chapter2/ch2.aux) (./chapters/chapter3/ch3.aux) (./bu.aux) (./chapters/chapter4/ch4.aux) (./bu.aux) (./chapters/chapter5/ch5.aux)) LaTeX Font Warning: Size substitutions with differences (Font) up to 0.45pt have occurred. LaTeX Font Warning: Some font shapes were not available, defaults substituted. LaTeX Warning: There were undefined references. LaTeX Warning: There were multiply-defined labels. ) Here is how much of TeX's memory you used: 21351 strings out of 478994 436257 string characters out of 5862207 702646 words of memory out of 5000000 38316 multiletter control sequences out of 15000+600000 542355 words of font info for 370 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 87i,14n,89p,2350b,893s stack positions out of 5000i,500n,10000p,200000b,80000s {c:/texlive/2021/texmf-dist/fonts/enc/dvips/base/8r.enc} Output written on document.pdf (216 pages, 933318 bytes). PDF statistics: 784 PDF objects out of 1000 (max. 8388607) 540 compressed objects within 6 object streams 0 named destinations out of 1000 (max. 500000) 13 words of extra memory for PDF output out of 10000 (max. 10000000)