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[36] [37] [38] [39] Overfull \hbox (0.39136pt too wide) in paragraph at lines 465--468 []\OT1/cmr/m/n/10.95 However note also that since $\OML/cmm/m/it/10.95 G[]$ \OT 1/cmr/m/n/10.95 is up-per semi-continuous, fur-ther the fact that, $\OML/cmm/m/ it/10.95 ^^^[] \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/n/10.95 , [] Overfull \hbox (81.7409pt too wide) in paragraph at lines 465--468 \OT1/cmr/m/n/10.95 and then $([]3\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 2\OML/ cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 27[])$, and $([]3\OML/cmm/m/it/10.95 :\OT1/c mr/m/n/10.95 2\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 16[])$, im-ply for all $\ OML/cmm/m/it/10.95 " \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/1 0.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ we have that: $[][] [] = [] [40] [41] [42] [43] [44] [45] [46] [47] [48] Overfull \hbox (86.18391pt too wide) in paragraph at lines 743--744 \OT1/cmr/m/it/10.95 ery $\OML/cmm/m/it/10.95 r \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m /n/10.95 (0\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ \OT 1/cmr/m/it/10.95 sat-isfy the con-di-tion that $\OML/cmm/m/it/10.95 O[] \OMS/cm sy/m/n/10.95 ^^R O$\OT1/cmr/m/it/10.95 , where $\OML/cmm/m/it/10.95 O[] \OT1/cm r/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2 O \OT1/cmr/m/n/10.95 : []\OMS/cmsy/m/n/10.95 g$ [] [49] [50] [51] [52] [53] Overfull \hbox (2.98135pt too wide) in paragraph at lines 894--895 []\OT1/cmr/bx/n/10.95 Corollary 3.3.1.1. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/ it/10.95 T \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OM S/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$\OT1/cmr/m/it/10.95 , let $[]$ be a prob -a-bil-ity space, let $\OML/cmm/m/it/10.95 u[] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m /it/10.95 C[] []$\OT1/cmr/m/it/10.95 , [] [54] [55] [56] [57]) [58] \openout2 = `brownian_motion_monte_carlo_non_linear_case.aux'. (./brownian_motion_monte_carlo_non_linear_case.tex Chapter 4. LaTeX Warning: Reference `3.3.21' on page 59 undefined on input line 3. [59 ]) [60] [61 ] \openout2 = `neural_network_introduction.aux'. (./neural_network_introduction.tex Chapter 5. [62 ] [63] [64] Missing character: There is no ; in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no ; in font nullfont! 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Fur-ther-more let $[] \OMS/cmsy/m/n/10.95 2 []$ [] Overfull \hbox (3.52759pt too wide) in paragraph at lines 327--329 \OT1/cmr/m/n/10.95 See Re-mark 5.3.2. Let $\OML/cmm/m/it/10.95 x[]; x[]; :::; x [] \OMS/cmsy/m/n/10.95 2 []$ \OT1/cmr/m/n/10.95 sat-isfy for all $\OML/cmm/m/it /10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/ cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$ [] [71] [72] [73] Overfull \hbox (20.19258pt too wide) detected at line 410 [] [] [74] [75] [76] [77] LaTeX Warning: Reference `5.4.4' on page 78 undefined on input line 546. LaTeX Warning: Reference `5.4.5' on page 78 undefined on input line 546. LaTeX Warning: Reference `5.4.4' on page 78 undefined on input line 559. LaTeX Warning: Reference `5.4.5' on page 78 undefined on input line 610. Overfull \hbox (24.45033pt too wide) detected at line 657 [] [] [78] [79] [80] [81] [82] [83] LaTeX Warning: Command \l invalid in math mode on input line 914. [84] [85] [86] [87] [88] LaTeX Warning: Reference `5.5.14' on page 89 undefined on input line 1095. [89] [90] [91] [92] [93]) [94] \openout2 = `ann_product.aux'. (./ann_product.tex Chapter 6. [95 ] [96] [97] [98] [99] Overfull \hbox (13.65341pt too wide) detected at line 188 [] [] [100] [101] Overfull \hbox (7.17166pt too wide) in paragraph at lines 226--228 \OT1/cmr/m/n/10.95 This and the as-sump-tion that $^^H \OMS/cmsy/m/n/10.95 2 \O ML/cmm/m/it/10.95 C []$ \OT1/cmr/m/n/10.95 along with the as-sump-tion that $[] [] \OMS/cmsy/m/n/10.95 j [] [] ^^@ [] Overfull \hbox (17.15163pt too wide) in paragraph at lines 237--239 \OT1/cmr/m/n/10.95 The as-sump-tion that for all $[][] \OMS/cmsy/m/n/10.95 j [] [] ^^@ [] [] j \OT1/cmr/m/n/10.95 = 0$ and the as-sump-tion that $[][] \OMS/cm sy/m/n/10.95 j\OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@ [] [102] [103] [104] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 317. 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[114] Underfull \hbox (badness 10000) in paragraph at lines 700--706 [] [115] Underfull \hbox (badness 10000) in paragraph at lines 709--714 [] [116] [117] [118] [119] [120] [121] [122] Overfull \hbox (7.31064pt too wide) in paragraph at lines 911--913 \OT1/cmr/m/n/10.95 Note that since $\U/euf/m/n/10.95 p[] \OMS/cmsy/m/n/10.95 2 O []$ \OT1/cmr/m/n/10.95 for $\OML/cmm/m/it/10.95 n \U/msa/m/n/10.95 > \OT1/cmr /m/n/10.95 2$, it is the case for all $\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.9 5 2 \U/msb/m/n/10.95 R$ \OT1/cmr/m/n/10.95 then that $[] \OMS/cmsy/m/n/10.95 2 [] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 915. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathop' on input line 915. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 915. 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[123] [124] Overfull \hbox (34.92166pt too wide) in paragraph at lines 961--966 []$[] [] \U/msa/m/n/10.95 6 []$ [] Underfull \hbox (badness 10000) in paragraph at lines 961--966 [] Underfull \hbox (badness 10000) in paragraph at lines 969--971 [] [125] Overfull \hbox (7.30336pt too wide) detected at line 1009 [] [] [126] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1042. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `superscript' on input line 1042. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1042. [127]) [128] \openout2 = `modified_mlp_associated_nn.aux'. (./modified_mlp_associated_nn.tex Chapter 7. [129 ] [130] Overfull \hbox (29.14606pt too wide) in paragraph at lines 54--55 []\OT1/cmr/m/it/10.95 that for all $\OML/cmm/m/it/10.95 ^^R \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 ^^B$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 n \OMS/cms y/m/n/10.95 2 \U/msb/m/n/10.95 N[]$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 t \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/it/10.95 , that $[][] \U/msa/m/n/10.95 6 [ ][] [] []$ [] ) [131] \openout2 = `categorical_neural_network.aux'. (./categorical_neural_network.tex Chapter 8. ) [132 ] Chapter 9. Overfull \hbox (5.90657pt too wide) in paragraph at lines 169--170 []\OT1/cmr/bx/n/10.95 Definition 9.1.1 \OT1/cmr/m/n/10.95 (Ac-ti-va-tion ANN)\O T1/cmr/bx/n/10.95 . []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 n \OMS/cmsy/ m/n/10.95 2 \U/msb/m/n/10.95 N$\OT1/cmr/m/it/10.95 . We de-note by $\U/euf/m/n/ 10.95 i[] \OMS/cmsy/m/n/10.95 2 [] ^^R [] [133 ] [134] [135] Overfull \hbox (4.2524pt too wide) in paragraph at lines 263--264 []\OT1/cmr/bx/n/10.95 Lemma 9.2.3. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/it/10. 95 ^^W \OMS/cmsy/m/n/10.95 2 []$ \OT1/cmr/m/it/10.95 with end-widths $\OML/cmm/ m/it/10.95 d$\OT1/cmr/m/it/10.95 . It is then the case that $[][] [] [] \OT1/cm r/m/n/10.95 = [][] [] = [] [136] Overfull \hbox (2.78886pt too wide) detected at line 337 [] [] [137] Overfull \hbox (3.18243pt too wide) detected at line 411 [] [] [138] [139] Overfull \hbox (7.72708pt too wide) detected at line 517 [] [] [140] [141] [142] [143] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 628. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathop' on input line 628. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 628. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 628. 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[144] Overfull \hbox (32.59174pt too wide) in paragraph at lines 670--672 []\OT1/cmr/m/n/10.95 9.4.4[] then tells us that for all $\OML/cmm/m/it/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 0\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/1 0.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; N\OM S/cmsy/m/n/10.95 g$\OT1/cmr/m/n/10.95 , $[] [] = []$, $[][] [] \OMS/cmsy/m/n/10 .95 2 [] [145] [146] [147] [148] Overfull \hbox (15.04857pt too wide) detected at line 761 [] [] [149] Overfull \hbox (14.36722pt too wide) in paragraph at lines 797--798 []\OT1/cmr/bx/n/10.95 Corollary 9.5.2.1. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/ it/10.95 " \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OM S/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10. 95 L; a \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 R$\OT1/cmr/m/it/10.95 , $\OML/cm m/m/it/10.95 b \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10 .95 N \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N[] \OMS/cmsy/m/n/10.95 \ []$\OT1/ cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 x[]; x[]; :::; x[] \OMS/cmsy/m/n/10.95 2 [] [150] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 817. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathop' on input line 817. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 817. 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[151] [152] [153] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathop' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathchardef' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\std@minus' on input line 908. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 908. 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[154] Overfull \hbox (0.97372pt too wide) in paragraph at lines 953--955 \OT1/cmr/m/n/10.95 lay-ers re-mains un-changed un-der par-al-leliza-tion it is then the case that $[] [] = [] [] = [] [155] [156] [157] [158] [159] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1116. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathop' on input line 1116. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 1116. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 1116. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathcode' on input line 1116. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `\mathchardef' on input line 1116. 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[160] [161] [162] [163] [164] [165] [166] Overfull \hbox (19.2441pt too wide) in paragraph at lines 1302--1303 \OT1/cmr/m/it/10.95 for all $\OML/cmm/m/it/10.95 x[]; y[] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 E$ \OT1/cmr/m/it/10.95 that $[] \U/msa/m/n/10.95 6 \OML/cm m/m/it/10.95 L [][]$ \OT1/cmr/m/it/10.95 and let $[] \OMS/cmsy/m/n/10.95 2 []$ \OT1/cmr/m/it/10.95 and $\OML/cmm/m/it/10.95 y \OT1/cmr/m/n/10.95 = []$ [] [167] [168] Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1341. Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing `math shift' on input line 1341. [169 ] \openout2 = `ann_rep_brownian_motion_monte_carlo.aux'. (./ann_rep_brownian_motion_monte_carlo.tex Chapter 10. [170 ] [171] [172] Overfull \hbox (15.57579pt too wide) in paragraph at lines 95--101 []$[] [] \U/msa/m/n/10.95 6 []$ [] [173] [174] [175] [176]) [177] (./main.bbl [178 ]) [179] [180 ] (/usr/local/texlive/2023/texmf-dist/tex/latex/listings/lstlang1.sty File: lstlang1.sty 2023/02/27 1.9 listings language file ) Package hyperref Info: bookmark level for unknown lstlisting defaults to 0 on i nput line 1353. File: Initial_Brownian_Motion.png Graphic file (type png) Package pdftex.def Info: Initial_Brownian_Motion.png used on input line 1375. (pdftex.def) Requested size: 461.72388pt x 450.68263pt. [181] [182 <./Initial_Brownian_Motion.png>] LaTeX Font Info: Trying to load font information for TS1+cmtt on input line 1411. 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Package rerunfilecheck Info: File `main.out' has not changed. (rerunfilecheck) Checksum: 966BEF3E4E950BBA4D809EE637146306;14811. ) Here is how much of TeX's memory you used: 34475 strings out of 476025 670396 string characters out of 5790018 2120388 words of memory out of 5000000 52896 multiletter control sequences out of 15000+600000 548725 words of font info for 186 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 99i,25n,97p,1071b,1622s stack positions out of 10000i,1000n,20000p,200000b,200000s < /usr/local/texlive/2023/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb>< /usr/local/texlive/2023/texmf-dist/fonts/type1/public/rsfs/rsfs10.pfb> Output written on main.pdf (185 pages, 1048897 bytes). PDF statistics: 3296 PDF objects out of 3580 (max. 8388607) 3005 compressed objects within 31 object streams 1357 named destinations out of 1440 (max. 500000) 487 words of extra memory for PDF output out of 10000 (max. 10000000)