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[]\TU/lmr/m/n/10.95 However note also that since $\OML/cmm/m/it/10.95 G[]$ \TU/lmr/m/n/10.95 is upper semi-continuous, further the fact that, $\OML/cmm/m/it/10.95 ^^^[] \OMS/cmsy/m/n/10.95 2 []$\TU/lmr/m/n/10.95 ,
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[]\TU/lmr/m/n/10.95 Since it is the case that for all $\OML/cmm/m/it/10.95 j \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$ \TU/lmr/m/n/10.95 that: $[] [] \OT1/cmr/m/n/10.95 = []$\TU/lmr/m/n/10.95 , where $\OML/cmm/m/it/10.95 l[]; l[]; :::; l[]; L \OMS/cmsy/m/n/10.95 2
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\TU/lmr/m/n/10.95 By symmetry note also that $[] [] \OT1/cmr/m/n/10.95 = [] []$ \TU/lmr/m/n/10.95 and also that $[] [] \OT1/cmr/m/n/10.95 =
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\TU/lmr/m/n/10.95 Note that since $\U/euf/m/n/10.95 p[] \OMS/cmsy/m/n/10.95 2 O []$ \TU/lmr/m/n/10.95 for $\OML/cmm/m/it/10.95 n \U/msa/m/n/10.95 > \OT1/cmr/m/n/10.95 2$\TU/lmr/m/n/10.95 , it is the case for all $\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 R$ \TU/lmr/m/n/10.95 then that $[] \OMS/cmsy/m/n/10.95 2
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[]\TU/lmr/m/it/10.95 that for all $\OML/cmm/m/it/10.95 ^^R \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 ^^B$\TU/lmr/m/it/10.95 , $\OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N[]$\TU/lmr/m/it/10.95 , $\OML/cmm/m/it/10.95 t \OMS/cmsy/m/n/10.95 2 []$\TU/lmr/m/it/10.95 , that $[][] \U/msa/m/n/10.95 6 [][] [] []$
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[]\TU/lmr/bx/n/10.95 Definition 8.1.1 \TU/lmr/m/n/10.95 (Activation ANN)\TU/lmr/bx/n/10.95 . []\TU/lmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N$\TU/lmr/m/it/10.95 . We denote by $\U/euf/m/n/10.95 i[] \OMS/cmsy/m/n/10.95 2 [] ^^R
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\OMS/cmsy/m/n/10.95 f\OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 3\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 4\OML/cmm/m/it/10.95 ; :::\OMS/cmsy/m/n/10.95 g$\TU/lmr/m/n/10.95 , $\OML/cmm/m/it/10.95 x \OT1/cmr/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x[]; x[]; :::; x[]\OMS/cmsy/m/n/10.95 g 2 \U/msb/m/n/10.95 R[]$ \TU/lmr/m/n/10.95 it holds that $[] [] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 C []$\TU/lmr/m/n/10.95 . and $[] [] \OT1/cmr/m/n/10.95 =
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[]\TU/lmr/m/it/10.95 it holds that $[] [] \U/msa/m/n/10.95 6 [] [] \OT1/cmr/m/n/10.95 + 3 [] \OMS/cmsy/m/n/10.95 ^^A [] \OT1/cmr/m/n/10.95 + 7 + 3\OML/cmm/m/it/10.95 N \OT1/cmr/m/n/10.95 + \OML/cmm/m/it/10.95 d[] [] \OT1/cmr/m/n/10.95 + \OML/cmm/m/it/10.95 d []$
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[]\TU/lmr/m/n/10.95 Note that for all $\OML/cmm/m/it/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; [] ; \U/euf/m/n/10.95 n\OMS/cmsy/m/n/10.95 g$\TU/lmr/m/n/10.95 , Lemma [][]9.0.3[][] tells us that $[][] [] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 C[] []$\TU/lmr/m/n/10.95 .
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[]$\TU/lmr/m/it/10.95 , $\OML/cmm/m/it/10.95 j \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N \OMS/cmsy/m/n/10.95 \ []$\TU/lmr/m/it/10.95 , is given by $\OML/cmm/m/it/10.95 c \U/msa/m/n/10.95 ~[] \OML/cmm/m/it/10.95 ^^W \OT1/cmr/m/n/10.95 = []$
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(/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/main.aux (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/Introduction.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/Brownian_motion_monte_carlo.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/u_visc_sol.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/brownian_motion_monte_carlo_non_linear_case.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/neural_network_introduction.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/ann_product.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/modified_mlp_associated_nn.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/ann_first_approximations.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/ann_rep_brownian_motion_monte_carlo.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/conclusions-further-research.aux) (/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/appendices.aux))
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LaTeX Font Warning: Some font shapes were not available, defaults substituted.
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LaTeX Warning: There were undefined references.
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LaTeX Warning: There were multiply-defined labels.
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Package rerunfilecheck Info: File `main.out' has not changed.
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(rerunfilecheck) Checksum: 4D3B80E0A07334E405D495AE556C4B5B;17945.
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Here is how much of TeX's memory you used:
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36233 strings out of 476683
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2311018 words of memory out of 5000000
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55628 multiletter control sequences out of 15000+600000
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540453 words of font info for 208 fonts, out of 8000000 for 9000
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1348 hyphenation exceptions out of 8191
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106i,25n,99p,1066b,2078s stack positions out of 10000i,1000n,20000p,200000b,200000s
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Output written on /Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/.texpadtmp/main.pdf (230 pages).
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