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Chapter 1.
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[]\OT1/cmr/m/it/10.95 A strong so-lu-tion to the stochas-tic dif-fer-en-tial eq
ua-tion ([]1.1.8[]) on prob-a-bil-ity space $\OT1/cmr/m/n/10.95 (
\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 F\OML/cmm/m/it/10.95 ; \U/msb/m/n/10.
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Chapter 2.
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Chapter 3.
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[]\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 ,
[]
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\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: $[][] [] =
[]
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\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 $\OML/cmm/m/it/10.95 O[] \OT1/cm
r/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2
O \OT1/cmr/m/n/10.95 : []\OMS/cmsy/m/n/10.95 g$
[]
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[]\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 ,
[]
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Chapter 4.
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Chapter 5.
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[] \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/it/10.95 , and $\OML/cmm/m/it/10.95 ^^W[
] \OT1/cmr/m/n/10.95 = [] \OMS/cmsy/m/n/10.95 2
[]
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[]\OT1/cmr/m/n/10.95 Since it is the case that for all $\OML/cmm/m/it/10.95 j \
OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10
.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$ \OT1/cmr/m/n/10.95 that
: $[] [] = []$, where $\OML/cmm/m/it/10.95 l[]; l[]; :::; l[]; L \OMS/cmsy/m/n/
10.95 2
[]
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[]$\OT1/cmr/m/n/10.95 . Fur-ther-more let $[] \OMS/cmsy/m/n/10.95 2 []$
[]
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\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 $\OML/cmm/m/it
/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/
cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$
[]
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Chapter 6.
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\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 [] [] ^^@
[]
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\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
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\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 $[] \OMS/cmsy/m/n/10.95 2
[]
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[]$[] [] \U/msa/m/n/10.95 6 []$
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\openout2 = `modified_mlp_associated_nn.aux'.
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Chapter 7.
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[]\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'.
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Chapter 9.
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[]\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
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[]\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 = [][] [] =
[]
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[]\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
[]
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[]\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
[]
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\OT1/cmr/m/n/10.95 lay-ers re-mains un-changed un-der par-al-leliza-tion it is
then the case that $[] [] = [] [] =
[]
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\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 $\OML/cmm/m/it/10.95 y \OT1/cmr/m/n/10.95 = []$
[]
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