dissertation_work/Dissertation/.texpadtmp/ann_product.aux

\relax 
\providecommand\hyper@newdestlabel[2]{}
\citation{grohs2019spacetime}
\@writefile{toc}{\contentsline {chapter}{\numberline {6}ANN Product Approximations}{106}{chapter.6}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Approximation for Products of Two Real Numbers}{106}{section.6.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1.1}The squares of real numbers}{107}{subsection.6.1.1}\protected@file@percent }
\newlabel{lem:6.1.1}{{6.1.1}{107}{}{theorem.6.1.1}{}}
\newlabel{lem:6.1.1@cref}{{[lemma][1][6,1]6.1.1}{[1][106][]107}}
\newlabel{(6.0.1)}{{6.1.1}{107}{}{equation.6.1.1}{}}
\newlabel{(6.0.1)@cref}{{[equation][1][6,1]6.1.1}{[1][106][]107}}
\newlabel{(6.0.3)}{{6.1.4}{108}{The squares of real numbers}{equation.6.1.4}{}}
\newlabel{(6.0.3)@cref}{{[equation][4][6,1]6.1.4}{[1][107][]108}}
\newlabel{(6.0.4.2)}{{6.1.5}{108}{The squares of real numbers}{equation.6.1.5}{}}
\newlabel{(6.0.4.2)@cref}{{[equation][5][6,1]6.1.5}{[1][108][]108}}
\newlabel{(6.0.5)}{{6.1.6}{108}{The squares of real numbers}{equation.6.1.6}{}}
\newlabel{(6.0.5)@cref}{{[equation][6][6,1]6.1.6}{[1][108][]108}}
\newlabel{6.0.6}{{6.1.7}{108}{The squares of real numbers}{equation.6.1.7}{}}
\newlabel{6.0.6@cref}{{[equation][7][6,1]6.1.7}{[1][108][]108}}
\newlabel{6.0.7}{{6.1.8}{109}{The squares of real numbers}{equation.6.1.8}{}}
\newlabel{6.0.7@cref}{{[equation][8][6,1]6.1.8}{[1][108][]109}}
\newlabel{6.0.8}{{6.1.9}{109}{The squares of real numbers}{equation.6.1.9}{}}
\newlabel{6.0.8@cref}{{[equation][9][6,1]6.1.9}{[1][109][]109}}
\newlabel{6.0.9}{{6.1.10}{109}{The squares of real numbers}{equation.6.1.10}{}}
\newlabel{6.0.9@cref}{{[equation][10][6,1]6.1.10}{[1][109][]109}}
\newlabel{6.0.11}{{6.1.12}{109}{The squares of real numbers}{equation.6.1.12}{}}
\newlabel{6.0.11@cref}{{[equation][12][6,1]6.1.12}{[1][109][]109}}
\newlabel{6.0.12}{{6.1.13}{110}{The squares of real numbers}{equation.6.1.13}{}}
\newlabel{6.0.12@cref}{{[equation][13][6,1]6.1.13}{[1][109][]110}}
\newlabel{6.0.13}{{6.1.14}{110}{The squares of real numbers}{equation.6.1.14}{}}
\newlabel{6.0.13@cref}{{[equation][14][6,1]6.1.14}{[1][110][]110}}
\newlabel{6.0.14}{{6.1.15}{110}{The squares of real numbers}{equation.6.1.15}{}}
\newlabel{6.0.14@cref}{{[equation][15][6,1]6.1.15}{[1][110][]110}}
\newlabel{(6.0.17)}{{6.1.17}{111}{The squares of real numbers}{equation.6.1.17}{}}
\newlabel{(6.0.17)@cref}{{[equation][17][6,1]6.1.17}{[1][110][]111}}
\newlabel{6.1.1.1}{{6.1.2.1}{112}{}{corollary.6.1.2.1}{}}
\newlabel{6.1.1.1@cref}{{[corollary][1][]6.1.2.1}{[1][112][]112}}
\citation{grohs2019spacetime}
\newlabel{6.0.3}{{6.1.4}{113}{}{theorem.6.1.4}{}}
\newlabel{6.0.3@cref}{{[lemma][4][6,1]6.1.4}{[1][113][]113}}
\newlabel{6.0.21}{{6.1.29}{114}{The squares of real numbers}{equation.6.1.29}{}}
\newlabel{6.0.21@cref}{{[equation][29][6,1]6.1.29}{[1][114][]114}}
\newlabel{6.0.23}{{6.1.31}{114}{The squares of real numbers}{equation.6.1.31}{}}
\newlabel{6.0.23@cref}{{[equation][31][6,1]6.1.31}{[1][114][]114}}
\newlabel{6.0.24}{{6.1.32}{115}{The squares of real numbers}{equation.6.1.32}{}}
\newlabel{6.0.24@cref}{{[equation][32][6,1]6.1.32}{[1][114][]115}}
\newlabel{6.0.25}{{6.1.33}{115}{The squares of real numbers}{equation.6.1.33}{}}
\newlabel{6.0.25@cref}{{[equation][33][6,1]6.1.33}{[1][115][]115}}
\newlabel{6.0.25}{{6.1.34}{115}{The squares of real numbers}{equation.6.1.34}{}}
\newlabel{6.0.25@cref}{{[equation][34][6,1]6.1.34}{[1][115][]115}}
\newlabel{6.0.26}{{6.1.35}{116}{The squares of real numbers}{equation.6.1.35}{}}
\newlabel{6.0.26@cref}{{[equation][35][6,1]6.1.35}{[1][115][]116}}
\newlabel{(6.1.42)}{{6.1.43}{117}{The squares of real numbers}{equation.6.1.43}{}}
\newlabel{(6.1.42)@cref}{{[equation][43][6,1]6.1.43}{[1][117][]117}}
\citation{grohs2019spacetime}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1.2}The $\prd  $ network}{118}{subsection.6.1.2}\protected@file@percent }
\newlabel{prd_network}{{6.1.7}{118}{}{theorem.6.1.7}{}}
\newlabel{prd_network@cref}{{[lemma][7][6,1]6.1.7}{[1][118][]118}}
\newlabel{6.0.33}{{6.1.47}{119}{The $\prd $ network}{equation.6.1.47}{}}
\newlabel{6.0.33@cref}{{[equation][47][6,1]6.1.47}{[1][118][]119}}
\newlabel{(6.1.49)}{{6.1.50}{120}{The $\prd $ network}{equation.6.1.50}{}}
\newlabel{(6.1.49)@cref}{{[equation][50][6,1]6.1.50}{[1][120][]120}}
\newlabel{(6.1.50)}{{6.1.51}{121}{The $\prd $ network}{equation.6.1.51}{}}
\newlabel{(6.1.50)@cref}{{[equation][51][6,1]6.1.51}{[1][120][]121}}
\newlabel{cor_prd}{{6.1.7.1}{122}{}{corollary.6.1.7.1}{}}
\newlabel{cor_prd@cref}{{[corollary][1][]6.1.7.1}{[1][122][]122}}
\@writefile{toc}{\contentsline {section}{\numberline {6.2}Higher Approximations}{123}{section.6.2}\protected@file@percent }
\newlabel{sec_tun}{{6.2}{123}{Higher Approximations}{section.6.2}{}}
\newlabel{sec_tun@cref}{{[section][2][6]6.2}{[1][123][]123}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.1}The $\tun  $ Neural Networks and Their Properties}{124}{subsection.6.2.1}\protected@file@percent }
\newlabel{6.2.2}{{6.2.3}{124}{}{theorem.6.2.3}{}}
\newlabel{6.2.2@cref}{{[lemma][3][6,2]6.2.3}{[1][124][]124}}
\newlabel{tun_mult}{{6.2.6}{127}{}{theorem.6.2.6}{}}
\newlabel{tun_mult@cref}{{[lemma][6][6,2]6.2.6}{[1][127][]127}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.2}The $\pwr  $ Neural Networks and Their Properties}{129}{subsection.6.2.2}\protected@file@percent }
\newlabel{6.2.4}{{6.2.9}{130}{}{theorem.6.2.9}{}}
\newlabel{6.2.4@cref}{{[lemma][9][6,2]6.2.9}{[1][130][]130}}
\newlabel{(6.2.31)}{{6.2.17}{131}{The $\pwr $ Neural Networks and Their Properties}{equation.6.2.17}{}}
\newlabel{(6.2.31)@cref}{{[equation][17][6,2]6.2.17}{[1][131][]131}}
\newlabel{power_prop}{{6.2.12}{131}{R---,2023}{theorem.6.2.12}{}}
\newlabel{power_prop@cref}{{[lemma][12][6,2]6.2.12}{[1][131][]131}}
\newlabel{(6.2.19)}{{6.2.31}{135}{The $\pwr $ Neural Networks and Their Properties}{equation.6.2.31}{}}
\newlabel{(6.2.19)@cref}{{[equation][31][6,2]6.2.31}{[1][135][]135}}
\newlabel{(6.2.39)}{{6.2.43}{138}{The $\pwr $ Neural Networks and Their Properties}{equation.6.2.43}{}}
\newlabel{(6.2.39)@cref}{{[equation][43][6,2]6.2.43}{[1][138][]138}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.3}The $\tay  $ Neural Networks and Their Properties}{139}{subsection.6.2.3}\protected@file@percent }
\newlabel{6.2.14}{{6.2.48}{139}{}{equation.6.2.48}{}}
\newlabel{6.2.14@cref}{{[equation][48][6,2]6.2.48}{[1][139][]139}}
\newlabel{6.2.8}{{6.2.17}{140}{}{theorem.6.2.17}{}}
\newlabel{6.2.8@cref}{{[lemma][17][6,2]6.2.17}{[1][140][]140}}
\newlabel{6.2.9}{{6.2.18}{140}{R---,2023}{theorem.6.2.18}{}}
\newlabel{6.2.9@cref}{{[lemma][18][6,2]6.2.18}{[1][140][]140}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.4}Neural Network Approximations For $e^x$.}{144}{subsection.6.2.4}\protected@file@percent }
\@setckpt{ann_product}{
\setcounter{page}{145}
\setcounter{equation}{63}
\setcounter{enumi}{6}
\setcounter{enumii}{0}
\setcounter{enumiii}{0}
\setcounter{enumiv}{0}
\setcounter{footnote}{1}
\setcounter{mpfootnote}{0}
\setcounter{part}{2}
\setcounter{chapter}{6}
\setcounter{section}{2}
\setcounter{subsection}{4}
\setcounter{subsubsection}{0}
\setcounter{paragraph}{0}
\setcounter{subparagraph}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{@pps}{0}
\setcounter{@ppsavesec}{0}
\setcounter{@ppsaveapp}{0}
\setcounter{AM@survey}{0}
\setcounter{parentequation}{0}
\setcounter{section@level}{2}
\setcounter{Item}{162}
\setcounter{Hfootnote}{1}
\setcounter{Hy@AnnotLevel}{0}
\setcounter{bookmark@seq@number}{42}
\setcounter{NAT@ctr}{0}
\setcounter{ALG@line}{0}
\setcounter{ALG@rem}{0}
\setcounter{ALG@nested}{0}
\setcounter{ALG@Lnr}{1}
\setcounter{ALG@blocknr}{1}
\setcounter{ALG@storecount}{0}
\setcounter{ALG@tmpcounter}{0}
\setcounter{@stackindex}{0}
\setcounter{ROWcellindex@}{0}
\setcounter{lstnumber}{1}
\setcounter{theorem}{19}
\setcounter{corollary}{0}
\setcounter{lstlisting}{0}
\setcounter{nn@layerindex@counter}{4}
\setcounter{nn@lastlayerstart@counter}{0}
\setcounter{nn@thislayerstart@counter}{1}
\setcounter{nn@lastlayercount@counter}{3}
\setcounter{nn@thislayercount@counter}{2}
\setcounter{nn@lastlayerindex@counter}{2}
\setcounter{nn@thislayerindex@counter}{3}
}