\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} }