\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{dudtuck}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction and results}{1}{section.1}}
\citation{chen}
\citation{chen}
\citation{spilker}
\citation{spilker}
\citation{spilker}
\citation{wall}
\citation{spilker}
\newlabel{sp1}{{B}{2}{\cite {spilker}}{theoremA.2}{}}
\newlabel{sp2}{{C}{2}{\cite {spilker}}{theoremA.3}{}}
\newlabel{sp3}{{D}{2}{\cite {spilker}}{theoremA.4}{}}
\newlabel{thm1}{{1}{3}{}{theorem.1}{}}
\newlabel{thm2}{{2}{3}{}{theorem.2}{}}
\citation{koshy}
\newlabel{thm3}{{3}{4}{}{theorem.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Preliminaries}{4}{section.2}}
\newlabel{lemggt}{{4}{4}{}{theorem.4}{}}
\newlabel{divrules}{{5}{4}{\cite {koshy}}{theorem.5}{}}
\newlabel{lem2}{{6}{4}{}{theorem.6}{}}
\newlabel{lem4}{{7}{5}{}{theorem.7}{}}
\newlabel{lem3}{{8}{5}{}{theorem.8}{}}
\newlabel{lem5}{{9}{5}{}{theorem.9}{}}
\citation{chen}
\newlabel{lem6}{{10}{6}{}{theorem.10}{}}
\newlabel{lem7}{{11}{6}{}{theorem.11}{}}
\newlabel{chen}{{12}{6}{\cite {chen}}{theorem.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Proof of Theorem \ref  {thm1} and Theorem \ref  {thm2}}{6}{section.3}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Proof of Theorem \ref  {thm3}}{9}{section.4}}
\citation{wall}
\citation{spilker}
\citation{chen}
\citation{spilker}
\newlabel{eqsp1}{{65}{10}{Proof of Theorem \ref {thm3}}{equation.4.65}{}}
\newlabel{eqsp2}{{68}{10}{Proof of Theorem \ref {thm3}}{equation.4.68}{}}
\bibcite{chen}{1}
\bibcite{dudtuck}{2}
\bibcite{koshy}{3}
\bibcite{spilker}{4}
\bibcite{wall}{5}
\@writefile{toc}{\contentsline {section}{\numberline {5}Remarks and future work}{11}{section.5}}
\newlabel{secweiter}{{5}{11}{Remarks and future work}{section.5}{}}