This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4) (format=pdflatex 2012.11.20) 12 DEC 2013 08:45 entering extended mode **dr.tex (./dr.tex LaTeX2e <2003/12/01> Babel and hyphenation patterns for american, french, german, ngerman, b ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur kish, ukrainian, nohyphenation, loaded. ! Undefined control sequence. l.1 \chapter {Dimensionality Reduction} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.1 \chapter{D imensionality Reduction} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no D in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no [ in font nullfont! \citation{DG02} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.3 ...ter is from Dasgupta and Gupta \cite{DG02} , Dubhashi \& Panconesi \c... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `DG02' on page undefined on input line 3. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no & in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no [ in font nullfont! \citation{DP09} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.3 ...te{DG02}, Dubhashi \& Panconesi \cite{DP09} , Matousek \cite{Mat96,Mat... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `DP09' on page undefined on input line 3. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no [ in font nullfont! \citation{Mat96} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.3 ...si \cite{DP09}, Matousek \cite{Mat96,Mat02} , Bourgain \cite{Bou85}, a... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Mat96' on page undefined on input line 3. Missing character: There is no , in font nullfont! \citation{Mat02} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.3 ...si \cite{DP09}, Matousek \cite{Mat96,Mat02} , Bourgain \cite{Bou85}, a... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Mat02' on page undefined on input line 3. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{Bou85} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.3 ...k \cite{Mat96,Mat02}, Bourgain \cite{Bou85} , and the lecture notes of... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bou85' on page undefined on input line 3. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 1--5 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 1--5 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 1--5 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 1--5 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 1--5 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 1--5 [] [] ! Undefined control sequence. l.6 \section {Preliminaries: Metric spaces and embeddings} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.6 \section{P reliminaries: Metric spaces and embeddings} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no P in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no : in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! 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Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! LaTeX Font Info: External font `cmex10' loaded for size (Font) <7> on input line 8. LaTeX Font Info: External font `cmex10' loaded for size (Font) <5> on input line 8. Missing character: There is no - in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 6--10 [] [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 6--10 \OML/cmm/m/it/10 n$ [] Overfull \hbox (25.94095pt too wide) in paragraph at lines 6--10 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X; d\OT1/cmr/m/n/10 )$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 6--10 \OML/cmm/m/it/10 n$ [] Overfull \hbox (15.27782pt too wide) in paragraph at lines 6--10 \OML/cmm/m/it/10 L[]$ [] ! Undefined control sequence. l.11 \section {Embeddings into $L_\infty$-normed spaces} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.11 \section{E mbeddings into $L_\infty$-normed spaces} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no E in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! \citation{Mat96} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.12 ...em due to Matousek, see \cite{Mat96,Mat02} , using the proof techniqu... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Mat96' on page undefined on input line 12. Missing character: There is no , in font nullfont! \citation{Mat02} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.12 ...em due to Matousek, see \cite{Mat96,Mat02} , using the proof techniqu... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Mat02' on page undefined on input line 12. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{Bou85} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.12 ...e proof technique of Bourgain \cite{Bou85} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bou85' on page undefined on input line 12. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! Undefined control sequence. l.13 We show that $\overset {O(\log^2n)}{\longrightarrow}L_\infty^{O(\l... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 11--14 [] [] Overfull \hbox (15.27782pt too wide) in paragraph at lines 11--14 \OML/cmm/m/it/10 L[]$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 11--14 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 11--14 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 11--14 [] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 11--14 \OML/cmm/m/it/10 < [] Overfull \hbox (28.71866pt too wide) in paragraph at lines 11--14 \OML/cmm/m/it/10 X; d > [] Overfull \hbox (97.27538pt too wide) in paragraph at lines 11--14 [][]\OML/cmm/m/it/10 L[]$ [] ! Undefined control sequence. l.15 \section {Embeddings into $L_p$-normed spaces} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.15 \section{E mbeddings into $L_p$-normed spaces} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no E in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{Bou85} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.16 ...on covers results of Bourgain \cite{Bou85} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bou85' on page undefined on input line 16. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! Undefined control sequence. l.17 We show that $\overset {O(\log n)}{\longrightarrow}L_p^{O(\log^2n)}$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 15--18 [] [] Overfull \hbox (11.42792pt too wide) in paragraph at lines 15--18 \OML/cmm/m/it/10 L[]$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 15--18 [] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 15--18 \OML/cmm/m/it/10 < [] Overfull \hbox (28.71866pt too wide) in paragraph at lines 15--18 \OML/cmm/m/it/10 X; d > [] Overfull \hbox (92.78925pt too wide) in paragraph at lines 15--18 [][]\OML/cmm/m/it/10 L[]$ [] ! Undefined control sequence. l.19 \section {Johnson and Lindenstrauss Theorem} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.19 \section{J ohnson and Lindenstrauss Theorem} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no J in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no J in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no 9 in font nullfont! Missing character: There is no 8 in font nullfont! Missing character: There is no 4 in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.21 \begin{theorem} \label{jl-thm} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! ! Undefined control sequence. l.22 ...ts in $d$-dimensions. A mapping $f:\mathbb {R}^d\rightarrow \mathbb{R... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.22 ...mapping $f:\mathbb{R}^d\rightarrow \mathbb {R}^k$ can be computed, i... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 19--23 [] [] Overfull \hbox (8.05556pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 V$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 n$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 d$ [] Overfull \hbox (11.52776pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 f \OT1/cmr/m/n/10 : [] Overfull \hbox (25.11075pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 R[] \OMS/cmsy/m/n/10 ! [] Overfull \hbox (12.59032pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (24.8194pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 u; v \OMS/cmsy/m/n/10 2 [] Overfull \hbox (8.05556pt too wide) in paragraph at lines 19--23 \OML/cmm/m/it/10 V$ [] Overfull \hbox (209.09065pt too wide) detected at line 23 \OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\O MS/cmsy/m/n/10 jj\OML/cmm/m/it/10 u \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 v\OMS /cmsy/m/n/10 jj[] ^^T jj\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 u\O T1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cm m/m/it/10 v\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 jj[] ^^T \OT1/cmr/m/n/10 (1 + \OML /cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 u \OMS/cms y/m/n/10 ^^@ \OML/cmm/m/it/10 v\OMS/cmsy/m/n/10 jj[]\OML/cmm/m/it/10 ; [] Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.25 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (15.55553pt too wide) in paragraph at lines 23--27 $\OT1/cmr/m/n/10 0 \OML/cmm/m/it/10 < [] Overfull \hbox (14.61455pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 ^^O < [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 23--27 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (15.65163pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 n; d$ [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (26.30254pt too wide) in paragraph at lines 23--27 \OT1/cmr/m/n/10 4([] \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (41.92982pt too wide) in paragraph at lines 23--27 []\OT1/cmr/m/n/10 )[] [] \OML/cmm/m/it/10 n$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 d$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 d$ [] Overfull \hbox (8.05556pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 V$ [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 23--27 \OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^F [] Overfull \hbox (7.94794pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k$ [] Overfull \hbox (36.17186pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 O\OT1/cmr/m/n/10 (1\OML/cmm/m/it/10 =n[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 23--27 \OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^F [] Overfull \hbox (7.94794pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 23--27 \OML/cmm/m/it/10 k$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.28 R ecall Standardized Normal Distributions ${\cal N}(0,1)$. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! ! Undefined control sequence. l.28 ...l Standardized Normal Distributions ${\cal N}(0,1)$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! ! LaTeX Error: Environment lemma undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.29 \begin{lemma} \label{rndm-pt-lem} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! ! Undefined control sequence. l.30 ...$d$-independent standardized normal ${\cal N}(0,1)$ random variables... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{lemma}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.31 \end{lemma} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.32 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.34 \end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 28--35 [] [] Overfull \hbox (31.34721pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1)$ [] Overfull \hbox (47.94052pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 X[]; [] ; X[]$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 d$ [] Overfull \hbox (31.34721pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1)$ [] Overfull \hbox (19.62494pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (72.9336pt too wide) in paragraph at lines 28--35 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; X[]; [] ; X[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (18.5833pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 Y \OT1/cmr/m/n/10 = [] Overfull \hbox (28.07153pt too wide) in paragraph at lines 28--35 []\OML/cmm/m/it/10 X$ [] Overfull \hbox (8.02779pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 Y$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 d$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (33.35013pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 ) = [] Overfull \hbox (45.81027pt too wide) in paragraph at lines 28--35 []\OML/cmm/m/it/10 e[]$ [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 X$ [] Overfull \hbox (82.8757pt too wide) in paragraph at lines 28--35 \OT1/cmr/m/n/10 ^^E[][[]\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 ] = [] Overfull \hbox (73.23892pt too wide) in paragraph at lines 28--35 []\OML/cmm/m/it/10 e[]$ [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 X$ [] Overfull \hbox (55.71832pt too wide) in paragraph at lines 28--35 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; [] ; X[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (8.02779pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 Y$ [] Overfull \hbox (20.18057pt too wide) in paragraph at lines 28--35 \OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 X\OMS/cmsy/m/n/10 jj$ [] Overfull \hbox (8.02779pt too wide) in paragraph at lines 28--35 \OML/cmm/m/it/10 Y$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.36 F rom now on let $Y=(Y_1,\ldots,Y_d)$ be a random unit vector as stated ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no F in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...ference `rndm-pt-lem' on page \thepage \space undefined\on@line . l.36 ...ector as stated in Lemma \ref{rndm-pt-lem} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `rndm-pt-lem' on page undefined on input line 36. Missing character: There is no . in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! ! Undefined control sequence. l.37 Define a vector $Z\in \mathbb {R}^K$, which is obtained by projecting $Y... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 36--38 [] [] Overfull \hbox (18.5833pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 Y \OT1/cmr/m/n/10 = [] Overfull \hbox (50.75998pt too wide) in paragraph at lines 36--38 \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 Y[]; [] ; Y[]\OT1/cmr/m/n/10 )$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 36--38 [] [] Overfull \hbox (16.98607pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 Z \OMS/cmsy/m/n/10 2 [] Overfull \hbox (15.37991pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (8.02779pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 Y$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 k$ [] Overfull \hbox (6.80557pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 L$ [] Overfull \hbox (7.54167pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 Z$ [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 36--38 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (23.13895pt too wide) in paragraph at lines 36--38 \OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 Z\OMS/cmsy/m/n/10 jj[]$ [] ! LaTeX Error: Environment lemma undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.39 \begin{lemma} \label{length-lem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.40 $ E[L]=k/d.$ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! LaTeX Error: \begin{document} ended by \end{lemma}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.41 \end{lemma} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.42 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (50.87497pt too wide) in paragraph at lines 40--43 []$\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 ] = [] Overfull \hbox (18.50351pt too wide) in paragraph at lines 40--43 \OML/cmm/m/it/10 k=d:$ [] Overfull \hbox (314.90102pt too wide) detected at line 43 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 ] = \OML/c mm/m/it/10 E\OT1/cmr/m/n/10 [\OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 Z[]\OMS/cmsy/m /n/10 jj\OT1/cmr/m/n/10 ] = []\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/ 10 X[] \OT1/cmr/m/n/10 + [] + \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ] = [] [] \OM L/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ] = \OML/cm m/m/it/10 k[]: [] Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (116.46928pt too wide) detected at line 44 \OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 Y\OMS/cmsy/m/n/10 jj[] \OT1/cmr/m/n/10 = 1 = [] [] \OML/cmm/m/it/10 X[]: [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (55.78532pt too wide) detected at line 45 \OML/cmm/m/it/10 d[] \OT1/cmr/m/n/10 = 1\OML/cmm/m/it/10 : [] Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (92.6748pt too wide) detected at line 46 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 ] = \OML/c mm/m/it/10 k[] \OT1/cmr/m/n/10 = []\OML/cmm/m/it/10 : [] ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.47 \end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.49 N ext we state the main lemma, first without proving it, and show how... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...g: Reference `jl-thm' on page \thepage \space undefined\on@line . l.49 ...ng it, and show how Theorem \ref{jl-thm} follows from it. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `jl-thm' on page undefined on input line 49. Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 49--50 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 49--50 [] [] ! LaTeX Error: Environment lemma undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.51 \begin{lemma} \label{main-lemma-dr} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.52 L et $k to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 52--53 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 52--53 \OML/cmm/m/it/10 k < [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 52--53 \OML/cmm/m/it/10 d$ [] Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (16.73953pt too wide) in paragraph at lines 54--55 \OML/cmm/m/it/10 ^^L < [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 54--55 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 54--55 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (27.4486pt too wide) in paragraph at lines 54--55 \OML/cmm/m/it/10 ^^L[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (54.6653pt too wide) in paragraph at lines 54--55 \OML/cmm/m/it/10 e[]:$ [] Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (16.73953pt too wide) in paragraph at lines 55--56 \OML/cmm/m/it/10 ^^L > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 55--56 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 55--56 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (27.4486pt too wide) in paragraph at lines 55--56 \OML/cmm/m/it/10 ^^L[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (54.6653pt too wide) in paragraph at lines 55--56 \OML/cmm/m/it/10 e[]:$ [] ! LaTeX Error: \begin{document} ended by \end{lemma}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.57 \end{lemma} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.59 \begin{proof} (Proof of Theorem \ref{jl-thm}) Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no ( in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...g: Reference `jl-thm' on page \thepage \space undefined\on@line . l.59 \begin{proof} (Proof of Theorem \ref{jl-thm} ) The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `jl-thm' on page undefined on input line 59. Missing character: There is no ) in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! ! Undefined control sequence. \mathbb l.60 ...andom $k$-dimensional subspace of $\mathbb {R}^d$ (for example we cho... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no ( in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. \mathbb l.60 ...v\in V$ and let its projection in $\mathbb {R}^k$ be $v'$. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...eference `length-lem' on page \thepage \space undefined\on@line . l.61 ...aves as required in Lemma \ref{length-lem} . Hence $E[L]=k/d$. For $... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `length-lem' on page undefined on input line 61. Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.62 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no P in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no | in font nullfont! Missing character: There is no | in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no | in font nullfont! Missing character: There is no | in font nullfont! ! Missing $ inserted. $ l.63 P(||v'-w'||^ 2 \le (1-\epsilon)||v-w||^2k/d) & = P(L\le (1-\epsilon)k/... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.63 ...v'-w'||^2 \le (1-\epsilon)||v-w||^2k/d) & = P(L\le (1-\epsilon)k/d)\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.64 & \le e^{\frac{k}{2}(1-(1-\... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.65 & \le e^{\frac{k}{2}(\epsil... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.66 & = e^{-\frac{k\epsilon^2}{... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.67 & \le e^{-2\ln n}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.68 & = \frac{1}{n^2} I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.69 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.69 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ... Reference `lnn-ineq' on page \thepage \space undefined\on@line . l.70 Note that for Equation \ref{lnn-ineq} we used that $\ln(1-\epsilon)\le ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `lnn-ineq' on page undefined on input line 70. Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ... Reference `ineq-eqn' on page \thepage \space undefined\on@line . l.70 ...0<\epsilon<1$. For Equation \ref{ineq-eqn} (and \ref{ineq1-eqn}), we... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `ineq-eqn' on page undefined on input line 70. Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! ! Undefined control sequence. ...Reference `ineq1-eqn' on page \thepage \space undefined\on@line . l.70 ...uation \ref{ineq-eqn} (and \ref{ineq1-eqn} ), we use the fact that $k... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `ineq1-eqn' on page undefined on input line 70. Missing character: There is no ) in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 59--71 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 59--71 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 d$ [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 k < [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 d$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 S$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 k$ [] Overfull \hbox (12.33302pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 k$ [] Overfull \hbox (14.6504pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 v \OMS/cmsy/m/n/10 2 [] Overfull \hbox (8.05556pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 V$ [] Overfull \hbox (12.59032pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 R[]$ [] Overfull \hbox (8.01157pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 v[]$ [] Overfull \hbox (21.36108pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 Y \OT1/cmr/m/n/10 := [] Overfull \hbox (28.3746pt too wide) in paragraph at lines 59--71 []$ [] Overfull \hbox (20.87497pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 Z \OT1/cmr/m/n/10 := [] Overfull \hbox (28.3746pt too wide) in paragraph at lines 59--71 []$ [] Overfull \hbox (17.36108pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 L \OT1/cmr/m/n/10 = [] Overfull \hbox (23.13895pt too wide) in paragraph at lines 59--71 \OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 Z\OMS/cmsy/m/n/10 jj[]$ [] Overfull \hbox (7.54167pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 Z$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 59--71 [] [] Overfull \hbox (30.87497pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 L\OT1/cmr/m/n/10 ] = [] Overfull \hbox (15.72572pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 k=d$ [] Overfull \hbox (15.55553pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 0 \OML/cmm/m/it/10 < [] Overfull \hbox (14.61455pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O < [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (16.73953pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^L \OT1/cmr/m/n/10 = [] Overfull \hbox (12.77782pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (4.05904pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O$ [] Overfull \hbox (15.04164pt too wide) in paragraph at lines 59--71 [] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (26.48732pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 v \OMS /cmsy/m/n/10 ^^@ [] Overfull \hbox (47.64009pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 w\OMS/cmsy/m/n/10 jj[]\OML/cmm/m/it/10 k=d\OT1/cmr/m/n/10 ) = [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (27.56256pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 k=d\OT1/cmr/m/n/10 )[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 59--71 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (80.37184pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (10.55551pt too wide) in paragraph at lines 59--71 [] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (52.35335pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 = [] Overfull \hbox (24.64839pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (10.55551pt too wide) in paragraph at lines 59--71 [] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (29.7478pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 = [] Overfull \hbox (11.24615pt too wide) in paragraph at lines 59--71 []$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 59--71 [] [] Overfull \hbox (25.00008pt too wide) in paragraph at lines 59--71 []\OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (18.50345pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (19.61464pt too wide) in paragraph at lines 59--71 \OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 ^^O \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (18.5452pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O[]=\OT1/cmr/m/n/10 2$ [] Overfull \hbox (15.55553pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 0 \OML/cmm/m/it/10 < [] Overfull \hbox (14.61455pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 ^^O < [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 59--71 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 59--71 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 59--71 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (26.30254pt too wide) in paragraph at lines 59--71 \OT1/cmr/m/n/10 4([] \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (41.92982pt too wide) in paragraph at lines 59--71 []\OT1/cmr/m/n/10 )[] [] \OML/cmm/m/it/10 n$ [] ! LaTeX Error: The font size command \normalsize is not defined: there is probably something wrong with the class file. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.71 Your command was ignored. Type I to replace it with another command, or to continue without it. [1 ! Undefined control sequence. \newlabel{chp:dr}{{}{\thepage }} l.71 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{chp:dr}{{}{}} {/usr/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map} ! Undefined control sequence. \newlabel{jl-thm}{{}{\thepage }} l.71 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{jl-thm}{{}{}} ! Undefined control sequence. \newlabel{rndm-pt-lem}{{}{\thepage }} l.71 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{rndm-pt-lem}{{}{}} ! Undefined control sequence. \newlabel{length-lem}{{}{\thepage }} l.71 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{length-lem}{{}{}} ! Undefined control sequence. \newlabel{main-lemma-dr}{{}{\thepage }} l.71 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{main-lemma-dr}{{}{}} ] ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.72 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no P in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no | in font nullfont! Missing character: There is no | in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no | in font nullfont! Missing character: There is no | in font nullfont! ! Missing $ inserted. $ l.73 P(||v'-w'||^ 2 \ge (1+\epsilon)||v-w||^2k/d) & = P(L\ge (1+\epsilon)k/... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.73 ...v'-w'||^2 \ge (1+\epsilon)||v-w||^2k/d) & = P(L\ge (1+\epsilon)k/d)\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.74 & \le e^{\frac{k}{2}(1-(1+\... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.75 & \le e^{\frac{k}{2}(-\epsi... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.76 & = e^{-\frac{k(\epsilon^2/... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.77 & \le e^{-2\ln n}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.78 & = \frac{1}{n^2} I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.79 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.79 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.82 \end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (35.04164pt too wide) in paragraph at lines 73--83 []$[] \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 (1 + [] Overfull \hbox (26.48732pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 v \OMS /cmsy/m/n/10 ^^@ [] Overfull \hbox (47.64009pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 w\OMS/cmsy/m/n/10 jj[]\OML/cmm/m/it/10 k=d\OT1/cmr/m/n/10 ) = [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 (1 + [] Overfull \hbox (27.56256pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 k=d\OT1/cmr/m/n/10 )[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 73--83 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (80.14964pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (10.55551pt too wide) in paragraph at lines 73--83 [] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (74.11282pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 = [] Overfull \hbox (56.61923pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (10.55551pt too wide) in paragraph at lines 73--83 [] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (29.7478pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 e[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 = [] Overfull \hbox (11.24615pt too wide) in paragraph at lines 73--83 []$ [] Overfull \hbox (14.6504pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 v \OMS/cmsy/m/n/10 2 [] Overfull \hbox (8.05556pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 V$ [] Overfull \hbox (29.51158pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v\OT1/cmr/m/n/10 ) = [] Overfull \hbox (24.83174pt too wide) in paragraph at lines 73--83 []\OML/cmm/m/it/10 v[]$ [] Overfull \hbox (24.8194pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 u; v \OMS/cmsy/m/n/10 2 [] Overfull \hbox (8.05556pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 V$ [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 (1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (27.00589pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 u \OMS /cmsy/m/n/10 ^^@ [] Overfull \hbox (25.80322pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 v\OMS/cmsy/m/n/10 jj[] ^^T [] Overfull \hbox (32.80801pt too wide) in paragraph at lines 73--83 \OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 u\OT1/cm r/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (39.55328pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 v\OT1/cmr/m/n/10 )\OMS/cmsy /m/n/10 jj[] ^^T [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 (1 + [] Overfull \hbox (27.00589pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 jj\OML/cmm/m/it/10 u \OMS /cmsy/m/n/10 ^^@ [] Overfull \hbox (15.24771pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 v\OMS/cmsy/m/n/10 jj[]$ [] Overfull \hbox (20.48851pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 =n[]$ [] Overfull \hbox (32.00888pt too wide) in paragraph at lines 73--83 [][] \OT1/cmr/m/n/10 = [] Overfull \hbox (12.77782pt too wide) in paragraph at lines 73--83 \OT1/cmr/m/n/10 1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.34332pt too wide) in paragraph at lines 73--83 []$ [] Overfull \hbox (5.97226pt too wide) in paragraph at lines 73--83 \OML/cmm/m/it/10 f$ [] Missing character: There is no B in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...rence `main-lemma-dr' on page \thepage \space undefined\on@line . l.84 Before we prove Lemma \ref{main-lemma-dr} , we need the following. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `main-lemma-dr' on page undefined on input line 84. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Environment lemma undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.85 \begin{lemma} \label{ineq-binomial-lem} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. l.86 Let $X$ be ${\cal N}(0,1)$. Then $E[e^{sX^2}]=\frac{1}{\sqrt{1-2s}}$, f... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{lemma}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.87 \end{lemma} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.88 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.90 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no E in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no e in font nullfont! ! Missing $ inserted. $ l.91 E[e^ {sX^2}] & = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{sx^2}e... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.91 E[e^{sX^2}] & = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{sx^2}e... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.92 & = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{-(1-2s... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.93 & \mbox{set }y=x\sqrt{1-2s}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! ! Misplaced alignment tab character &. & l.94 & = \frac{1}{\sqrt{1-2s}}\frac{1}{\sqrt{2\pi}}\int_{-\inft... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.95 & = \frac{1}{\sqrt{1-2s}} {\mbox{ (since }} \frac{1}{\sqrt... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.96 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.96 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (20.0pt too wide) in paragraph at lines 84--97 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 84--97 [] [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 X$ [] Overfull \hbox (31.34721pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 N\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1)$ [] Overfull \hbox (44.00493pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 ] = [] Overfull \hbox (26.96606pt too wide) in paragraph at lines 84--97 []$ [] Overfull \hbox (15.24301pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 s < [] Overfull \hbox (15.00005pt too wide) in paragraph at lines 84--97 \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 2$ [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 X$ [] Overfull \hbox (30.02084pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) = [] Overfull \hbox (39.55054pt too wide) in paragraph at lines 84--97 []\OML/cmm/m/it/10 e[]$ [] Overfull \hbox (25.9722pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 )$ [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 X$ [] Overfull \hbox (52.26382pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 X\OT1/cmr/m/n/10 )) = [] Overfull \hbox (75.94797pt too wide) in paragraph at lines 84--97 [][] \OML/cmm/m/it/10 H\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML /cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML/cmm/m/it /10 dx$ [] Overfull \hbox (28.61256pt too wide) in paragraph at lines 84--97 []\OT1/cmr/m/n/10 ] = [] Overfull \hbox (92.4499pt too wide) in paragraph at lines 84--97 [] [][] \OML/cmm/m/it/10 e[]e[]dx[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 84--97 \OT1/cmr/m/n/10 = [] Overfull \hbox (105.58812pt too wide) in paragraph at lines 84--97 [] [][] \OML/cmm/m/it/10 e[]dx[] [] Overfull \hbox (15.81711pt too wide) in paragraph at lines 84--97 []\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 = [] Overfull \hbox (40.9583pt too wide) in paragraph at lines 84--97 \OML/cmm/m/it/10 x[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 84--97 \OT1/cmr/m/n/10 = [] Overfull \hbox (107.6259pt too wide) in paragraph at lines 84--97 [][] [][] \OML/cmm/m/it/10 e[]dy[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 84--97 \OT1/cmr/m/n/10 = [] Overfull \hbox (118.18141pt too wide) in paragraph at lines 84--97 [][][] [][] \OML/cmm/m/it/10 e[]dy \OT1/cmr/m/n/10 = [] Overfull \hbox (8.88892pt too wide) in paragraph at lines 84--97 \OT1/cmr/m/n/10 1)$ [] ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.98 \end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.100 \begin{proof} (Proof of Lemma \ref{main-lemma-dr}) Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no ( in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...rence `main-lemma-dr' on page \thepage \space undefined\on@line . l.100 ...proof}(Proof of Lemma \ref{main-lemma-dr} ) The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `main-lemma-dr' on page undefined on input line 100. Missing character: There is no ) in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 100--102 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 100--102 [] [] Overfull \hbox (16.73953pt too wide) in paragraph at lines 100--102 \OML/cmm/m/it/10 ^^L < [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 100--102 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 100--102 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (27.4486pt too wide) in paragraph at lines 100--102 \OML/cmm/m/it/10 ^^L[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (54.6653pt too wide) in paragraph at lines 100--102 \OML/cmm/m/it/10 e[]:$ [] Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 103--104 [] [] Overfull \hbox (238.05057pt too wide) detected at line 104 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 d\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 X[] \OT1/cmr/m/n/10 + [] + \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ) \OMS/c msy/m/n/10 ^^T \OML/cmm/m/it/10 k^^L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[] \OT1/ cmr/m/n/10 + [] + \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 )) \OMS/cmsy/m/n/10 ^^T \ OML/cmm/m/it/10 e[]: [] Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (42.12506pt too wide) in paragraph at lines 104--106 $\OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 d\OT1/cmr/m/n/10 (\OML/c mm/m/it/10 X[] \OT1/cmr/m/n/10 + [] Overfull \hbox (19.44441pt too wide) in paragraph at lines 104--106 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (27.99997pt too wide) in paragraph at lines 104--106 \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (36.92712pt too wide) in paragraph at lines 104--106 \OML/cmm/m/it/10 k^^L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[] \OT1/cmr/m/n/10 + [] Overfull \hbox (19.44441pt too wide) in paragraph at lines 104--106 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (21.33336pt too wide) in paragraph at lines 104--106 \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ))$ [] ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.107 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Misplaced alignment tab character &. l.108 & = P(k\beta(X_1^2+\cdots+X_d^2) - d(X_1^2+\cdots+X_k^2)\ge 0)\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no = in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no k in font nullfont! ! Missing $ inserted. $ l.108 & = P(k\beta (X_1^2+\cdots+X_d^2) - d(X_1^2+\cdots+X_k^2)\ge 0)\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. & l.109 & = P(e^{t(k\beta(X_1^2+\cdots+X_d^2) - d(X_1^2+\cdots+X_k^2)} \ge e^{... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! ! Misplaced alignment tab character &. & l.110 & \le E[e^{t(k\beta(X_1^2+\cdots+X_d^2) - d(X_1^2+\cdots+X_k^2)}] \mbo... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no U in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ) in font nullfont! ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.111 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.111 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.113 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Misplaced alignment tab character &. l.114 & = E[e^{tk\beta X_1^2}]^{d-k}E[e^{t(k\beta-d)X_1^2}]^k\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no = in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no e in font nullfont! ! Missing $ inserted. $ l.114 & = E[e^ {tk\beta X_1^2}]^{d-k}E[e^{t(k\beta-d)X_1^2}]^k\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. & l.115 & = {\left( \frac{1}{\sqrt{1-2tk\beta}}\right)}^{d-k}{\left( \frac{1}... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no U in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...e `ineq-binomial-lem' on page \thepage \space undefined\on@line . l.115 ...x{ (Using Lemma \ref{ineq-binomial-lem})} \label{last-eqn-minimize} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `ineq-binomial-lem' on page undefined on input line 1 15. Missing character: There is no ) in font nullfont! ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.116 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.116 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. ...e `last-eqn-minimize' on page \thepage \space undefined\on@line . l.117 ...ity in Inequality \ref{last-eqn-minimize} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `last-eqn-minimize' on page undefined on input line 1 17. Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 108--119 [] [] Overfull \hbox (31.40628pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 ^^L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[] \OT1/cmr/m/n/10 + [] Overfull \hbox (19.44441pt too wide) in paragraph at lines 108--119 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (25.22226pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (30.42712pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 d\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[] \OT1/cmr/m/n/10 + [] Overfull \hbox (19.44441pt too wide) in paragraph at lines 108--119 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (27.99997pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (8.88892pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 0)[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 = [] Overfull \hbox (148.60826pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 e[] \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (25.09375pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 e[] \OT1/cmr/m/n/10 = [] Overfull \hbox (23.05554pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 1)[]\OML/cmm/m/it/10 t > [] Overfull \hbox (8.88892pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 0)[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 108--119 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (139.8687pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 e[]\OT1/cmr/m/n/10 ][]$ [] Overfull \hbox (7.50002pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 A$ [] Overfull \hbox (8.0868pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 B$ [] Overfull \hbox (39.65622pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 AB\OT1/cmr/m/n/10 ] = [] Overfull \hbox (42.6146pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 A\OT1/cmr/m/n/10 ]\OML/cmm/ m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 B\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (36.05554pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 A[]\OT1/cmr/m/n/10 ] = [] Overfull \hbox (25.50003pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 A\OT1/cmr/m/n/10 ][]$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (12.77086pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (105.54425pt too wide) in paragraph at lines 108--119 []\OT1/cmr/m/n/10 ][]\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 e[]\OT 1/cmr/m/n/10 ][][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 = [] Overfull \hbox (151.80331pt too wide) in paragraph at lines 108--119 [][][][][][]$ [] Overfull \hbox (15.55553pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 0 \OML/cmm/m/it/10 < [] Overfull \hbox (14.16663pt too wide) in paragraph at lines 108--119 \OML/cmm/m/it/10 t < [] Overfull \hbox (26.70491pt too wide) in paragraph at lines 108--119 \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 k^^L$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 108--119 [] [] Overfull \hbox (155.07974pt too wide) detected at line 119 \OML/cmm/m/it/10 q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = (1 \OM S/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 tk^^L\OT1/cmr/m/n/10 )[](1 \OMS/cmsy/m/n/10 ^^@ \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/c mm/m/it/10 k^^L \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 d\OT1/cmr/m/n/10 ))[]\OML /cmm/m/it/10 : [] Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. ...e `last-eqn-minimize' on page \thepage \space undefined\on@line . l.120 ...ack in Inequality \ref{last-eqn-minimize} , as $1/\sqrt{q(t)}$, we o... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `last-eqn-minimize' on page undefined on input line 1 20. Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (14.16663pt too wide) in paragraph at lines 119--123 $\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = [] Overfull \hbox (37.25748pt too wide) in paragraph at lines 119--123 []$ [] Overfull \hbox (22.87843pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 q\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = [] Overfull \hbox (51.70189pt too wide) in paragraph at lines 119--123 []\OT1/cmr/m/n/10 ) = [] Overfull \hbox (73.56345pt too wide) in paragraph at lines 119--123 [][][][]$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 119--123 [] [] Overfull \hbox (36.21187pt too wide) in paragraph at lines 119--123 \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =[]$ [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (27.4486pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 ^^L[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (99.9891pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 ^^L[][][]$ [] Overfull \hbox (25.00008pt too wide) in paragraph at lines 119--123 []\OT1/cmr/m/n/10 (1 + [] Overfull \hbox (20.15968pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (5.71527pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 x$ [] Overfull \hbox (110.54462pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 ^^L[][][] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (51.88751pt too wide) in paragraph at lines 119--123 \OML/cmm/m/it/10 e[]$ [] Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.125 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no P in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no X in font nullfont! ! Missing $ inserted. $ l.126 P(d(X_ 1^2+\cdots+X_k^2) \ge k\beta(X_1^2+\cdots+X_d^2)) & \le {\left(... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.126 ...+X_k^2) \ge k\beta(X_1^2+\cdots+X_d^2)) & \le {\left( \frac{1}{\sq... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. & l.127 & = \frac{1}{\sqrt{q(-t)}} I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.128 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.128 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.130 \end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 124--131 [] [] Overfull \hbox (16.73953pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 ^^L > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 124--131 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (29.05902pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 P\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 L \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (27.4486pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 ^^L[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (54.6653pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 e[]:$ [] Overfull \hbox (12.26393pt too wide) in paragraph at lines 124--131 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (19.44441pt too wide) in paragraph at lines 124--131 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (27.99997pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (36.92712pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 k^^L\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X[] \OT1/cmr/m/n/10 + [] Overfull \hbox (19.44441pt too wide) in paragraph at lines 124--131 [] \OT1/cmr/m/n/10 + [] Overfull \hbox (31.88887pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 )) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (140.72006pt too wide) in paragraph at lines 124--131 [][][][][][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 124--131 \OT1/cmr/m/n/10 = [] Overfull \hbox (30.17058pt too wide) in paragraph at lines 124--131 []$ [] Overfull \hbox (23.98962pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 q\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@\OML/cmm/m/it/10 t\OT1/c mr/m/n/10 )$ [] Overfull \hbox (14.16663pt too wide) in paragraph at lines 124--131 \OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = [] Overfull \hbox (45.03528pt too wide) in paragraph at lines 124--131 \OMS/cmsy/m/n/10 ^^@[]$ [] ! Undefined control sequence. l.133 \section {Applications of Metric Embeddings} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no A in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 133--134 [] [] ! Undefined control sequence. l.136 \section {Exercises} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! ! LaTeX Error: Environment exercise undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.137 \begin{exercise} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no S in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{exercise}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.139 \end{exercise} Your command was ignored. Type I to replace it with another command, or to continue without it. ) ! Emergency stop. <*> dr.tex *** (job aborted, no legal \end found) Here is how much of TeX's memory you used: 39 strings out of 94500 418 string characters out of 1176765 58348 words of memory out of 1000000 3304 multiletter control sequences out of 10000+50000 3948 words of font info for 15 fonts, out of 500000 for 2000 580 hyphenation exceptions out of 1000 20i,7n,26p,811b,120s stack positions out of 1500i,500n,5000p,200000b,5000s PDF statistics: 34 PDF objects out of 300000 0 named destinations out of 131072 1 words of extra memory for PDF output out of 65536 ! ==> Fatal error occurred, the output PDF file is not finished!