chapters/langevin-equation.log

This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2024.11.23)  11 JAN 2025 21:13
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**langevin-equation.tex
(./langevin-equation.tex
LaTeX2e <2023-11-01> patch level 1
L3 programming layer <2024-01-22>
! Undefined control sequence.
l.1 \section
            {Langevin Equation}
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 <return>  for immediate help.
 ...                                              
                                                  
l.1 \section{L
              angevin Equation}
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no L in font nullfont!
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! LaTeX Error: Missing \begin{document}.

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l.3     \frame{\sectionpage}
                            
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

! Undefined control sequence.
<argument> \sectionpage 
                        
l.3     \frame{\sectionpage}
                            
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.


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! LaTeX Error: Missing \begin{document}.

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 ...                                              
                                                  
l.5 ...gin{frame}{Ordinary Differential Equations}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no O in font nullfont!
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! Undefined control sequence.
l.6         \uncover
                    <+->{\begin{equation*}
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,
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Missing character: There is no < in font nullfont!
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! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.6         \uncover<+->{\begin{equation*}
                                          
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Undefined control sequence.
l.7             \dv
                   {x} y(x) + \frac{1}{CR} y(x) = 0
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,
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Missing character: There is no x in font nullfont!
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! Missing $ inserted.
<inserted text> 
                $
l.7             \dv{x} y(x) + \frac{1}{CR}
                                           y(x) = 0
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <7> on input line 7.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <5> on input line 7.
! Extra }, or forgotten $.
\frac  #1#2->{\begingroup #1\endgroup \over #2}
                                               
l.7             \dv{x} y(x) + \frac{1}{CR}
                                           y(x) = 0
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.


! LaTeX Error: \begin{frame} on input line 5 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
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 ...                                              
                                                  
l.8         \end{equation*}
                           }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.8         \end{equation*}
                           }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 } inserted.
<inserted text> 
                }
l.8         \end{equation*}
                           }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.


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! Undefined control sequence.
l.10         \uncover
                     <+->{\begin{equation*}
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 <return>  for immediate help.
 ...                                              
                                                  
l.10         \uncover<
                      +->{\begin{equation*}
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no + in font nullfont!
Missing character: There is no - in font nullfont!
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! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.10         \uncover<+->{\begin{equation*}
                                           
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Undefined control sequence.
l.11             \dv
                    [2]{x} y(x) + \gamma \dv{x} y(x) + \omega_0^2 y(x) = f(x)
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 2 in font nullfont!
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! Missing $ inserted.
<inserted text> 
                $
l.11             \dv[2]{x} y(x) + \gamma
                                         \dv{x} y(x) + \omega_0^2 y(x) = f(x)
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.11             \dv[2]{x} y(x) + \gamma \dv
                                            {x} y(x) + \omega_0^2 y(x) = f(x)
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: \begin{frame} on input line 5 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.12         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.12         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.


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! LaTeX Error: Missing \begin{document}.

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 ...                                              
                                                  
l.15     \begin{frame}{Non-equilibrium system}
                                              
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If that doesn't work, type  X <return>  to quit.

Missing character: There is no N in font nullfont!
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Missing character: There is no m in font nullfont!
! Undefined control sequence.
l.16         \uncover
                     <+>{\begin{equation*}
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 + in font nullfont!
Missing character: There is no > in font nullfont!

! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.16         \uncover<+>{\begin{equation*}
                                          
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Undefined control sequence.
l.17             \dv
                    [2]{x} y(x) + \gamma \dv{x} y(x) + \omega_0^2 y(x) = f(x)
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 2 in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no x 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 x in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no + in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.17             \dv[2]{x} y(x) + \gamma
                                         \dv{x} y(x) + \omega_0^2 y(x) = f(x)
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.17             \dv[2]{x} y(x) + \gamma \dv
                                            {x} y(x) + \omega_0^2 y(x) = f(x)
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: \begin{frame} on input line 15 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.18         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.18         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.19         \uncover
                     <+>{\[ \Downarrow \]
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 + in font nullfont!
Missing character: There is no > in font nullfont!

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Overfull \hbox (37.98059pt too wide) in paragraph at lines 15--19
\OML/cmm/m/it/10 ^^Mxy\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) +
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Overfull \hbox (6.11111pt too wide) detected at line 19
\OMS/cmsy/m/n/10 +
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! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.20         \begin{equation*}
                              
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Undefined control sequence.
l.21             \colch
                       {\dv[2]{x} + \gamma \dv{x} + \omega_0^2} y(x) = f(x)
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.21             \colch{\dv
                           [2]{x} + \gamma \dv{x} + \omega_0^2} y(x) = f(x)
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 2 in font nullfont!
Missing character: There is no ] in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no + in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.21             \colch{\dv[2]{x} + \gamma
                                           \dv{x} + \omega_0^2} y(x) = f(x)
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.21             \colch{\dv[2]{x} + \gamma \dv
                                              {x} + \omega_0^2} y(x) = f(x)
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.

! Extra }, or forgotten $.
l.21 ...ch{\dv[2]{x} + \gamma \dv{x} + \omega_0^2}
                                                   y(x) = f(x)
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.


! LaTeX Error: \begin{frame} on input line 15 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.22         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.22         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 } inserted.
<inserted text> 
                }
l.22         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.23         \uncover
                     <+>{\[ \Downarrow \]
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 + in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (19.22592pt too wide) in paragraph at lines 19--23
   $\OML/cmm/m/it/10 ^^Mx \OT1/cmr/m/n/10 +
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Overfull \hbox (40.37962pt too wide) in paragraph at lines 19--23
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Overfull \hbox (19.46533pt too wide) in paragraph at lines 19--23
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Overfull \hbox (6.11111pt too wide) detected at line 23
\OMS/cmsy/m/n/10 +
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! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.24         \begin{equation*}
                              
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

Missing character: There is no y in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ) 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 ( in font nullfont!
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Missing character: There is no ) in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.25 ...)}{\dv[2]{x} + \gamma \dv{x} + \omega_0^2}
                                                  
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
<argument> \dv 
               [2]{x} + \gamma \dv {x} + \omega _0^2
l.25 ...)}{\dv[2]{x} + \gamma \dv{x} + \omega_0^2}
                                                  
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.
<argument> \dv [2]{x} + \gamma \dv 
                                   {x} + \omega _0^2
l.25 ...)}{\dv[2]{x} + \gamma \dv{x} + \omega_0^2}
                                                  
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.

! Extra }, or forgotten $.
\frac  #1#2->{\begingroup #1\endgroup \over #2}
                                               
l.25 ...)}{\dv[2]{x} + \gamma \dv{x} + \omega_0^2}
                                                  
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.


! LaTeX Error: \begin{frame} on input line 15 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.26         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.26         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 } inserted.
<inserted text> 
                }
l.26         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 (46.08405pt too wide) in paragraph at lines 23--28
   $[]$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.29     \begin{frame}{Non-equilibrium system}
                                              
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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 - 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 l 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 r in font nullfont!
Missing character: There is no i 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 s 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 e in font nullfont!
Missing character: There is no m in font nullfont!
! Undefined control sequence.
l.31         \includegraphics
                             [width = 0.8\textwidth]{images/coke.jpg}
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 i 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 = in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 8 in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   ]
l.31 ...   \includegraphics[width = 0.8\textwidth]
                                                  {images/coke.jpg}
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   ]
l.31 ...   \includegraphics[width = 0.8\textwidth]
                                                  {images/coke.jpg}
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Missing character: There is no ] 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 g 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 c in font nullfont!
Missing character: There is no o 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 . in font nullfont!
Missing character: There is no j in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no g in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 29--33
[][]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.34 ...ural Network: Learning cross-correlations}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  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 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 N 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 o 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 : 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 r 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 g in font nullfont!
Missing character: There is no c 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 s 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 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 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!
! Undefined control sequence.
l.36         \includegraphics
                             [width = 0.9\textwidth]{images/fourier-1.pdf}
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 i 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 = in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 9 in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   ]
l.36 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-1.pdf}
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   ]
l.36 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-1.pdf}
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Missing character: There is no ] 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 g 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 f 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 i 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 1 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 d in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 34--38
[][]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.39 ...ural Network: Learning cross-correlations}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  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 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 N 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 o 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 : 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 r 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 g in font nullfont!
Missing character: There is no c 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 s 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 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 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!
! Undefined control sequence.
l.41         \includegraphics
                             [width = 0.9\textwidth]{images/fourier-2.pdf}
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 i 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 = in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 9 in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   ]
l.41 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-2.pdf}
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   ]
l.41 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-2.pdf}
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Missing character: There is no ] 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 g 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 f 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 i 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 2 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 d in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 39--43
[][]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.44 ...ural Network: Learning cross-correlations}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  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 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 N 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 o 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 : 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 r 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 g in font nullfont!
Missing character: There is no c 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 s 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 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 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!
! Undefined control sequence.
l.46         \includegraphics
                             [width = 0.9\textwidth]{images/fourier-3.pdf}
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 i 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 = in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 9 in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   ]
l.46 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-3.pdf}
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   ]
l.46 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-3.pdf}
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Missing character: There is no ] 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 g 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 f 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 i 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 3 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 d in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 44--48
[][]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.49 ...ural Network: Learning cross-correlations}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  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 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 N 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 o 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 : 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 r 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 g in font nullfont!
Missing character: There is no c 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 s 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 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 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!
! Undefined control sequence.
l.51         \includegraphics
                             [width = 0.9\textwidth]{images/fourier-4.pdf}
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 i 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 = in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 9 in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   ]
l.51 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-4.pdf}
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   ]
l.51 ...   \includegraphics[width = 0.9\textwidth]
                                                  {images/fourier-4.pdf}
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Missing character: There is no ] 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 g 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 f 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 i 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 4 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 d in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 49--53
[][]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.54 ...frame}{Ergodic Test: Quantifying $T_{eq}$}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E 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 o 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 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 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 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 f 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 g in font nullfont!
! Undefined control sequence.
l.56         \includegraphics
                             [height = 0.7\textheight]{images/fourier-5.pdf}
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 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 g 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 = in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no 7 in font nullfont!
! Missing number, treated as zero.
<to be read again> 
                   ]
l.56 ... \includegraphics[height = 0.7\textheight]
                                                  {images/fourier-5.pdf}
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
                   ]
l.56 ... \includegraphics[height = 0.7\textheight]
                                                  {images/fourier-5.pdf}
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Missing character: There is no ] 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 g 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 f 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 i 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 5 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 d in font nullfont!
Missing character: There is no f in font nullfont!

Overfull \hbox (34.0542pt too wide) in paragraph at lines 54--58
[][]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.59 ...frame}{Ergodic Test: Quantifying $T_{eq}$}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E 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 o 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 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 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 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 f 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 g in font nullfont!
! Undefined control sequence.
l.60         \uncover
                     <+>{\begin{equation*}
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 + in font nullfont!
Missing character: There is no > in font nullfont!

! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.60         \uncover<+>{\begin{equation*}
                                          
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: \mathcal allowed only in math mode.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.61             \mathcal
                         {F}[f](\xi) = \hat{f}(\xi) = \frac{1}{\sqrt{2\pi}} ...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no F 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 ] in font nullfont!
Missing character: There is no ( in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.61             \mathcal{F}[f](\xi
                                   ) = \hat{f}(\xi) = \frac{1}{\sqrt{2\pi}} ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.61 ..._{-\infty}^{+\infty} f(x) e^{-i x \xi} \dd
                                                  {x}
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: \begin{frame} on input line 59 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.62         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.62         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.63         \uncover
                     <+>{\begin{equation*}
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 + in font nullfont!
Missing character: There is no > in font nullfont!

! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.63         \uncover<+>{\begin{equation*}
                                          
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: \mathcal allowed only in math mode.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.64             \mathcal
                         {F}^{-1}[\hat{f}](x) = f(x) = \frac{1}{\sqrt{2\pi}}...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no F in font nullfont!
! Missing $ inserted.
<inserted text> 
                $
l.64             \mathcal{F}^
                             {-1}[\hat{f}](x) = f(x) = \frac{1}{\sqrt{2\pi}}...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.64 ...ty}^{+\infty} \hat{f}(\xi) e^{i x \xi} \dd
                                                  {\xi}
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: \begin{frame} on input line 59 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.65         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.65         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 (34.0542pt too wide) in paragraph at lines 59--67
[][]
 []


Overfull \hbox (19.2795pt too wide) in paragraph at lines 59--67
\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (29.14066pt too wide) in paragraph at lines 59--67
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (90.32266pt too wide) in paragraph at lines 59--67
[] [][] \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 e[]x$
 []


Overfull \hbox (46.31256pt too wide) in paragraph at lines 59--67
[]\OT1/cmr/m/n/10 [[]](\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 59--67
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (82.31227pt too wide) in paragraph at lines 59--67
[] [][] []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )\OML/cmm/m/it/1
0 e[]^^X$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.68 ...gin{frame}{Deterministic and Random Force}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D 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 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 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 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 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 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 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 c in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.69         \uncover
                     <+>{\begin{equation*}
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 + in font nullfont!
Missing character: There is no > in font nullfont!

! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.69         \uncover<+>{\begin{equation*}
                                          
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.70             \widehat
                         {(f + \alpha g)}(\xi) = \frac{1}{\sqrt{2\pi}} \int_...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.70 ...sqrt{2\pi}} \int_{-\infty}^{+\infty} \prnt
                                                  {f(x) + \alpha g(x)} e^{-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.

! Undefined control sequence.
l.70 ...\prnt{f(x) + \alpha g(x)} e^{-i x \xi} \dd
                                                  {x}
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: \begin{frame} on input line 68 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.71         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.71         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.72         \uncover
                     <+>{\[ \Downarrow \]
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 + in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 68--72
[][]
 []


Overfull \hbox (60.7031pt too wide) in paragraph at lines 68--72
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (127.60037pt too wide) in paragraph at lines 68--72
[] [][] []\OML/cmm/m/it/10 e[]x$ 
 []


Overfull \hbox (6.11111pt too wide) detected at line 72
\OMS/cmsy/m/n/10 +
 []


! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.73         \begin{equation*}
                              
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.74             \widehat
                         {(f + \alpha g)}(\xi) = \frac{1}{\sqrt{2\pi}} \int_...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.74 ..._{-\infty}^{+\infty} f(x) e^{-i x \xi} \dd
                                                  {x} + \frac{\alpha}{\sqrt{...
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.74 ..._{-\infty}^{+\infty} g(x) e^{-i x \xi} \dd
                                                  {x}
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: \begin{frame} on input line 68 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.75         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.75         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.76         \uncover
                     <+>{\[ \Downarrow \]
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 + in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (60.7031pt too wide) in paragraph at lines 72--76
 $[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (98.10046pt too wide) in paragraph at lines 72--76
[] [][] \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 e[]x \OT1/cmr/m/n/10 +
 []


Overfull \hbox (89.47887pt too wide) in paragraph at lines 72--76
[] [][] \OML/cmm/m/it/10 g\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\
OML/cmm/m/it/10 e[]x$ 
 []


Overfull \hbox (6.11111pt too wide) detected at line 76
\OMS/cmsy/m/n/10 +
 []


! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.77         \begin{equation*}
                              
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.78             \widehat
                         {(f + \alpha g)}(\xi) = \hat{f}(\xi) + \alpha \hat{...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


! LaTeX Error: \begin{frame} on input line 68 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.79         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.79         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 (60.7031pt too wide) in paragraph at lines 76--81
 $[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (26.36295pt too wide) in paragraph at lines 76--81
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) +
 []


Overfull \hbox (24.17538pt too wide) in paragraph at lines 76--81
\OML/cmm/m/it/10 []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.82 ...gin{frame}{Deterministic and Random Force}
                                                  
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D 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 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 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 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 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 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 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 c in font nullfont!
Missing character: There is no e in font nullfont!
! Undefined control sequence.
l.83         \uncover
                     <+>{\begin{equation*}
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 + in font nullfont!
Missing character: There is no > in font nullfont!

! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.83         \uncover<+>{\begin{equation*}
                                          
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.84             \widehat
                         {f'}(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.84 ...{-\infty}^{+\infty} f'(x) e^{-i x \xi} \dd
                                                  {x}
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: \begin{frame} on input line 82 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.85         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.85         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.86         \uncover
                     <+>{\[ \Downarrow \]
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 + in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 82--86
[][]
 []


Overfull \hbox (31.94621pt too wide) in paragraph at lines 82--86
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (93.12822pt too wide) in paragraph at lines 82--86
[] [][] \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 e[]x$ 
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Overfull \hbox (6.11111pt too wide) detected at line 86
\OMS/cmsy/m/n/10 +
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! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.87         \begin{equation*}
                              
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.88             \widehat
                         {f'}(\xi) = \eval{\frac{f(x) e^{-ix\xi}}{\sqrt{2\pi...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
l.88             \widehat{f'}(\xi) = \eval
                                          {\frac{f(x) e^{-ix\xi}}{\sqrt{2\pi...
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.88 ..._{-\infty}^{+\infty} f(x) e^{-i x \xi} \dd
                                                  {x}
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: \begin{frame} on input line 82 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.89         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.89         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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.

! Undefined control sequence.
l.90         \uncover
                     <+>{\[ \Downarrow \]
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 + in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (31.94621pt too wide) in paragraph at lines 86--90
 $[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
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Overfull \hbox (60.13045pt too wide) in paragraph at lines 86--90
[][] \OT1/cmr/m/n/10 +
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\OML/cmm/m/it/10 i^^X \OMS/cmsy/m/n/10 ^^A
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Overfull \hbox (90.32266pt too wide) in paragraph at lines 86--90
[] [][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\
OML/cmm/m/it/10 e[]x$ 
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Overfull \hbox (6.11111pt too wide) detected at line 90
\OMS/cmsy/m/n/10 +
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! LaTeX Error: Environment equation* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.91         \begin{equation*}
                              
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.92             \widehat
                         {f'}(\xi) = i\xi \widehat{f}(\xi)
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


! LaTeX Error: \begin{frame} on input line 82 ended by \end{equation*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.93         \end{equation*}
                            }
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.93         \end{equation*}
                            }
I've inserted something that you may have forgotten.
(See the <inserted text> 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 (31.94621pt too wide) in paragraph at lines 90--95
 $[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (26.86536pt too wide) in paragraph at lines 90--95
\OML/cmm/m/it/10 i^^X[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^X\OT1/cmr/m/n/10 )$
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.96     \begin{frame}{Technology Implications}
                                               
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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!
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! Undefined control sequence.
l.98         \huge
                  {The inverse does work}
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 T in font nullfont!
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Missing character: There is no k in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 96--99
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! 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 <return>  for immediate help.
 ...                                              
                                                  
l.100         \normalsize
                         {for appropriate functions}
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.100         \normalsize{f
                           or appropriate functions}
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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 p in font nullfont!
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Missing character: There is no r in font nullfont!
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Missing character: There is no p in font nullfont!
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Missing character: There is no i 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 f in font nullfont!
Missing character: There is no u in font nullfont!
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Missing character: There is no c in font nullfont!
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Missing character: There is no s in font nullfont!
! Undefined control sequence.
l.102         \tiny
                   {and, sometimes, the Fourier Transform of a function is n...
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 <return>  for immediate help.
 ...                                              
                                                  
l.102         \tiny{a
                     nd, sometimes, the Fourier Transform of a function is n...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no a in font nullfont!
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Missing character: There is no n in font nullfont!
)
! Emergency stop.
<*> langevin-equation.tex
                         
*** (job aborted, no legal \end found)

 
Here is how much of TeX's memory you used:
 26 strings out of 474222
 607 string characters out of 5748732
 1928975 words of memory out of 5000000
 22380 multiletter control sequences out of 15000+600000
 558069 words of font info for 36 fonts, out of 8000000 for 9000
 1141 hyphenation exceptions out of 8191
 19i,5n,20p,252b,73s stack positions out of 10000i,1000n,20000p,200000b,200000s
!  ==> Fatal error occurred, no output PDF file produced!