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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 22:00, 9 September 2013

5,160 bytes added ,  22:00, 9 September 2013
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| <math>f_0\!</math>
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| <math>f_{0}\!</math>
 
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<br>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
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|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
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|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
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|- style="background:#f0f0ff"
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|- style="background:ghostwhite"
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| width="10%" | &nbsp;
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| style="width:10%; border-bottom:1px solid black" | &nbsp;
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| width="18%" | <math>f\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
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| width="18%" |  
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| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
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<p><math>\mathrm{T}_{11} f</math></p>
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<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x~\mathrm{d}y}\end{matrix}</math>
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<p><math>\mathrm{E}f|_{\mathrm{d}x~\mathrm{d}y}</math></p>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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| width="18%" |
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<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
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<p><math>\mathrm{T}_{10} f</math></p>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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<p><math>\mathrm{E}f|_{\mathrm{d}x(\mathrm{d}y)}</math></p>
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<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
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| width="18%" |
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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<p><math>\mathrm{T}_{01} f</math></p>
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<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{ \texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
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<p><math>\mathrm{E}f|_{(\mathrm{d}x)\mathrm{d}y}</math></p>
  −
| width="18%" |
  −
<p><math>\mathrm{T}_{00} f</math></p>
  −
<p><math>\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}</math></p>
   
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|-
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| <math>f_0\!</math>
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| style="border-top:1px solid black" | <math>f_{0}\!</math>
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| <math>(~)</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
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| <math>(~)</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{(~)}\!</math>
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| <math>(~)</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
 
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|-
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<math>\begin{matrix}
 
<math>\begin{matrix}
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f_1
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f_{1}\\f_{2}\\f_{4}\\f_{8}
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\\[4pt]
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\end{matrix}</math>
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f_2
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\\[4pt]
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f_4
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\\[4pt]
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f_8
  −
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
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(x)(y)
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\[4pt]
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\\
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(x)~y~
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\texttt{(} x \texttt{)~} y \texttt{~}
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\\
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~x~(y)
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\[4pt]
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\\
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~x~~y~
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\texttt{~} x \texttt{~~} y \texttt{~}
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\end{matrix}\!</math>
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\end{matrix}</math>
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| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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~x~~y~
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\texttt{~} x \texttt{~~} y \texttt{~}
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\\[4pt]
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\\
−
~x~(y)
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\[4pt]
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\\
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(x)~y~
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\texttt{(} x \texttt{)~} y \texttt{~}
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\\[4pt]
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\\
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(x)(y)
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\texttt{(} x \texttt{)(} y \texttt{)}
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\end{matrix}\!</math>
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\end{matrix}</math>
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|
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| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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~x~(y)
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\[4pt]
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\\
−
~x~~y~
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\texttt{~} x \texttt{~~} y \texttt{~}
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\\[4pt]
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\\
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(x)(y)
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\[4pt]
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\\
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(x)~y~
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\texttt{(} x \texttt{)~} y \texttt{~}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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|
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| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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(x)~y~
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\texttt{(} x \texttt{)~} y \texttt{~}
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\\[4pt]
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\\
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(x)(y)
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\[4pt]
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\\
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~x~~y~
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\texttt{~} x \texttt{~~} y \texttt{~}
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\\[4pt]
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\\
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~x~(y)
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\texttt{~} x \texttt{~(} y \texttt{)}
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\end{matrix}\!</math>
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\end{matrix}</math>
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| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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(x)(y)
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\[4pt]
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\\
−
(x)~y~
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\texttt{(} x \texttt{)~} y \texttt{~}
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\\[4pt]
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\\
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~x~(y)
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\[4pt]
+
\\
−
~x~~y~
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\texttt{~} x \texttt{~~} y \texttt{~}
−
\end{matrix}\!</math>
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\end{matrix}</math>
 
|-
 
|-
−
|
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| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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f_3
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f_{3}\\f_{12}
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\\[4pt]
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\end{matrix}</math>
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f_{12}
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| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
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(x)
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\texttt{(} x \texttt{)}
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\\[4pt]
+
\\
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~x~
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\texttt{~} x \texttt{~}
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\end{matrix}\!</math>
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\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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~x~
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\texttt{~} x \texttt{~}
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\\[4pt]
+
\\
−
(x)
+
\texttt{(} x \texttt{)}
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\end{matrix}\!</math>
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\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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~x~
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\texttt{~} x \texttt{~}
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\\[4pt]
+
\\
−
(x)
+
\texttt{(} x \texttt{)}
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\end{matrix}\!</math>
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\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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(x)
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
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~x~
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\texttt{~} x \texttt{~}
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\end{matrix}\!</math>
+
\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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(x)
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
−
~x~
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\texttt{~} x \texttt{~}
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\end{matrix}\!</math>
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\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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f_6
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f_{6}\\f_{9}
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\\[4pt]
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\end{matrix}</math>
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f_9
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| style="border-top:1px solid black; border-left:1px solid black" |
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\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
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~(x,~y)~
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\texttt{~(} x \texttt{,~} y \texttt{)~}
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\\[4pt]
+
\\
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((x,~y))
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\texttt{((} x \texttt{,~} y \texttt{))}
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\end{matrix}\!</math>
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\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(x,~y)~
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
−
\\[4pt]
+
\\
−
((x,~y))
+
\texttt{((} x \texttt{,~} y \texttt{))}
−
\end{matrix}\!</math>
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\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
((x,~y))
+
\texttt{((} x \texttt{,~} y \texttt{))}
−
\\[4pt]
+
\\
−
~(x,~y)~
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
((x,~y))
+
\texttt{((} x \texttt{,~} y \texttt{))}
−
\\[4pt]
+
\\
−
~(x,~y)~
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(x,~y)~
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
−
\\[4pt]
+
\\
−
((x,~y))
+
\texttt{((} x \texttt{,~} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_5
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f_{5}\\f_{10}
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\\[4pt]
+
\end{matrix}</math>
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f_{10}
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| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(y)
+
\texttt{(} y \texttt{)}
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\\[4pt]
+
\\
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~y~
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\texttt{~} y \texttt{~}
−
\end{matrix}\!</math>
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\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~y~
+
\texttt{~} y \texttt{~}
−
\\[4pt]
+
\\
−
(y)
+
\texttt{(} y \texttt{)}
−
\end{matrix}\!</math>
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\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(y)
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
~y~
+
\texttt{~} y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~y~
+
\texttt{~} y \texttt{~}
−
\\[4pt]
+
\\
−
(y)
+
\texttt{(} y \texttt{)}
−
\end{matrix}\!</math>
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\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(y)
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
~y~
+
\texttt{~} y \texttt{~}
−
\end{matrix}\!</math>
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\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_7
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f_{7}\\f_{11}\\f_{13}\\f_{14}
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\\[4pt]
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\end{matrix}</math>
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f_{11}
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| style="border-top:1px solid black; border-left:1px solid black" |
−
\\[4pt]
  −
f_{13}
  −
\\[4pt]
  −
f_{14}
  −
\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(~x~~y~)
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
−
\\[4pt]
+
\\
−
(~x~(y))
+
\texttt{(~} x \texttt{~(} y \texttt{))}
−
\\[4pt]
+
\\
−
((x)~y~)
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\texttt{((} x \texttt{)~} y \texttt{~)}
−
\\[4pt]
+
\\
−
((x)(y))
+
\texttt{((} x \texttt{)(} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
((x)(y))
+
\texttt{((} x \texttt{)(} y \texttt{))}
−
\\[4pt]
+
\\
−
((x)~y~)
+
\texttt{((} x \texttt{)~} y \texttt{~)}
−
\\[4pt]
+
\\
−
(~x~(y))
+
\texttt{(~} x \texttt{~(} y \texttt{))}
−
\\[4pt]
+
\\
−
(~x~~y~)
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
((x)~y~)
+
\texttt{((} x \texttt{)~} y \texttt{~)}
−
\\[4pt]
+
\\
−
((x)(y))
+
\texttt{((} x \texttt{)(} y \texttt{))}
−
\\[4pt]
+
\\
−
(~x~~y~)
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
−
\\[4pt]
+
\\
−
(~x~(y))
+
\texttt{(~} x \texttt{~(} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(~x~(y))
+
\texttt{(~} x \texttt{~(} y \texttt{))}
−
\\[4pt]
+
\\
−
(~x~~y~)
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
−
\\[4pt]
+
\\
−
((x)(y))
+
\texttt{((} x \texttt{)(} y \texttt{))}
−
\\[4pt]
+
\\
−
((x)~y~)
+
\texttt{((} x \texttt{)~} y \texttt{~)}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(~x~~y~)
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
−
\\[4pt]
+
\\
−
(~x~(y))
+
\texttt{(~} x \texttt{~(} y \texttt{))}
−
\\[4pt]
+
\\
−
((x)~y~)
+
\texttt{((} x \texttt{)~} y \texttt{~)}
−
\\[4pt]
+
\\
−
((x)(y))
+
\texttt{((} x \texttt{)(} y \texttt{))}
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\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
| <math>f_{15}\!</math>
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| style="border-top:1px solid black" | <math>f_{15}\!</math>
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| <math>((~))</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
−
| <math>((~))</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{((~))}</math>
−
| <math>((~))</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
−
| <math>((~))</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
−
| <math>((~))</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
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|- style="background:#f0f0ff"
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|- style="background:ghostwhite"
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| colspan="2" | <math>\text{Fixed Point Total}\!</math>
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| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math>
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| <math>4\!</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math>
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| <math>4\!</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
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| <math>4\!</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
−
| <math>16\!</math>
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| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math>
 
|}
 
|}
  
Jon Awbrey
12,080

edits

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