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

Revision as of 20:30, 10 September 2013

4,660 bytes added ,  20:30, 10 September 2013
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| style="border-top:1px solid black" | <math>f_{0}\!</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>0\!</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{(~)}\!</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</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>0\!</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>0\!</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>0\!</math>
 
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| style="border-top:1px solid black" | <math>f_{15}\!</math>
 
| style="border-top:1px solid black" | <math>f_{15}\!</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>1\!</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{((~))}</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</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>1\!</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>1\!</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>1\!</math>
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math>
 
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math>
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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 A5.}~~\mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
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|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
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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%" | <math>\mathrm{E}f|_{xy}</math>
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| width="18%" | <math>\mathrm{E}f|_{x(y)}</math>
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<math>\mathrm{E}f|_{xy}\!</math>
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| width="18%" | <math>\mathrm{E}f|_{(x)y}</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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| width="18%" | <math>\mathrm{E}f|_{(x)(y)}</math>
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<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math>
 
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<math>\begin{matrix}
 
<math>\begin{matrix}
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f_4
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f_8
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\end{matrix}\!</math>
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<math>\begin{matrix}
 
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<math>\begin{matrix}
 
<math>\begin{matrix}
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\\
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<math>\begin{matrix}
 
<math>\begin{matrix}
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\\[4pt]
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\\
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\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
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\\[4pt]
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\\
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\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
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\\
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\\
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−
\\[4pt]
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\\
−
~\mathrm{d}x~~\mathrm{d}y~
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
−
\end{matrix}\!</math>
+
\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
+
f_{3}\\f_{12}
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\\[4pt]
+
\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{)}
−
\\[4pt]
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\\
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~x~
+
\texttt{~} x \texttt{~}
−
\end{matrix}\!</math>
+
\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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~\mathrm{d}x~
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\texttt{~} \mathrm{d}x \texttt{~}
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\\[4pt]
+
\\
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(\mathrm{d}x)
+
\texttt{(} \mathrm{d}x \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}
−
~\mathrm{d}x~
+
\texttt{~} \mathrm{d}x \texttt{~}
−
\\[4pt]
+
\\
−
(\mathrm{d}x)
+
\texttt{(} \mathrm{d}x \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}
−
(\mathrm{d}x)
+
\texttt{(} \mathrm{d}x \texttt{)}
−
\\[4pt]
+
\\
−
~\mathrm{d}x~
+
\texttt{~} \mathrm{d}x \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}
−
(\mathrm{d}x)
+
\texttt{(} \mathrm{d}x \texttt{)}
−
\\[4pt]
+
\\
−
~\mathrm{d}x~
+
\texttt{~} \mathrm{d}x \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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f_6
+
f_{6}\\f_{9}
−
\\[4pt]
+
\end{matrix}</math>
−
f_9
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}~</math>
  −
|
   
<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:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(\mathrm{d}x,~\mathrm{d}y)~
+
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
−
\\[4pt]
+
\\
−
((\mathrm{d}x,~\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}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}
−
((\mathrm{d}x,~\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
~(\mathrm{d}x,~\mathrm{d}y)~
+
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}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}
−
((\mathrm{d}x,~\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
~(\mathrm{d}x,~\mathrm{d}y)~
+
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}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}
−
~(\mathrm{d}x,~\mathrm{d}y)~
+
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
−
\\[4pt]
+
\\
−
((\mathrm{d}x,~\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
−
\end{matrix}~</math>
+
\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_5
+
f_{5}\\f_{10}
−
\\[4pt]
+
\end{matrix}</math>
−
f_{10}
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}\!</math>
  −
|
   
<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:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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~\mathrm{d}y~
+
\texttt{~} \mathrm{d}y \texttt{~}
−
\\[4pt]
+
\\
−
(\mathrm{d}y)
+
\texttt{(} \mathrm{d}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}
−
(\mathrm{d}y)
+
\texttt{(} \mathrm{d}y \texttt{)}
−
\\[4pt]
+
\\
−
~\mathrm{d}y~
+
\texttt{~} \mathrm{d}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}
−
~\mathrm{d}y~
+
\texttt{~} \mathrm{d}y \texttt{~}
−
\\[4pt]
+
\\
−
(\mathrm{d}y)
+
\texttt{(} \mathrm{d}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}
−
(\mathrm{d}y)
+
\texttt{(} \mathrm{d}y \texttt{)}
−
\\[4pt]
+
\\
−
~\mathrm{d}y~
+
\texttt{~} \mathrm{d}y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{7}\\f_{11}\\f_{13}\\f_{14}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_7
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
−
\\[4pt]
+
\\
−
f_{11}
+
\texttt{(~} x \texttt{~(} y \texttt{))}
−
\\[4pt]
+
\\
−
f_{13}
+
\texttt{((} x \texttt{)~} y \texttt{~)}
−
\\[4pt]
+
\\
−
f_{14}
+
\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{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
(~x~(y))
+
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
−
\\[4pt]
+
\\
−
((x)~y~)
+
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
((x)(y))
+
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}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}
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
−
\\[4pt]
+
\\
−
((\mathrm{d}x)~\mathrm{d}y~)
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
(~\mathrm{d}x~(\mathrm{d}y))
+
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
−
\\[4pt]
+
\\
−
(~\mathrm{d}x~~\mathrm{d}y~)
+
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}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}
−
((\mathrm{d}x)~\mathrm{d}y~)
+
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
−
\\[4pt]
+
\\
−
(~\mathrm{d}x~~\mathrm{d}y~)
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
(~\mathrm{d}x~(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}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}
−
(~\mathrm{d}x~(\mathrm{d}y))
+
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
−
\\[4pt]
+
\\
−
(~\mathrm{d}x~~\mathrm{d}y~)
+
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
−
\\[4pt]
+
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
−
\\[4pt]
+
\\
−
((\mathrm{d}x)~\mathrm{d}y~)
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
−
\end{matrix}~\!</math>
+
\end{matrix}</math>
−
|
  −
<math>\begin{matrix}
  −
(~\mathrm{d}x~~\mathrm{d}y~)
  −
\\[4pt]
  −
(~\mathrm{d}x~(\mathrm{d}y))
  −
\\[4pt]
  −
((\mathrm{d}x)~\mathrm{d}y~)
  −
\\[4pt]
  −
((\mathrm{d}x)(\mathrm{d}y))
  −
\end{matrix}\!</math>
   
|-
 
|-
−
| <math>f_{15}\!</math>
+
| style="border-top:1px solid black" | <math>f_{15}\!</math>
−
| <math>((~))</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
−
| <math>((~))</math>
+
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
−
| <math>((~))</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
−
| <math>((~))</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
−
| <math>((~))</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 
|}
 
|}
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:80%"
 
|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
 
|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
Jon Awbrey
12,080

edits

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