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

Revision as of 04:30, 10 September 2013

3,431 bytes added ,  04:30, 10 September 2013
update
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
 
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
 
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
 
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
−
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x~\mathrm{d}y}\end{matrix}</math>
+
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math>
 
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
 
<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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<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
 
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
 
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
−
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{ \texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
+
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
 
|-
 
|-
 
| style="border-top:1px solid black" | <math>f_{0}\!</math>
 
| style="border-top:1px solid black" | <math>f_{0}\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
 
| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{(~)}\!</math>
 
| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{(~)}\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{(~)}\!</math>
 
|-
 
|-
 
| style="border-top:1px solid black" |
 
| style="border-top:1px solid black" |
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|-
 
|-
 
| style="border-top:1px solid black" | <math>f_{15}\!</math>
 
| style="border-top:1px solid black" | <math>f_{15}\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
 
| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{((~))}</math>
 
| style="border-top:1px solid black; border-left:4px double black" | <math>\texttt{((~))}</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((~))}</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>
 
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math>
 
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math>
−
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math>
 
|}
 
|}
    
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+
{| 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%"
−
|+ <math>\text{Table A4.}~~\mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
+
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
−
|- style="background:#f0f0ff"
+
|- style="background:ghostwhite"
−
| width="10%" | &nbsp;
+
| style="width:10%; border-bottom:1px solid black" | &nbsp;
−
| width="18%" | <math>f\!</math>
+
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
−
| width="18%" |
+
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
−
<math>\mathrm{D}f|_{\mathrm{d}x~\mathrm{d}y}</math>
+
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math>
−
| width="18%" |
+
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
−
<math>\mathrm{D}f|_{\mathrm{d}x(\mathrm{d}y)}</math>
+
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math>
−
| width="18%" |
+
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
−
<math>\mathrm{D}f|_{(\mathrm{d}x)\mathrm{d}y}</math>
+
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math>
−
| width="18%" |
+
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
−
<math>\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}</math>
+
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math>
 
|-
 
|-
−
| <math>f_0\!</math>
+
| style="border-top:1px solid black" | <math>f_{0}\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_1
+
f_{1}\\f_{2}\\f_{4}\\f_{8}
−
\\[4pt]
+
\end{matrix}</math>
−
f_2
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\\[4pt]
  −
f_4
  −
\\[4pt]
  −
f_8
  −
\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)
+
\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}
−
(y)
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
~y~
+
y
−
\\[4pt]
+
\\
−
(y)
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
~y~
+
y
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(x)
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
−
(x)
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
−
~x~
+
x
−
\\[4pt]
+
\\
−
~x~
+
x
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
<math>\begin{matrix}
+
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
−
(~)
  −
\\[4pt]
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\end{matrix}\!</math>
   
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_3
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
−
f_{12}
+
x
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:4px double black" |
−
<math>\begin{matrix}
+
<math>\begin{matrix}1\\1\end{matrix}</math>
−
(x)
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\\[4pt]
+
<math>\begin{matrix}1\\1\end{matrix}</math>
−
~x~
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}\!</math>
+
<math>\begin{matrix}0\\0\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
<math>\begin{matrix}
+
<math>\begin{matrix}0\\0\end{matrix}</math>
−
((~))
  −
\\[4pt]
  −
((~))
  −
\end{matrix}~\!</math>
  −
|
  −
<math>\begin{matrix}
  −
((~))
  −
\\[4pt]
  −
((~))
  −
\end{matrix}~\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\end{matrix}\!</math>
   
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_6
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
−
\\[4pt]
+
\\
−
f_9
+
\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}0\\0\end{matrix}</math>
−
~(x,~y)~
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\\[4pt]
+
<math>\begin{matrix}1\\1\end{matrix}</math>
−
((x,~y))
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}\!</math>
+
<math>\begin{matrix}1\\1\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
<math>\begin{matrix}
+
<math>\begin{matrix}0\\0\end{matrix}</math>
−
(~)
  −
\\[4pt]
  −
(~)
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
((~))
  −
\\[4pt]
  −
((~))
  −
\end{matrix}~\!</math>
  −
|
  −
<math>\begin{matrix}
  −
((~))
  −
\\[4pt]
  −
((~))
  −
\end{matrix}~\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\end{matrix}\!</math>
   
|-
 
|-
−
|
+
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_5
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
f_{10}
+
\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}1\\1\end{matrix}</math>
−
(y)
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\\[4pt]
+
<math>\begin{matrix}0\\0\end{matrix}</math>
−
~y~
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
\end{matrix}\!</math>
+
<math>\begin{matrix}1\\1\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
<math>\begin{matrix}
+
<math>\begin{matrix}0\\0\end{matrix}</math>
−
((~))
  −
\\[4pt]
  −
((~))
  −
\end{matrix}~\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
((~))
  −
\\[4pt]
  −
((~))
  −
\end{matrix}~\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\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{((} 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))
+
y
−
\\[4pt]
+
\\
−
~(x,~y)~
+
\texttt{(} y \texttt{)}
−
\\[4pt]
+
\\
−
~(x,~y)~
+
y
−
\\[4pt]
+
\\
−
((x,~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~
+
x
−
\\[4pt]
+
\\
−
(y)
+
x
−
\\[4pt]
+
\\
−
~y~
+
\texttt{(} x \texttt{)}
−
\\[4pt]
+
\\
−
(y)
+
\texttt{(} x \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
<math>\begin{matrix}
+
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
−
~x~
  −
\\[4pt]
  −
~x~
  −
\\[4pt]
  −
(x)
  −
\\[4pt]
  −
(x)
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\\[4pt]
  −
(~)
  −
\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>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 
|}
 
|}
  
Jon Awbrey
12,080

edits

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