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

Revision as of 21:48, 10 September 2013

247 bytes removed ,  21:48, 10 September 2013
fix copy errors in old version of Table A6
Line 8,503: Line 8,503:  
| width="10%" |  
 
| width="10%" |  
 
| width="18%" | <math>f\!</math>
 
| width="18%" | <math>f\!</math>
−
| width="18%" | <math>\mathrm{D}f|_{xy}</math>
+
| width="18%" | <math>\mathrm{D}f|_{xy}\!</math>
−
| width="18%" | <math>\mathrm{D}f|_{x(y)}</math>
+
| width="18%" | <math>\mathrm{D}f|_{x(y)}\!</math>
−
| width="18%" | <math>\mathrm{D}f|_{(x)y}</math>
+
| width="18%" | <math>\mathrm{D}f|_{(x)y}~\!</math>
−
| width="18%" | <math>\mathrm{D}f|_{(x)(y)}</math>
+
| width="18%" | <math>\mathrm{D}f|_{(x)(y)}\!</math>
 
|-
 
|-
−
| <math>f_0\!</math>
+
| <math>f_{0}\!</math>
−
| <math>(~)</math>
+
| <math>(~)\!</math>
−
| <math>(~)</math>
+
| <math>(~)\!</math>
−
| <math>(~)</math>
+
| <math>(~)\!</math>
−
| <math>(~)</math>
+
| <math>(~)\!</math>
−
| <math>(~)</math>
+
| <math>(~)\!</math>
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
f_1
 
f_1
−
\\[4pt]
+
\\
 
f_2
 
f_2
−
\\[4pt]
+
\\
 
f_4
 
f_4
−
\\[4pt]
+
\\
 
f_8
 
f_8
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(x)(y)
 
(x)(y)
−
\\[4pt]
+
\\
 
(x)~y~
 
(x)~y~
−
\\[4pt]
+
\\
 
~x~(y)
 
~x~(y)
−
\\[4pt]
+
\\
 
~x~~y~
 
~x~~y~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
Line 8,538: Line 8,538:  
<math>\begin{matrix}
 
<math>\begin{matrix}
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
−
\\[4pt]
+
\\
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
−
\\[4pt]
+
\\
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
−
\\[4pt]
+
\\
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
f_3
 
f_3
−
\\[4pt]
+
\\
 
f_{12}
 
f_{12}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(x)
 
(x)
−
\\[4pt]
+
\\
 
~x~
 
~x~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}x
 
\mathrm{d}x
−
\\[4pt]
+
\\
 
\mathrm{d}x
 
\mathrm{d}x
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}x
 
\mathrm{d}x
−
\\[4pt]
+
\\
 
\mathrm{d}x
 
\mathrm{d}x
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}x
 
\mathrm{d}x
−
\\[4pt]
+
\\
 
\mathrm{d}x
 
\mathrm{d}x
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}x
 
\mathrm{d}x
−
\\[4pt]
+
\\
 
\mathrm{d}x
 
\mathrm{d}x
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
f_6
 
f_6
−
\\[4pt]
+
\\
 
f_9
 
f_9
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
~(x,~y)~
 
~(x,~y)~
−
\\[4pt]
+
\\
 
((x,~y))
 
((x,~y))
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
−
\\[4pt]
+
\\
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
−
\\[4pt]
+
\\
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
−
\\[4pt]
+
\\
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
−
\\[4pt]
+
\\
 
(\mathrm{d}x,~\mathrm{d}y)
 
(\mathrm{d}x,~\mathrm{d}y)
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
f_5
 
f_5
−
\\[4pt]
+
\\
 
f_{10}
 
f_{10}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(y)
 
(y)
−
\\[4pt]
+
\\
 
~y~
 
~y~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}y
 
\mathrm{d}y
−
\\[4pt]
+
\\
 
\mathrm{d}y
 
\mathrm{d}y
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}y
 
\mathrm{d}y
−
\\[4pt]
+
\\
 
\mathrm{d}y
 
\mathrm{d}y
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}y
 
\mathrm{d}y
−
\\[4pt]
+
\\
 
\mathrm{d}y
 
\mathrm{d}y
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
Line 8,683: Line 8,683:  
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\mathrm{d}y
 
\mathrm{d}y
−
\\[4pt]
+
\\
 
\mathrm{d}y
 
\mathrm{d}y
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
f_7
 
f_7
−
\\[4pt]
+
\\
 
f_{11}
 
f_{11}
−
\\[4pt]
+
\\
 
f_{13}
 
f_{13}
−
\\[4pt]
+
\\
 
f_{14}
 
f_{14}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
(~x~~y~)
 
(~x~~y~)
−
\\[4pt]
+
\\
 
(~x~(y))
 
(~x~(y))
−
\\[4pt]
+
\\
 
((x)~y~)
 
((x)~y~)
−
\\[4pt]
+
\\
 
((x)(y))
 
((x)(y))
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
−
\\[4pt]
+
\\
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
Line 8,720: Line 8,720:  
<math>\begin{matrix}
 
<math>\begin{matrix}
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
Line 8,730: Line 8,730:  
<math>\begin{matrix}
 
<math>\begin{matrix}
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
−
\\[4pt]
+
\\
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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<math>\begin{matrix}
 
<math>\begin{matrix}
 
~~\mathrm{d}x~~\mathrm{d}y~~
 
~~\mathrm{d}x~~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
~~\mathrm{d}x~(\mathrm{d}y)~
 
~~\mathrm{d}x~(\mathrm{d}y)~
−
\\[4pt]
+
\\
 
~(\mathrm{d}x)~\mathrm{d}y~~
 
~(\mathrm{d}x)~\mathrm{d}y~~
−
\\[4pt]
+
\\
 
((\mathrm{d}x)(\mathrm{d}y))
 
((\mathrm{d}x)(\mathrm{d}y))
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
 
|-
 
|-
 
| <math>f_{15}\!</math>
 
| <math>f_{15}\!</math>
−
| <math>((~))</math>
+
| <math>((~))\!</math>
−
| <math>((~))</math>
+
| <math>(~)\!</math>
−
| <math>((~))</math>
+
| <math>(~)\!</math>
−
| <math>((~))</math>
+
| <math>(~)\!</math>
−
| <math>((~))</math>
+
| <math>(~)\!</math>
 
|}
 
|}
  
Jon Awbrey
12,080

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