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

Revision as of 16:41, 22 September 2013

9,380 bytes added ,  16:41, 22 September 2013
update
Line 10,677: Line 10,677:     
=====Computation of &epsilon;''f''<sub>9</sub>=====
 
=====Computation of &epsilon;''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
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|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math>
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|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{9}
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& = && f_{9}(u, v)
 +
\\[4pt]
 +
& = && \texttt{((} u \texttt{,~} v \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{9}(1, 1)
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)
 +
\\[4pt]
 +
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{9}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
    
=====Computation of E''f''<sub>9</sub>=====
 
=====Computation of E''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{9}
 +
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{E}f_{9}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
    
=====Computation of D''f''<sub>9</sub>=====
 
=====Computation of D''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{9}
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& = && \mathrm{E}f_{9}
 +
& + &  \boldsymbol\varepsilon f_{9}
 +
\\[4pt]
 +
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{9}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
 +
& + &  \texttt{((} u \texttt{,} v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{9}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[20pt]
 +
\mathrm{D}f_{9}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{*{9}{l}}
 +
\mathrm{D}f_{9}
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& = & 0 \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
 +
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
    
=====Computation of d''f''<sub>9</sub>=====
 
=====Computation of d''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{9}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{9}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
    
=====Computation of r''f''<sub>9</sub>=====
 
=====Computation of r''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{9}
 +
& = & \mathrm{D}f_{9}
 +
& + & \mathrm{d}f_{9}
 +
\\[20pt]
 +
\mathrm{D}f_{9}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{9}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[20pt]
 +
\mathrm{r}f_{9}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\end{array}</math>
 +
|}
 +
 +
<br>
    
=====Computation Summary for Equality=====
 
=====Computation Summary for Equality=====
Line 10,695: Line 10,901:  
<math>\begin{array}{c*{8}{l}}
 
<math>\begin{array}{c*{8}{l}}
 
\boldsymbol\varepsilon f_{9}
 
\boldsymbol\varepsilon f_{9}
−
& = & u \!\cdot\! v \cdot 1
+
& = & uv \cdot 1
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
Line 10,701: Line 10,907:  
\\[6pt]
 
\\[6pt]
 
\mathrm{E}f_{9}
 
\mathrm{E}f_{9}
−
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
+
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
Line 10,707: Line 10,913:  
\\[6pt]
 
\\[6pt]
 
\mathrm{D}f_{9}
 
\mathrm{D}f_{9}
−
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
+
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
Line 10,713: Line 10,919:  
\\[6pt]
 
\\[6pt]
 
\mathrm{d}f_{9}
 
\mathrm{d}f_{9}
−
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
+
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
Line 10,719: Line 10,925:  
\\[6pt]
 
\\[6pt]
 
\mathrm{r}f_{9}
 
\mathrm{r}f_{9}
−
& = & u \!\cdot\! v \cdot 0
+
& = & uv \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
Jon Awbrey
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