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

Revision as of 05:15, 7 September 2013

50 bytes added ,  05:15, 7 September 2013
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& + & \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]
 
\\[6pt]
−
\mathrm{D}J
+
\mathrm{D}f
 
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
 
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
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& + & \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]
 
\\[6pt]
−
\mathrm{d}J
+
\mathrm{d}f
 
& = & u \!\cdot\! v \cdot 0
 
& = & u \!\cdot\! v \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
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& + & \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]
 
\\[6pt]
−
\mathrm{r}J
+
\mathrm{r}f
 
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
 
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
 
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 
\\[6pt]
 
\\[6pt]
−
\mathrm{E}f
+
\mathrm{E}g
 
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
 
& = & u \!\cdot\! v \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{)}
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& + & \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]
 
\\[6pt]
−
\mathrm{D}J
+
\mathrm{D}g
 
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& = & u \!\cdot\! v \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{)}
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& + & \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]
 
\\[6pt]
−
\mathrm{d}J
+
\mathrm{d}g
 
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 
& = & u \!\cdot\! v \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{)}
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& + & \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]
 
\\[6pt]
−
\mathrm{r}J
+
\mathrm{r}g
 
& = & u \!\cdot\! v \cdot 0
 
& = & u \!\cdot\! v \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot 0
 
& + & u \texttt{(} v \texttt{)} \cdot 0
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<br>
 
<br>
   −
<font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
   
|
 
|
−
{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{array}{c*{8}{l}}
−
| &nbsp;
+
\mathrm{D}f
−
|-
+
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
−
| D''f''
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
−
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
+
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
−
| + || ''u''(''v'')   || <math>\cdot</math> || d''u'' (d''v'')
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
−
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'') d''v''
+
\\[8pt]
−
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
+
\mathrm{D}g
−
|-
+
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
| &nbsp;
+
& + & 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{)}
−
| D''g''
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
+
\end{array}</math>
−
| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
  −
|-
  −
| &nbsp;
   
|}
 
|}
−
|}
  −
</font>
      
<br>
 
<br>
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<br>
 
<br>
   −
<font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
   
|
 
|
−
{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{array}{c*{8}{l}}
−
| &nbsp;
+
\mathrm{d}f
−
|-
+
& = & u \!\cdot\! v \cdot 0
−
| d''f''
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
−
| = || ''uv''        || <math>\cdot</math> || 0
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
−
| + || ''u''(''v'')   || <math>\cdot</math> || d''u''
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
| + || (''u'')''v''  || <math>\cdot</math> || d''v''
+
\\[8pt]
−
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
+
\mathrm{d}g
−
|-
+
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
| &nbsp;
+
& + & 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{)}
−
| d''g''
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
+
\end{array}</math>
−
| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
  −
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
  −
|-
  −
| &nbsp;
  −
|}
   
|}
 
|}
−
</font>
      
<br>
 
<br>
 +
    
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
 
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
Jon Awbrey
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