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

Revision as of 03:30, 7 September 2013

435 bytes added ,  03:30, 7 September 2013
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
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|+ '''Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))'''
   
|
 
|
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{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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<math>\begin{array}{c*{8}{l}}
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| <math>\epsilon</math>''f''
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\boldsymbol\varepsilon f
−
| = || ''uv''        || <math>\cdot</math> || 1
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& = & u \!\cdot\! v \cdot 1
−
| + || ''u''(''v'')   || <math>\cdot</math> || 1
+
& + & u \texttt{(} v \texttt{)} \cdot 1
−
| + || (''u'')''v''  || <math>\cdot</math> || 1
+
& + & \texttt{(} u \texttt{)} v \cdot 1
−
| + || (''u'')(''v'') || <math>\cdot</math> || 0
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
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|-
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\\[6pt]
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| E''f''
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\mathrm{E}f
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| = || ''uv''        || <math>\cdot</math> || (d''u'' d''v'')
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& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}
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| + || ''u''(''v'')   || <math>\cdot</math> || (d''u (d''v''))
+
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
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| + || (''u'')''v''  || <math>\cdot</math> || ((d''u'') d''v'')
+
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
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| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
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|-
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\\[6pt]
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| D''f''
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\mathrm{D}J
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| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
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& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
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| + || ''u''(''v'')   || <math>\cdot</math> || d''u'' (d''v'')
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
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| + || (''u'')''v''  || <math>\cdot</math> || (d''u'') d''v''
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& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
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| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
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|-
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\\[6pt]
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| d''f''
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\mathrm{d}J
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| = || ''uv''        || <math>\cdot</math> || 0
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& = & u \!\cdot\! v \cdot 0
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| + || ''u''(''v'')   || <math>\cdot</math> || d''u''
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& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
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| + || (''u'')''v''  || <math>\cdot</math> || d''v''
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& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
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| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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|-
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\\[6pt]
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| r''f''
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\mathrm{r}J
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| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
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& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
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| + || ''u''(''v'')   || <math>\cdot</math> || d''u'' d''v''
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& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
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| + || (''u'')''v''  || <math>\cdot</math> || d''u'' d''v''
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
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| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
\end{array}</math>
 
|}
 
|}
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|}
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</font>
      
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<br>
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<font face="courier new">
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
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|+ '''Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))'''
   
|
 
|
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{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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<math>\begin{array}{c*{8}{l}}
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| <math>\epsilon</math>''g''
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\boldsymbol\varepsilon g
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| = || ''uv''        || <math>\cdot</math> || 1
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& = & u \!\cdot\! v \cdot 1
−
| + || ''u''(''v'')   || <math>\cdot</math> || 0
+
& + & u \texttt{(} v \texttt{)} \cdot 0
−
| + || (''u'')''v''  || <math>\cdot</math> || 0
+
& + & \texttt{(} u \texttt{)} v \cdot 0
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| + || (''u'')(''v'') || <math>\cdot</math> || 1
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
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|-
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\\[6pt]
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| E''g''
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\mathrm{E}f
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| = || ''uv''        || <math>\cdot</math> || ((d''u'', d''v''))
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& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
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| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'', d''v'')
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& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
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& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
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|-
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\\[6pt]
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| D''g''
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\mathrm{D}J
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| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
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& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'', d''v'')
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& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
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& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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|-
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\\[6pt]
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| d''g''
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\mathrm{d}J
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| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
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& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'', d''v'')
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& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || (''u'')''v''  || <math>\cdot</math> || (d''u'', d''v'')
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& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
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|-
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\\[6pt]
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| r''g''
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\mathrm{r}J
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| = || ''uv''        || <math>\cdot</math> || 0
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& = & u \!\cdot\! v \cdot 0
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| + || ''u''(''v'')   || <math>\cdot</math> || 0
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& + & u \texttt{(} v \texttt{)} \cdot 0
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| + || (''u'')''v''  || <math>\cdot</math> || 0
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& + & \texttt{(} u \texttt{)} v \cdot 0
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| + || (''u'')(''v'') || <math>\cdot</math> || 0
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& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\end{array}</math>
 
|}
 
|}
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|}
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<br>
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 67.  Computation of an Analytic Series in Terms of Coordinates'''
 
|+ '''Table 67.  Computation of an Analytic Series in Terms of Coordinates'''
 
|
 
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 68.  Computation of an Analytic Series in Symbolic Terms'''
 
|+ '''Table 68.  Computation of an Analytic Series in Symbolic Terms'''
 
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|- style="background:ghostwhite"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
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{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
 
{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
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{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
 
{| align="left" border="0" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
Jon Awbrey
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