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

Revision as of 22:18, 29 August 2013

1,452 bytes added ,  22:18, 29 August 2013
update
Line 5,134: Line 5,134:  
=====Summary of Conjunction=====
 
=====Summary of Conjunction=====
   −
To establish a convenient reference point for further discussion, Table 49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition ''J''.
+
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math>
   −
<font face="courier new">
+
<br>
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
 
−
|+ Table 49. Computation Summary for ''J''
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math>
 
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{array}{c*{8}{l}}
−
| <math>\epsilon</math>''J''
+
\boldsymbol\varepsilon J
−
| = || ''uv''        || <math>\cdot</math> || 1
+
& = & 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
−
| + || (''u'')(''v'') || <math>\cdot</math> || 0
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
−
|-
+
\\[6pt]
−
| E''J''
+
\mathrm{E}J
−
| = || ''uv''        || <math>\cdot</math> || (d''u'')(d''v'')
+
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
−
| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'')d''v''
+
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
−
| + || (''u'')''v''  || <math>\cdot</math> || d''u''(d''v'')
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
−
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
−
|-
+
\\[6pt]
−
| D''J''
+
\mathrm{D}J
−
| = || ''uv''        || <math>\cdot</math> || ((d''u'')(d''v''))
+
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
−
| + || ''u''(''v'')   || <math>\cdot</math> || (d''u'')d''v''
+
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
−
| + || (''u'')''v''  || <math>\cdot</math> || d''u''(d''v'')
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
−
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
−
|-
+
\\[6pt]
−
| d''J''
+
\mathrm{d}J
−
| = || ''uv''        || <math>\cdot</math> || (d''u'', d''v'')
+
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
−
| + || ''u''(''v'')   || <math>\cdot</math> || d''v''
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
−
| + || (''u'')''v''  || <math>\cdot</math> || d''u''
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
−
| + || (''u'')(''v'') || <math>\cdot</math> || 0
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
−
|-
+
\\[6pt]
−
| r''J''
+
\mathrm{r}J
−
| = || ''uv''        || <math>\cdot</math> || d''u'' d''v''
+
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
−
| + || ''u''(''v'')   || <math>\cdot</math> || d''u'' d''v''
+
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
−
| + || (''u'')''v''  || <math>\cdot</math> || d''u'' d''v''
+
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
−
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
+
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
\end{array}</math>
 
|}
 
|}
−
|}
+
 
−
</font><br>
+
<br>
    
====Analytic Series : Coordinate Method====
 
====Analytic Series : Coordinate Method====
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<br>
 
<br>
   −
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:70%"
+
{| align="center" cellpadding="1" 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:70%"
−
|+ Table 50. Computation of an Analytic Series in Terms of Coordinates
+
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math>
−
|
+
| width="50%" |
−
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
{| 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:100%"
−
|
+
|- style="height:35px; background:ghostwhite"
−
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
+
| style="width:16%; border-bottom:1px solid black" | <math>u\!</math>
−
| ''u''
+
| style="width:16%; border-bottom:1px solid black" | <math>v\!</math>
−
| ''v''
+
| style="width:16%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math>
−
|}
+
| style="width:16%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math>
−
|
+
| style="width:16%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math>
−
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
+
| style="width:16%; border-bottom:1px solid black" | <math>v'\!</math>
−
| d''u''
  −
| d''v''
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
  −
| ''u''<font face="courier new">’</font>
  −
| ''v''<font face="courier new">’</font>
  −
|}
  −
|-
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0      || 0
  −
|-
  −
| &nbsp; || &nbsp;
  −
|-
  −
| &nbsp; || &nbsp;
  −
|-
  −
| &nbsp; || &nbsp;
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0 || 0
   
|-
 
|-
−
| 0 || 1
+
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
 +
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 
|-
 
|-
−
| 1 || 0
+
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
 +
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}1\\0\\1\\0\end{matrix}</math>
 
|-
 
|-
−
| 1 || 1
+
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
−
|}
+
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
−
|
+
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
−
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
−
| 0 || 0
+
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}1\\1\\0\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 
|-
 
|-
−
| 0 || 1
+
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
−
|-
+
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
−
| 1 || 0
+
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
−
|-
+
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
−
| 1 || 1
+
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}1\\1\\0\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}1\\0\\1\\0\end{matrix}</math>
 
|}
 
|}
 +
| width="50%" |
 +
{| 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:100%"
 +
|- style="height:35px; background:ghostwhite"
 +
| style="width:20%; border-bottom:1px solid black" | <math>\boldsymbol\varepsilon J\!</math>
 +
| style="width:20%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math>
 +
| style="width:20%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math>
 +
| style="width:20%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math>
 +
| style="width:20%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math>
 
|-
 
|-
−
|
+
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
−
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="vertical-align:top; border-top:1px solid black" |
−
| 0     || 1
+
<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>
 
|-
 
|-
−
| &nbsp; || &nbsp;
+
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\1\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>
 
|-
 
|-
−
| &nbsp; || &nbsp;
+
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\0\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\1\\0\\1\end{matrix}</math>
 +
| style="vertical-align:top; border-top:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>
 
|-
 
|-
−
| &nbsp; || &nbsp;
+
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math>
−
|}
+
| style="vertical-align:top; border-top:1px solid black" |
−
|
+
<math>\begin{matrix}1\\0\\0\\0\end{matrix}</math>
−
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
−
| 0 || 0
+
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
−
|-
+
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |
−
| 0 || 1
+
<math>\begin{matrix}0\\1\\1\\0\end{matrix}</math>
−
|-
+
| style="vertical-align:top; border-top:1px solid black" |
−
| 1 || 0
+
<math>\begin{matrix}0\\0\\0\\1\end{matrix}</math>
−
|-
  −
| 1 || 1
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0 || 1
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|-
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| 0 || 0
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|-
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| 1 || 1
  −
|-
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| 1 || 0
  −
|}
  −
|-
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 1      || 0
  −
|-
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| &nbsp; || &nbsp;
  −
|-
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| &nbsp; || &nbsp;
  −
|-
  −
| &nbsp; || &nbsp;
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0 || 0
  −
|-
  −
| 0 || 1
  −
|-
  −
| 1 || 0
  −
|-
  −
| 1 || 1
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 1 || 0
  −
|-
  −
| 1 || 1
  −
|-
  −
| 0 || 0
  −
|-
  −
| 0 || 1
  −
|}
  −
|-
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 1      || 1
  −
|-
  −
| &nbsp; || &nbsp;
  −
|-
  −
| &nbsp; || &nbsp;
  −
|-
  −
| &nbsp; || &nbsp;
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0 || 0
  −
|-
  −
| 0 || 1
  −
|-
  −
| 1 || 0
  −
|-
  −
| 1 || 1
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 1 || 1
  −
|-
  −
| 1 || 0
  −
|-
  −
| 0 || 1
  −
|-
  −
| 0 || 0
  −
|}
  −
|}
  −
|
  −
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
  −
| <math>\epsilon</math>''J''
  −
| E''J''
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
  −
| D''J''
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
  −
| d''J''
  −
| d<sup>2</sup>''J''
  −
|}
  −
|-
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0      || 0
  −
|-
  −
| &nbsp; || 0
  −
|-
  −
| &nbsp; || 0
  −
|-
  −
| &nbsp; || 1
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0
  −
|-
  −
| 0
  −
|-
  −
| 0
  −
|-
  −
| 1
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0 || 0
  −
|-
  −
| 0 || 0
  −
|-
  −
| 0 || 0
  −
|-
  −
| 0 || 1
  −
|}
  −
|-
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0      || 0
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 1
  −
|-
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| &nbsp; || 0
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0
  −
|-
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| 0
  −
|-
  −
| 1
  −
|-
  −
| 0
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
  −
| 0 || 0
  −
|-
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| 0 || 0
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|-
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| 1 || 0
  −
|-
  −
| 1 || 1
  −
|}
  −
|-
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0      || 0
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|-
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| &nbsp; || 1
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|-
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| &nbsp; || 0
  −
|-
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| &nbsp; || 0
  −
|}
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|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0
  −
|-
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| 1
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|-
  −
| 0
  −
|-
  −
| 0
  −
|}
  −
|
  −
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 1 || 0
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|-
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| 0 || 0
  −
|-
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| 1 || 1
  −
|}
  −
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====Analytic Series : Recap====
 
====Analytic Series : Recap====
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Let us now summarize the results of Table&nbsp;50 by writing down for each column, and for each block of constant ‹''u'',&nbsp;''v''›, a reasonably canonical symbolic expression for the function of ‹d''u'',&nbsp;d''v''› that appears there.  The synopsis formed in this way is presented in Table&nbsp;51.  As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.
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Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there.  The synopsis formed in this way is presented in Table&nbsp;51.  As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:70%"
 
|+ Table 51.  Computation of an Analytic Series in Symbolic Terms
 
|+ Table 51.  Computation of an Analytic Series in Symbolic Terms
 
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</font>
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Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.
   −
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of E''J''&nbsp;=&nbsp;''J''&nbsp;+&nbsp;D''J'' and D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'' in two different styles of diagram.
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{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
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| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]
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|-
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| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math>
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<br>
 
<br>
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<p>[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]</p>
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<p><center><font size="+1">'''Figure 52.  Decomposition of E''J'''''</font></center></p>
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{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
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<p>[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]</p>
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| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]
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<p><center><font size="+1">'''Figure 53. Decomposition of D''J'''''</font></center></p>
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|-
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| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math>
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====Terminological Interlude====
 
====Terminological Interlude====
Jon Awbrey
12,080

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