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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)
Revision as of 11:36, 2 June 2007
, 11:36, 2 June 2007→Difference Map of Conjunction: wiki table
With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'', ''v''].
With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'', ''v''].
<pre>
<font face="courier new">
Table 41. Computation of DJ (Method 1)
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|+ Table 41. Computation of D''J'' (Method 1)
| |
|
| DJ = EJ + !e!J |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| width="8%" | D''J''
| = J<u + du, v + dv> + J<u, v> |
| width="4%" | =
| |
| width="42%" | E''J''
| = (u, du)(v, dv) + u v |
| width="4%" | +
| |
| width="42%" | <math>\epsilon</math>''J''
|-
| |
| width="8%" |
| DJ = 0 |
| width="4%" | =
| |
| width="42%" | ''J''‹''u'' + d''u'', ''v'' + d''v''›
| + u v (du) dv + u (v)(du) dv |
| width="4%" | +
| |
| width="42%" | ''J''‹''u'', ''v''›
| + u v du (dv) + (u) v du (dv) |
|-
| |
| width="8%" |
| + u v du dv + (u)(v) du dv |
| width="4%" | =
| |
| width="42%" | (''u'', d''u'')(''v'', d''v'')
| width="4%" | +
| |
| width="42%" | ''u'' ''v''
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
|}
| |
|-
|
</pre>
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | D''J''
| width="3%" | =
| width="20%" align=center | 0
| width="23%" |
| width="23%" |
| width="23%" |
|-
| width="8%" |
| width="3%" | +
| width="20%" | ''u'' ''v'' (d''u'') d''v''
| width="23%" | + ''u'' (''v'')(d''u'') d''v''
| width="23%" |
| width="23%" |
|-
| width="8%" |
| width="3%" | +
| width="20%" | ''u'' ''v'' d''u'' (d''v'')
| width="23%" |
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')
| width="23%" |
|-
| width="8%" |
| width="3%" | +
| width="20%" | ''u'' ''v'' d''u'' d''v''
| width="23%" |
| width="23%" |
| width="23%" | + (''u'')(''v'') d''u'' d''v''
|}
|-
|
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| width="8%" | D''J''
| width="3%" | =
| width="22%" | ''u'' ''v'' ((d''u'')(d''v''))
| width="22%" | + ''u'' (''v'')(d''u'') d''v''
| width="22%" | + (''u'') ''v'' d''u'' (d''v'')
| width="22%" | + (''u'')(''v'') d''u'' d''v''
|}
|}
</font><br>
Alternatively, the difference map D''J'' can be expanded over the cells of [d''u'', d''v''] to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.
Alternatively, the difference map D''J'' can be expanded over the cells of [d''u'', d''v''] to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.