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

Revision as of 17:40, 9 July 2007

1,017 bytes added ,  17:40, 9 July 2007
→‎Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>: wiki table
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</pre>
 
</pre>
   −
<pre>
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
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Table 68.  Computation of an Analytic Series in Symbolic Terms
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|+ '''Table 68.  Computation of an Analytic Series in Symbolic Terms'''
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o-----o-----o------------o----------o----------o----------o----------o----------o
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|- style="background:paleturquoise"
−
| u v | f g |     Df    |   Dg    |   df    |   dg    |   rf    |   rf    |
+
| ''u''&nbsp;&nbsp;''v''
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o-----o-----o------------o----------o----------o----------o----------o----------o
+
| ''f''&nbsp;&nbsp;''g''
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|     |     |           |         |         |         |         |         |
+
| D''f''
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| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du  dv  |    ()   |
+
| D''g''
−
|     |     |            |          |          |          |          |          |
+
| d''f''
−
| 0 1 | 1 0 | (du) dv  | (du, dv) |   dv    | (du, dv) | du  dv  |   ()   |
+
| d''g''
−
|     |     |           |         |         |         |         |         |
+
| d<sup>2</sup>''f''
−
| 1 0 | 1 0 |   du (dv) | (du, dv) |   du    | (du, dv) | du  dv  |   ()   |
+
| d<sup>2</sup>''g''
−
|     |     |           |         |         |         |         |         |
+
|-
−
| 1 1 | 1 1 |   du  dv  | (du, dv) |   ()   | (du, dv) | du  dv  |   ()   |
+
|
−
|     |    |            |          |          |          |          |          |
+
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
−
o-----o-----o------------o----------o----------o----------o----------o----------o
+
| 0&nbsp;&nbsp;0
−
</pre>
+
|-
 +
| 0&nbsp;&nbsp;1
 +
|-
 +
| 1&nbsp;&nbsp;0
 +
|-
 +
| 1&nbsp;&nbsp;1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| 0&nbsp;&nbsp;1
 +
|-
 +
| 1&nbsp;&nbsp;0
 +
|-
 +
| 1&nbsp;&nbsp;0
 +
|-
 +
| 1&nbsp;&nbsp;1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| ((d''u'')(d''v''))
 +
|-
 +
| (d''u'')&nbsp;d''v''&nbsp;
 +
|-
 +
| &nbsp;d''u''&nbsp;(d''v'')
 +
|-
 +
| d''u''&nbsp;&nbsp;d''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (d''u'', d''v'')
 +
|-
 +
| d''v''
 +
|-
 +
| d''u''
 +
|-
 +
| (&nbsp;)
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|-
 +
| (d''u'', d''v'')
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| d''u'' d''v''
 +
|-
 +
| d''u'' d''v''
 +
|-
 +
| d''u'' d''v''
 +
|-
 +
| d''u'' d''v''
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
 +
| (&nbsp;)
 +
|-
 +
| (&nbsp;)
 +
|-
 +
| (&nbsp;)
 +
|-
 +
| (&nbsp;)
 +
|}
 +
|}
 +
<br>
    
Figure&nbsp;69 gives a graphical picture of the difference map D''F''&nbsp;=&nbsp;‹D''f'',&nbsp;D''g''› for the transformation ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›.  This depicts the same information about D''f'' and D''g'' that was given in the corresponding rows of the computation summary in Tables&nbsp;66-i and 66-ii, excerpted here:
 
Figure&nbsp;69 gives a graphical picture of the difference map D''F''&nbsp;=&nbsp;‹D''f'',&nbsp;D''g''› for the transformation ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›.  This depicts the same information about D''f'' and D''g'' that was given in the corresponding rows of the computation summary in Tables&nbsp;66-i and 66-ii, excerpted here:
Jon Awbrey
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