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

Revision as of 02:32, 10 July 2007

4,862 bytes added ,  02:32, 10 July 2007
→‎Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>: wiki table
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Table&nbsp;67 shows how to compute the analytic series for ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))› in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.
 
Table&nbsp;67 shows how to compute the analytic series for ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))› in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.
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<pre>
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
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Table 67.  Computation of an Analytic Series in Terms of Coordinates
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|+ Table 67.  Computation of an Analytic Series in Terms of Coordinates
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o--------o-------o-------o--------o-------o-------o-------o-------o
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|
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| u v | du dv | u' v' | f  g  | Ef Eg | Df Dg | df dg | rf rg |
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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o--------o-------o-------o--------o-------o-------o-------o-------o
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|
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|       |       |       |       |       |       |       |       |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| 0 0 | 0 0  | 0  0  | 0  1 | 0  1  | 0 0 | 0 0 | 0 0  |
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| ''u''
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|       |       |       |       |       |       |       |       |
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| ''v''
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|       | 0 1 | 0 1  |       | 1  0 | 1 1 | 1 1  | 0 0  |
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|}
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|       |       |       |       |       |       |       |       |
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|
−
|       | 1  0 | 1  0 |       | 1  0 | 1 1 | 1 1 | 0 0 |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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|       |       |       |       |       |       |       |       |
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| d''u''
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|       | 1 1 | 1 1 |       | 1 1 | 1  0 | 0 0 | 1  0 |
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| d''v''
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|       |       |       |       |       |       |       |       |
+
|}
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o--------o-------o-------o--------o-------o-------o-------o-------o
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|
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|       |       |       |       |       |       |       |       |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| 0  1  | 0 0 | 0 1 | 1  0  | 1  0  | 0 0 | 0 0  | 0  0  |
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| ''u''<font face="courier new">’</font>
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|       |       |       |       |       |       |       |       |
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| ''v''<font face="courier new">’</font>
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|       | 0  1 | 0 0 |       | 0 1 | 1 1 | 1 1 | 0 0  |
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|}
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|       |       |       |       |       |       |       |       |
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|-
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|       | 1 0  | 1 1  |       | 1  1  | 0 1  | 0 1  | 0 0 |
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| valign="top" |
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|       |       |       |       |       |       |       |      |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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|       | 1  1  | 1  0 |       | 1  0  | 0 0 | 1 0  | 1  0 |
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| 0 || 0
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|       |       |       |       |       |       |       |       |
+
|}
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o--------o-------o-------o--------o-------o-------o-------o-------o
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|
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|       |       |       |       |       |       |       |       |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1  0 | 0 0 | 1 0  | 1 0 | 1 0 | 0 0  | 0 0 | 0 0 |
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| 0 || 0
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|       |       |       |       |       |       |       |       |
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|-
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|       | 0 1  | 1  1  |       | 1  1  | 0 1  | 0 1  | 0 0 |
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| 0 || 1
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|       |       |       |       |       |       |       |       |
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|-
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|       | 1  0  | 0 0 |       | 0 1  | 1  1  | 1  1  | 0 0 |
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| 1 || 0
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|       |       |       |       |       |       |       |       |
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|-
−
|       | 1  1  | 0 1  |       | 1  0 | 0 0 | 1  0 | 1 0  |
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| 1 || 1
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|       |       |       |       |       |       |       |       |
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|}
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o--------o-------o-------o--------o-------o-------o-------o-------o
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|
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|       |       |       |       |       |       |       |      |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 1 | 0  0  | 1 1  | 1  1  | 1  1  | 0 0 | 0 0 | 0 0 |
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| 0 || 0
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|       |       |       |       |       |       |       |       |
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|-
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|       | 0 1 | 1 0 |       | 1 0  | 0  1 | 0 1 | 0 0  |
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| 0 || 1
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|       |       |       |       |       |       |       |       |
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|-
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|       | 1  0 | 0  1 |       | 1  0 | 0  1 | 0  1 | 0 0  |
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| 1 || 0
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|       |       |       |       |       |       |       |       |
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|-
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|       | 1 1  | 0 0 |       | 0 1  | 1  0 | 0 0 | 1  0 |
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| 1 || 1
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|       |       |       |       |       |       |       |       |
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|}
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o--------o-------o-------o--------o-------o-------o-------o-------o
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|-
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</pre>
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| 1 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 0 || 1
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| 1 || 0
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| 1 || 1
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 0
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| 1 || 1
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| 0 || 0
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| 0 || 1
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| valign="top" |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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|-
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| 1 || 0
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|-
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| 0 || 1
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| 0 || 0
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 1
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| 1 || 0
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| 1 || 1
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| <math>\epsilon</math>''f''
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| <math>\epsilon</math>''g''
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| E''f''
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| E''g''
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| D''f''
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| D''g''
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|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| d''f''
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| d''g''
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|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| d<sup>2</sup>''f''
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| d<sup>2</sup>''g''
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|}
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|-
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| valign="top" |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 1
 +
|}
 +
|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 1
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|-
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| 1 || 0
 +
|-
 +
| 1 || 0
 +
|-
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| 1 || 1
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|}
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|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 1 || 1
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|-
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| 1 || 1
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|-
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| 1 || 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 1 || 1
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| 1 || 1
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| 0 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 0 || 0
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| 1 || 0
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|-
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| valign="top" |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 1
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| 1 || 1
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| 1 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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| 0 || 1
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| 0 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 1 || 1
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| 0 || 1
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| 1 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 0 || 0
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| 0 || 0
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| 1 || 0
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| 1 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 0 || 0
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| 0 || 0
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| 1 || 0
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| valign="top" |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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|-
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| 1 || 0
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| 1 || 0
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|-
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| 0 || 1
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 1
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|-
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| 0 || 1
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| 1 || 0
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|}
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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| 0 || 1
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| 0 || 1
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| 0 || 0
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 0
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| 0 || 0
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| 1 || 0
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<br>
    
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
Jon Awbrey
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