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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)
Revision as of 19:23, 14 June 2007
, 19:23, 14 June 2007→Analytic Series : Recap: wiki table
Let us now summarize the results of Table 50 by writing down for each column, and for each block of constant ‹''u'', ''v''›, a reasonably canonical symbolic expression for the function of ‹d''u'', d''v''› that appears there. The synopsis formed in this way is presented in Table 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.
Let us now summarize the results of Table 50 by writing down for each column, and for each block of constant ‹''u'', ''v''›, a reasonably canonical symbolic expression for the function of ‹d''u'', d''v''› that appears there. The synopsis formed in this way is presented in Table 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.
<pre>
<font face="courier new">
Table 51. Computation of an Analytic Series in Symbolic Terms
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ Table 51. Computation of an Analytic Series in Symbolic Terms
| u v | J | EJ | DJ | dJ | d^2.J |
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| | | | | | |
| ''u'' || ''v''
| 0 0 | 0 | du dv | du dv | () | du dv |
|}
| | | | | | |
|
| 0 1 | 0 | du (dv) | du (dv) | du | du dv |
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| | | | | | |
| ''J''
| 1 0 | 0 | (du) dv | (du) dv | dv | du dv |
|}
| | | | | | |
|
| 1 1 | 1 | (du)(dv) | ((du)(dv)) | (du, dv) | du dv |
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| | | | | | |
| E''J''
|}
</pre>
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| D''J''
|}
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d''J''
|}
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| d<sup>2</sup>''J''
|}
|-
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0 || 0
|-
| 0 || 1
|-
| 1 || 0
|-
| 1 || 1
|}
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| 0
|-
| 0
|-
| 0
|-
| 1
|}
|
{| align="center" border="0" cellpadding="8" 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="8" 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="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| ()
|-
| d''u''
|-
| d''v''
|-
| (d''u'', d''v'')
|}
|
{| align="center" border="0" cellpadding="8" 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''
|}
|}
</font><br>
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of E''J'' = ''J'' + D''J'' and D''J'' = d''J'' + r''J'' in two different styles of diagram.
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of E''J'' = ''J'' + D''J'' and D''J'' = d''J'' + r''J'' in two different styles of diagram.