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Revision as of 20:46, 17 June 2009
, 20:46, 17 June 2009→Note 26: markup
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<pre>+ d[pq]+ =+
− p q . (dp, dq)
−
− +
−
− p (q) . dq
−
− +
−
− (p) q . dp
−
− +
−
− (p)(q) . 0
−
−</pre>
</pre>
|}
|}
−Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
−Just to be clear about what's being indicated here,
−it's a visual way of specifying the following data:
−{| align="center" cellspacing="10" style="text-align:center"
+|
+<math>\begin{array}{rcccccc}
+\operatorname{d}(pq)
+& = & p & \cdot & q & \cdot &
+\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
+\\[4pt]
+& + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q
+\\[4pt]
+& + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p
+\\[4pt]
+& + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & 0
+\end{array}</math>
+|}
−To understand the extended interpretations, that is, the conjunctions of basic and differential features that are being indicated here, it may help to note the following equivalences:
−To understand the extended interpretations, that is,
−the conjunctions of basic and differential features
−that are being indicated here, it may help to note
−the following equivalences:
+<pre>
(dp, dq) = dp + dq = dp(dq) + (dp)dq
(dp, dq) = dp + dq = dp(dq) + (dp)dq
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dq = dp dq + (dp)dq
dq = dp dq + (dp)dq
+</pre>
−Capping the series that analyzes the proposition pq
+Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure 26-2 shows the remainder map <math>\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.
−in terms of succeeding orders of linear propositions,
−Figure 26-2 shows the remainder map r[pq] : EX -> B,
−that happens to be linear in pairs of variables.
−{| align="center" cellspacing="10" style="text-align:center; width:90%"
{| align="center" cellspacing="10" style="text-align:center; width:90%"
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Reading the arrows off the map produces the following data:
Reading the arrows off the map produces the following data:
+<pre>
r[pq]
r[pq]
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(p)(q) . dp dq
(p)(q) . dp dq
+</pre>
−In short, r[pq] is a constant field,
+In short, <math>\operatorname{r}(pq)</math> is a constant field, having the value <math>\operatorname{d}p~\operatorname{d}q</math> at each cell.
−having the value dp dq at each cell.
A more detailed presentation of Differential Logic can be found here:
A more detailed presentation of Differential Logic can be found here:
−DLOG D. http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
+* DLOG D. http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
−DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
+* DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
−DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
+* DLOG D. http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
−==Document History==
==Document History==
