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Revision as of 16:40, 18 June 2009
, 16:40, 18 June 2009→Note 26: convert graphics
==Note 26==
==Note 26==
If we follow the classical line that singles out linear functions as ideals of simplicity, then we may complete the analytic series of the proposition <math>f = pq : X \to \mathbb{B}</math> in the following way.
If we follow the classical line that singles out linear functions
as ideals of simplicity, then we may complete the analytic series
of the proposition f = pq : X -> B in the following way.
Figure 26-1 shows the differential proposition df = d[pq] : EX -> B
Figure 26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\operatorname{d}f.\!</math>
that we get by extracting the cell-wise linear approximation to the
difference map Df = D[pq] : EX -> B. This is the logical analogue
of what would ordinarily be called 'the' differential of pq, but
since I've been attaching the adjective "differential" to just
about everything in sight, the distinction tends to be lost.
For the time being, I'll resort to using the alternative
name "tangent map" for df.
{| align="center" cellspacing="10" style="text-align:center"
| |
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
|-
| o-------------------o o-------------------o |
| <math>\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|}
| | / | | \ | |
Figure 26-1. Differential or Tangent d[pq] : EX -> B
Just to be clear about what's being indicated here,
Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
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:
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{matrix}
\texttt{(}
\operatorname{d}p
\texttt{,}
\operatorname{d}q
\texttt{)}
& = &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
& + &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\\[4pt]
dp
& = &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
& + &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{(} \operatorname{d}q \texttt{)}
\\[4pt]
\operatorname{d}q
& = &
\texttt{~} \operatorname{d}p \texttt{~}
\texttt{~} \operatorname{d}q \texttt{~}
& + &
\texttt{(} \operatorname{d}p \texttt{)}
\texttt{~} \operatorname{d}q \texttt{~}
\end{matrix}</math>
|}
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.
{| align="center" cellspacing="10" style="text-align:center"
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
|-
| <math>\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|}
| o-------------------o o-------------------o |
| / P o Q \ |
Figure 26-2. Remainder r[pq] : EX -> B
Reading the arrows off the map produces the following data:
Reading the arrows off the map produces the following data:
{| align="center" cellspacing="10" style="text-align:center"
|
<math>\begin{array}{rcccccc}
\operatorname{r}(pq)
& = &
p & \cdot & q & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
& + &
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
& + &
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\\[4pt]
& + &
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot &
\operatorname{d}p ~ \operatorname{d}q
\end{array}</math>
|}
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.
==Further Reading==
A more detailed presentation of Differential Logic can be found here:
A more detailed presentation of Differential Logic can be found here:
:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems]]
==Document History==
==Document History==
