Difference between revisions of "User:Jon Awbrey/GRAPHICS"
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& + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) | & + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) | ||
\\[4pt] | \\[4pt] | ||
| − | & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~\end{array}</math> | + | & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \end{array}</math> | ||
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& + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) | & + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) | ||
\\[4pt] | \\[4pt] | ||
| − | & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~\end{array}</math> | + | & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~ |
| + | \end{array}</math> | ||
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Revision as of 04:12, 17 June 2009
Differential Logic
ASCII Graphics
o-------------------------------------------------o | | | | | o-------------o o-------------o | | / \ / \ | | / o \ | | / /%\ \ | | / /%%%\ \ | | o o%%%%%o o | | | |%%%%%| | | | | P |%%%%%| Q | | | | |%%%%%| | | | o o%%%%%o o | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o | f = p q | o-------------------------------------------------o Figure 22-a. Conjunction pq : X -> B |
o-------------------------------------------------o | | | | | o-------------o o-------------o | | / \ / \ | | / P o Q \ | | / /%\ \ | | / /%%%\ \ | | o o.->-.o o | | | p(q)(dp)dq |%\%/%| (p)q dp(dq) | | | | o---------------|->o<-|---------------o | | | | |%%^%%| | | | o o%%|%%o o | | \ \%|%/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | | | | o | | (p)(q) dp dq | | | o-------------------------------------------------o | f = p q | o-------------------------------------------------o | | | Ef = p q (dp)(dq) | | | | + p (q) (dp) dq | | | | + (p) q dp (dq) | | | | + (p)(q) dp dq | | | o-------------------------------------------------o Figure 22-b. Enlargement E[pq] : EX -> B |
o-------------------------------------------------o | | | | | o-------------o o-------------o | | / \ / \ | | / P o Q \ | | / /%\ \ | | / /%%%\ \ | | o o%%%%%o o | | | (dp)dq |%%%%%| dp(dq) | | | | o<--------------|->o<-|-------------->o | | | | |%%^%%| | | | o o%%|%%o o | | \ \%|%/ / | | \ \|/ / | | \ o / | | \ /|\ / | | o-------------o | o-------------o | | | | | | | | v | | o | | dp dq | | | o-------------------------------------------------o | f = p q | o-------------------------------------------------o | | | Df = p q ((dp)(dq)) | | | | + p (q) (dp) dq | | | | + (p) q dp (dq) | | | | + (p)(q) dp dq | | | o-------------------------------------------------o Figure 22-c. Difference D[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | o o | | | | | | | | | | | | | | | G | | | | | | | | | | | | | | | o o | | \ / | | \ / | | \ T / | | \ o<------------/-------------o | | \ / | | \ / | | \ / | | o-------------------o | | | | | o---------------------------------------------------------------------o Figure 23. Elements of a Cybernetic System |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / /%\ \ | | / /%%%\ \ | | / /%%%%%\ \ | | / /%%%%%%%\ \ | | / /%%%%%%%%%\ \ | | o o%%%%%%%%%%%o o | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | | P |%%%%%%%%%%%| Q | | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | | |%%%%%%%%%%%| | | | o o%%%%%%%%%%%o o | | \ \%%%%%%%%%/ / | | \ \%%%%%%%/ / | | \ \%%%%%/ / | | \ \%%%/ / | | \ \%/ / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 24-1. Proposition pq : X -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (dp) (dq) o o | | | | o-->--o | | | | | | \ / | | | | | (dp) dq | \ / | dp (dq) | | | | o<-----------------o----------------->o | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 24-2. Tacit Extension !e![pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (dp) (dq) o o | | | | o-->--o | | | | | | \ / | | | | | (dp) dq | \ / | dp (dq) | | | | o----------------->o<-----------------o | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | | | | o | | | o---------------------------------------------------------------------o Figure 25-1. Enlargement E[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | (dp) dq | | dp (dq) | | | | o<---------------->o<---------------->o | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | dp | dq | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 25-2. Difference Map D[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / o \ \ | | / / ^ ^ \ \ | | o o / \ o o | | | | / \ | | | | | | / \ | | | | | |/ \| | | | | (dp)/ dq dp \(dq) | | | | /| |\ | | | | / | | \ | | | | / | | \ | | | o / o o \ o | | \ v \ dp dq / v / | | \ o<--------------------->o / | | \ \ / / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 26-1. Differential or Tangent d[pq] : EX -> B |
o---------------------------------------------------------------------o | | | X | | o-------------------o o-------------------o | | / \ / \ | | / P o Q \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | | dp dq | | | | | o<------------------------------->o | | | | | | | | | | | | | | | | | o | | | | o o ^ o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ dp | dq / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | | | | | | | v | | o | | | o---------------------------------------------------------------------o Figure 26-2. Remainder r[pq] : EX -> B |
JPEG Graphics
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| \(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\) |
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| \(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q) \\[4pt] & + & p & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] & + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~ \end{array}\) |
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| \(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & ((\operatorname{d}p)(\operatorname{d}q)) \\[4pt] & + & p & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~ \\[4pt] & + & (p) & \cdot & q & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~ \\[4pt] & + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~ \end{array}\) |
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| \(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\) |
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| \(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q) \\[4pt] & + & p & \cdot & q & \cdot & (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] & + & p & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] & + & p & \cdot & q & \cdot & ~\operatorname{d}p~~\operatorname{d}q~ \end{array}\) |
|
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| \(\text{Figure 25-1. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & (\operatorname{d}p)(\operatorname{d}q) \\[4pt] & + & p & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~ \\[4pt] & + & (p) & \cdot & q & \cdot & ~\operatorname{d}p~(\operatorname{d}q) \\[4pt] & + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~ \end{array}\) |
