Changes

MyWikiBiz, Author Your Legacy — Monday June 10, 2024
Jump to navigationJump to search

Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 03:05, 5 September 2013

882 bytes added ,  03:05, 5 September 2013
update
Line 2,389: Line 2,389:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
+
<math>\begin{matrix}
−
| <math>r(q)\!</math>
+
r(q)
−
| <math>=\!</math>
+
& = &
−
| <math>\sum_k d_k \cdot 2^{-k}\!</math>
+
\displaystyle\sum_k d_k \cdot 2^{-k}
−
| <math>=\!</math>
+
& = &
−
| <math>\sum_k \text{d}^k A(q) \cdot 2^{-k}\!</math>
+
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}
−
|-
+
\\[8pt]
−
| <math>=\!</math>
+
=
−
|-
+
\\[8pt]
−
| <math>\frac{s(q)}{t}\!</math>
+
\displaystyle\frac{s(q)}{t}
−
| <math>=\!</math>
+
& = &
−
| <math>\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}\!</math>
+
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}
−
| <math>=\!</math>
+
& = &
−
| <math>\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}\!</math>
+
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}
−
|}
+
\end{matrix}</math>
 
|}
 
|}
   Line 6,272: Line 6,272:  
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
   −
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''] to ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y''] that is defined by the following system of equations:
+
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:
   −
<br><font face="courier new">
+
<br>
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
 
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
 
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| &nbsp;
+
x & = & f(u, v) & = & \texttt{((} u \texttt{)(} v \texttt{))}
−
| ''x''
+
\\[8pt]
−
| =
+
y & = & g(u, v) & = & \texttt{((} u \texttt{,} v \texttt{))}
−
| ''f''‹''u'', ''v''›
+
\end{matrix}</math>
−
| =
  −
| ((''u'')(''v''))
  −
| &nbsp;
  −
|-
  −
| &nbsp;
  −
| ''y''
  −
| =
  −
| ''g''‹''u'', ''v''›
  −
| =
  −
| ((''u'', ''v''))
  −
| &nbsp;
   
|}
 
|}
−
|}
  −
</font><br>
     −
The component notation ''F''&nbsp;=&nbsp;‹''F''<sub>1</sub>,&nbsp;''F''<sub>2</sub>›&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows:
+
<br>
 +
 
 +
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:
 +
 
 +
<br>
   −
<br><font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
   
|
 
|
−
{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| &nbsp;
+
(x, y) & = & F(u, v) & = & ( \texttt{((} u \texttt{)(} v \texttt{))}, \texttt{((} u \texttt{,} v \texttt{))} )
−
| ‹''x'', ''y''›
+
\end{matrix}</math>
−
| =
  −
| ''F''‹''u'', ''v''›
  −
| =
  −
| ‹((''u'')(''v'')), ((''u'', ''v''))›
  −
| &nbsp;
   
|}
 
|}
−
|}
  −
</font><br>
     −
The information that defines the logical transformation ''F'' can be represented in the form of a truth table, as in Table&nbsp;60.  To cut down on subscripts in this example I continue to use plain letter equivalents for all components of spaces and maps.
+
<br>
 +
 
 +
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as in Table&nbsp;60.  To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.
 +
 
 +
<br>
   −
<font face="courier new">
+
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math>
−
|+ '''Table 60. Propositional Transformation'''
+
|- style="height:40px; background:ghostwhite; width:100%"
−
|- style="background:ghostwhite"
+
| style="width:25%" | <math>u\!</math>
−
| width="25%" | ''u''
+
| style="width:25%" | <math>v\!</math>
−
| width="25%" | ''v''
+
| style="width:25%; border-left:1px solid black" | <math>f\!</math>
−
| width="25%" | ''f''
+
| style="width:25%" | <math>g\!</math>
−
| width="25%" | ''g''
   
|-
 
|-
−
| width="25%" |
+
| style="border-top:1px solid black" |
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| 0
+
0
−
|-
+
\\[4pt]
−
| 0
+
0
−
|-
+
\\[4pt]
−
| 1
+
1
−
|-
+
\\[4pt]
−
| 1
+
1
−
|}
+
\end{matrix}</math>
−
| width="25%" |
+
| style="border-top:1px solid black" |
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| 0
+
0
−
|-
+
\\[4pt]
−
| 1
+
1
−
|-
+
\\[4pt]
−
| 0
+
0
−
|-
+
\\[4pt]
−
| 1
+
1
−
|}
+
\end{matrix}</math>
−
| width="25%" |
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| 0
+
0
−
|-
+
\\[4pt]
−
| 1
+
1
−
|-
+
\\[4pt]
−
| 1
+
1
−
|-
+
\\[4pt]
−
| 1
+
1
−
|}
+
\end{matrix}</math>
−
| width="25%" |
+
| style="border-top:1px solid black" |
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
<math>\begin{matrix}
−
| 1
+
1
−
|-
+
\\[4pt]
−
| 0
+
0
 +
\\[4pt]
 +
0
 +
\\[4pt]
 +
1
 +
\end{matrix}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| style="border-top:1px solid black" | <math>u\!</math>
 +
| style="border-top:1px solid black" | <math>v\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math>
 +
| style="border-top:1px solid black" |
 +
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
Figure&nbsp;61 shows how we might paint a picture of the logical transformation <math>F\!</math> on the canvass that was earlier primed for this purpose (way back in Figure&nbsp;30).
 +
 
 +
<br>
 +
 
 +
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
 +
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]
 
|-
 
|-
−
| 0
+
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.
 +
 
 +
<br>
 +
 
 +
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
 +
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]
 
|-
 
|-
−
| 1
+
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math>
 
|}
 
|}
 +
 +
<br>
 +
 +
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math>  The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.
 +
 +
<br>
 +
 +
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
 +
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]
 
|-
 
|-
−
| width="25%" | &nbsp;
+
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math>
−
| width="25%" | &nbsp;
  −
| width="25%" | ((''u'')(''v''))
  −
| width="25%" | ((''u'', ''v''))
   
|}
 
|}
−
</font><br>
  −
  −
Figure&nbsp;61 shows how one might paint a picture of the logical transformation ''F'' on the canvass that was earlier primed for this purpose (way back in Figure&nbsp;30).
      
<br>
 
<br>
−
<p>[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]</p>
  −
<p><center><font size="+1">'''Figure 61.  Propositional Transformation'''</font></center></p>
     −
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exemplifying a style of diagram that is adequate for most purposes.
+
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.
    
<br>
 
<br>
−
<p>[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]</p>
  −
<p><center><font size="+1">'''Figure 62.  Propositional Transformation (Short Form)'''</font></center></p>
  −
  −
Figure&nbsp;63 give a more complete picture of the transformation ''F'', showing how the points of ''U''<sup>&nbsp;&bull;</sup> are transformed into points of ''X''<sup>&nbsp;&bull;</sup>.  The lines that cross from one universe to the other trace the action that ''F'' induces on points, in other words, they depict the aspect of the transformation that acts as a mapping from points to points, and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.
  −
  −
<br>
  −
<p>[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]</p>
  −
<p><center><font size="+1">'''Figure 63.  Transformation of Positions'''</font></center></p>
  −
  −
Table&nbsp;64 shows how the action of the transformation ''F'' on cells or points is computed in terms of coordinates.
     −
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
−
|+ '''Table 64. Transformation of Positions'''
+
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math>
−
|- style="background:ghostwhite"
+
|- style="height:40px; background:ghostwhite; width:100%"
−
| ''u''&nbsp;&nbsp;''v''
+
| style="width:10%" | <math>u\!</math>
−
| ''x''
+
| style="width:10%" | <math>v\!</math>
−
| ''y''
+
| style="width:12%; border-left:1px solid black" | <math>x\!</math>
−
| ''x''&nbsp;''y''
+
| style="width:12%" | <math>y\!</math>
−
| ''x''&nbsp;(''y'')
+
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math>
−
| (''x'')&nbsp;''y''
+
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math>
−
| (''x'')(''y'')
+
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math>
−
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y''&nbsp;]
+
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
−
|-
+
| style="width:16%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math>
−
| width="12%" |
+
|-
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black" |
−
| 0&nbsp;&nbsp;0
+
<math>\begin{matrix}
−
|-
+
0
−
| 0&nbsp;&nbsp;1
+
\\[4pt]
−
|-
+
0
−
| 1&nbsp;&nbsp;0
+
\\[4pt]
−
|-
+
1
−
| 1&nbsp;&nbsp;1
+
\\[4pt]
−
|}
+
1
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black" |
−
| 0
+
<math>\begin{matrix}
−
|-
+
0
−
| 1
+
\\[4pt]
−
|-
+
1
−
| 1
+
\\[4pt]
−
|-
+
0
−
| 1
+
\\[4pt]
−
|}
+
1
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
| 1
+
<math>\begin{matrix}
−
|-
+
0
−
| 0
+
\\[4pt]
−
|-
+
1
−
| 0
+
\\[4pt]
−
|-
+
1
−
| 1
+
\\[4pt]
−
|}
+
1
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black" |
−
| 0
+
<math>\begin{matrix}
−
|-
+
1
−
| 0
+
\\[4pt]
−
|-
+
0
−
| 0
+
\\[4pt]
−
|-
+
0
−
| 1
+
\\[4pt]
−
|}
+
1
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
| 0
+
<math>\begin{matrix}
−
|-
+
0
−
| 1
+
\\[4pt]
−
|-
+
0
−
| 1
+
\\[4pt]
−
|-
+
0
−
| 0
+
\\[4pt]
−
|}
+
1
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black" |
−
| 1
+
<math>\begin{matrix}
−
|-
+
0
−
| 0
+
\\[4pt]
−
|-
+
1
−
| 0
+
\\[4pt]
−
|-
+
1
−
| 0
+
\\[4pt]
−
|}
+
0
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black" |
−
| 0
+
<math>\begin{matrix}
−
|-
+
1
−
| 0
+
\\[4pt]
−
|-
+
0
−
| 0
+
\\[4pt]
−
|-
+
0
−
| 0
+
\\[4pt]
−
|}
+
0
−
| width="12%" |
+
\end{matrix}</math>
−
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
+
| style="border-top:1px solid black" |
−
| &uarr;
+
<math>\begin{matrix}
−
|-
+
0
−
| ''F''
+
\\[4pt]
−
|-
+
0
−
| ‹''f'',&nbsp;''g''&nbsp;›
+
\\[4pt]
−
|-
+
0
−
| &uarr;
+
\\[4pt]
−
|}
+
0
−
|-
+
\end{matrix}</math>
−
| &nbsp;
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
| ((''u'')(''v''))
+
<math>\begin{matrix}
−
| ((''u'',&nbsp;''v''))
+
\uparrow
−
| ''u''&nbsp;''v''
+
\\[4pt]
−
| (''u'',&nbsp;''v'')
+
F =
−
| (''u'')(''v'')
+
\\[4pt]
−
| (&nbsp;)
+
(f, g)
−
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''&nbsp;]
+
\\[4pt]
−
|}
+
\uparrow
 +
\end{matrix}</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| style="border-top:1px solid black" | <math>u\!</math>
 +
| style="border-top:1px solid black" | <math>v\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math>
 +
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math>
 +
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math>
 +
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math>
 +
| style="border-top:1px solid black" | <math>0\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math>
 +
|}
 +
 
 
<br>
 
<br>
  
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/211138"

Navigation menu