Changes

MyWikiBiz, Author Your Legacy — Thursday May 02, 2024
Jump to navigationJump to search

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

Revision as of 20:48, 6 September 2013

680 bytes removed ,  20:48, 6 September 2013
update
Line 6,276: Line 6,276:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
 
|
 
|
−
<math>\begin{matrix}
+
<math>\begin{array}{lllll}
−
x & = & f(u, v) & = & \texttt{((} u \texttt{)(} v \texttt{))}
+
x
 +
& = & f(u, v)
 +
& = & \texttt{((} u \texttt{)(} v \texttt{))}
 
\\[8pt]
 
\\[8pt]
−
y & = & g(u, v) & = & \texttt{((} u \texttt{,} v \texttt{))}
+
y
−
\end{matrix}</math>
+
& = & g(u, v)
 +
& = & \texttt{((} u \texttt{,} v \texttt{))}
 +
\end{array}</math>
 
|}
 
|}
   Line 6,291: Line 6,295:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
 
|
 
|
−
<math>\begin{matrix}
+
<math>\begin{array}{lllll}
−
(x, y) & = & F(u, v) & = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)
+
(x, y)
−
\end{matrix}</math>
+
& = & F(u, v)
 +
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)
 +
\end{array}</math>
 
|}
 
|}
   Line 6,771: Line 6,777:  
<br>
 
<br>
   −
<font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
+
<math>\begin{array}{lll}
−
| width="8%"  | E''G''<sub>''i''</sub>
+
\mathrm{E}G_i
−
| width="4%"  | =
+
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)
−
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
+
\end{array}</math>
 
|}
 
|}
−
|}
  −
</font>
      
<br>
 
<br>
Line 6,788: Line 6,791:  
<br>
 
<br>
   −
<font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
+
<math>\begin{array}{lllll}
−
| width="8%"  | D''G''<sub>''i''</sub>
+
\mathrm{D}G_i
−
| width="4%"  | =
+
& = & G_i (u, v)
−
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
+
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)
−
| width="4%"  | +
+
\\[8pt]
−
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
+
& = & G_i (u, v)
−
|-
+
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)
−
| width="8%"  | &nbsp;
+
\end{array}</math>
−
| width="4%"  | =
  −
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
  −
| width="4%"  | +
  −
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
   
|}
 
|}
−
|}
  −
</font>
      
<br>
 
<br>
Line 6,817: Line 6,813:  
<br>
 
<br>
   −
<font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
+
<math>\begin{array}{lll}
−
| width="8%"  | E''f''
+
\mathrm{E}f
−
| width="4%"  | =
+
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}
−
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
+
\\[8pt]
−
|-
+
\mathrm{E}g
−
| width="8%"  | E''g''
+
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}
−
| width="4%"  | =
+
\end{array}</math>
−
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
  −
|}
   
|}
 
|}
−
</font>
      
<br>
 
<br>
   −
<font face="courier new">
+
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
−
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
|
 
|
−
{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
+
<math>\begin{array}{lllll}
−
| width="8%"  | D''f''
+
\mathrm{D}f
−
| width="4%"  | =
+
& = & \texttt{((} u \texttt{)(} v \texttt{))}
−
| width="20%" | ((''u'')(''v''))
+
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}
−
| width="4%"  | +
+
\\[8pt]
−
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
+
\mathrm{D}g
−
|-
+
& = & \texttt{((} u \texttt{,~} v \texttt{))}
−
| width="8%"  | D''g''
+
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}
−
| width="4%"  | =
+
\end{array}</math>
−
| width="20%" | ((''u'', ''v''))
  −
| width="4%"  | +
  −
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
   
|}
 
|}
−
|}
  −
</font>
      
<br>
 
<br>
Line 7,441: Line 7,427:  
<br>
 
<br>
   −
<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p>
+
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
−
<p><center><font size="+1">'''Figure 70-a. Tangent Functor Diagram for F‹u,&nbsp;v› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›'''</font></center></p>
+
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]
 +
|-
 +
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math>
 +
|}
 +
 
 +
<br>
    
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math>
 
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/211850"

Navigation menu