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

Revision as of 20:48, 6 September 2013

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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
 
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|
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<math>\begin{matrix}
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<math>\begin{array}{lllll}
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x & = & f(u, v) & = & \texttt{((} u \texttt{)(} v \texttt{))}
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x
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& = & f(u, v)
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& = & \texttt{((} u \texttt{)(} v \texttt{))}
 
\\[8pt]
 
\\[8pt]
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y & = & g(u, v) & = & \texttt{((} u \texttt{,} v \texttt{))}
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y
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\end{matrix}</math>
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& = & g(u, v)
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& = & \texttt{((} u \texttt{,} v \texttt{))}
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\end{array}</math>
 
|}
 
|}
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
 
|
 
|
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<math>\begin{matrix}
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<math>\begin{array}{lllll}
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(x, y) & = & F(u, v) & = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)
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(x, y)
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\end{matrix}</math>
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& = & F(u, v)
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& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)
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\end{array}</math>
 
|}
 
|}
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<font face="courier new">
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
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{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
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{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
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<math>\begin{array}{lll}
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| width="8%"  | E''G''<sub>''i''</sub>
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\mathrm{E}G_i
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| width="4%"  | =
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& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)
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| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
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\end{array}</math>
 
|}
 
|}
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|}
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</font>
      
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
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{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
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|
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{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
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<math>\begin{array}{lllll}
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| width="8%"  | D''G''<sub>''i''</sub>
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\mathrm{D}G_i
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| width="4%"  | =
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& = & G_i (u, v)
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| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
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& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)
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| width="4%"  | +
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\\[8pt]
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| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›
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& = & G_i (u, v)
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|-
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& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)
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| width="8%"  | &nbsp;
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\end{array}</math>
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| width="4%"  | =
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| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›
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| width="4%"  | +
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| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›
   
|}
 
|}
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|}
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</font>
      
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
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{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
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{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
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<math>\begin{array}{lll}
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| width="8%"  | E''f''
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\mathrm{E}f
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| width="4%"  | =
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& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}
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| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))
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\\[8pt]
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|-
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\mathrm{E}g
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| width="8%"  | E''g''
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& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}
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| width="4%"  | =
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\end{array}</math>
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| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))
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|}
   
|}
 
|}
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</font>
      
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{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"
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{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
   
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{| align="left" border="0" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
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<math>\begin{array}{lllll}
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| width="8%"  | D''f''
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\mathrm{D}f
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| width="4%"  | =
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& = & \texttt{((} u \texttt{)(} v \texttt{))}
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| width="20%" | ((''u'')(''v''))
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& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}
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| width="4%"  | +
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\\[8pt]
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| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))
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\mathrm{D}g
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|-
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& = & \texttt{((} u \texttt{,~} v \texttt{))}
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| width="8%"  | D''g''
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& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}
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| width="4%"  | =
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\end{array}</math>
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| width="20%" | ((''u'', ''v''))
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| width="4%"  | +
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| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))
   
|}
 
|}
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|}
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</font>
      
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<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p>
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{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
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<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>
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| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]
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|-
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| 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>
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<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

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