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

Revision as of 19:24, 15 September 2013

5,018 bytes added ,  19:24, 15 September 2013
restore original appendices ... with a slight permutation ...
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==Appendices==
 
==Appendices==
   −
===Appendix A===
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===Appendix 1. Propositional Forms and Differential Expansions===
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 +
====Table A1. Propositional Forms on Two Variables====
    
<br>
 
<br>
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\end{matrix}</math>
 
\end{matrix}</math>
 
|}
 
|}
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 +
<br>
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 +
====Table A2. Propositional Forms on Two Variables====
    
<br>
 
<br>
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| <math>1\!</math>
 
| <math>1\!</math>
 
|}
 
|}
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<br>
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====Table A3. E''f'' Expanded Over Differential Features====
    
<br>
 
<br>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>16\!</math>
 
| style="border-top:1px solid black; border-left:1px solid black"  | <math>16\!</math>
 
|}
 
|}
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<br>
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====Table A4. D''f'' Expanded Over Differential Features====
    
<br>
 
<br>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 
|}
 
|}
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<br>
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====Table A5. E''f'' Expanded Over Ordinary Features====
    
<br>
 
<br>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 
|}
 
|}
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<br>
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====Table A6. D''f'' Expanded Over Ordinary Features====
    
<br>
 
<br>
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<br>
 
<br>
   −
===Appendix B===
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====Table A12. Detail of Calculation for the Difference Map====
    
<br>
 
<br>
    
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"
 
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"
−
|+ style="height:30px" | <math>\text{Table B1.} ~~ \text{Detail of Calculation for}~ \mathrm{E}f + f = \mathrm{D}f\!</math>
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|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math>
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| style="width:6%" | &nbsp;
 
| style="width:6%" | &nbsp;
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<br>
 
<br>
   −
===Appendix C===
+
===Appendix 2. Computational Details===
 +
 
 +
====Operator Maps for the Logical Disjunction ''f''(u, v)====
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 +
=====Computation of &ldquo;&epsilon;f&rdquo;=====
 +
 
 +
=====Computation of &ldquo;Ef&rdquo;=====
 +
 
 +
=====Computation of &ldquo;Df&rdquo; (1)=====
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 +
=====Computation of &ldquo;Df&rdquo; (2)=====
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 +
=====Computation of &ldquo;df&rdquo;=====
 +
 
 +
=====Computation of &ldquo;rf&rdquo;=====
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 +
=====Computation Summary for Disjunction=====
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 +
<br>
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 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
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|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
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|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f
 +
& = & u \!\cdot\! v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 1
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\\[6pt]
 +
\mathrm{E}f
 +
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
\\[6pt]
 +
\mathrm{D}f
 +
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
\\[6pt]
 +
\mathrm{d}f
 +
& = & u \!\cdot\! v \cdot 0
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{r}f
 +
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
====Operator Maps for the Logical Equality ''g''(u, v)====
 +
 
 +
======Computation of &ldquo;&epsilon;g&rdquo;======
 +
 
 +
=====Computation of &ldquo;Eg&rdquo;=====
 +
 
 +
=====Computation of &ldquo;Dg&rdquo; (1)=====
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 +
=====Computation of &ldquo;Dg&rdquo; (2)=====
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 +
=====Computation of &ldquo;dg&rdquo;=====
 +
 
 +
=====Computation of &ldquo;rg&rdquo;=====
 +
 
 +
=====Computation Summary for Equality=====
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 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon g
 +
& = & u \!\cdot\! v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}g
 +
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
 +
\\[6pt]
 +
\mathrm{D}g
 +
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}g
 +
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{r}g
 +
& = & u \!\cdot\! v \cdot 0
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\end{array}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
===Appendix 3. Source Materials===
   −
===Appendix D===
+
===Appendix 4. Various Definitions of the Tangent Vector===
    
==References==
 
==References==
Jon Awbrey
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