12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Sunday April 06, 2025
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 19:24, 15 September 2013
, 19:24, 15 September 2013restore original appendices ... with a slight permutation ...
Line 7,643:
Line 7,643:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
==Appendices==
==Appendices==
−===Appendix A===
+===Appendix 1. Propositional Forms and Differential Expansions===
+====Table A1. Propositional Forms on Two Variables====
<br>
<br>
Line 7,801:
Line 7,803:
\end{matrix}</math>
\end{matrix}</math>
|}
|}
+
+<br>
+
+====Table A2. Propositional Forms on Two Variables====
<br>
<br>
Line 8,017:
Line 8,023:
| <math>1\!</math>
| <math>1\!</math>
|}
|}
+
+<br>
+
+====Table A3. E''f'' Expanded Over Differential Features====
<br>
<br>
Line 8,269:
Line 8,279:
| 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>
|}
|}
+
+<br>
+
+====Table A4. D''f'' Expanded Over Differential Features====
<br>
<br>
Line 8,443:
Line 8,457:
| 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>
|}
|}
+
+<br>
+
+====Table A5. E''f'' Expanded Over Ordinary Features====
<br>
<br>
Line 8,689:
Line 8,707:
| 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>
|}
|}
+
+<br>
+
+====Table A6. D''f'' Expanded Over Ordinary Features====
<br>
<br>
Line 8,887:
Line 8,909:
<br>
<br>
−===Appendix B===
+====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>
+|+ 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%" |
| style="width:6%" |
Line 9,450:
Line 9,472:
<br>
<br>
−===Appendix C===
+===Appendix 2. Computational Details===
+====Operator Maps for the Logical Disjunction ''f''(u, v)====
+=====Computation of “εf”=====
+=====Computation of “Ef”=====
+=====Computation of “Df” (1)=====
+=====Computation of “Df” (2)=====
+=====Computation of “df”=====
+=====Computation of “rf”=====
+=====Computation Summary for Disjunction=====
+<br>
+{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
+|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
+|
+<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 “εg”======
+=====Computation of “Eg”=====
+=====Computation of “Dg” (1)=====
+=====Computation of “Dg” (2)=====
+=====Computation of “dg”=====
+=====Computation of “rg”=====
+=====Computation Summary for Equality=====
+<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==
