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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 20:21, 19 September 2013
, 20:21, 19 September 2013update
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|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}\!</math>
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math>
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<math>\begin{array}{c*{8}{l}}
<math>\begin{array}{c*{8}{l}}
<br>
<br>
====Operator Maps for the Logical Disjunction ''f''(u, v)====
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====
=====Computation of ε''f''=====
=====Computation of ε''f''<sub>9</sub>=====
=====Computation of E''f''=====
=====Computation of E''f''<sub>9</sub>=====
=====Computation of D''f''=====
=====Computation of D''f''<sub>9</sub>=====
=====Computation of d''f''=====
=====Computation of d''f''<sub>9</sub>=====
=====Computation of r''f''=====
=====Computation of r''f''<sub>9</sub>=====
=====Computation Summary for Disjunction=====
=====Computation Summary for Equality=====
<br>
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
{| 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>
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
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<math>\begin{array}{c*{8}{l}}
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f
\boldsymbol\varepsilon f_{9}
& = & u \!\cdot\! v \cdot 1
& = & u \!\cdot\! v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\\[6pt]
\mathrm{E}f
\mathrm{E}f_{9}
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}
& = & 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{))}
& + & 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 \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{))}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
\\[6pt]
\\[6pt]
\mathrm{D}f
\mathrm{D}f_{9}
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \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{(} 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{))}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\\[6pt]
\mathrm{d}f
\mathrm{d}f_{9}
& = & u \!\cdot\! v \cdot 0
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \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{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
\\[6pt]
\\[6pt]
\mathrm{r}f
\mathrm{r}f_{9}
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
& = & u \!\cdot\! v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\end{array}</math>
\end{array}</math>
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<br>
<br>
====Operator Maps for the Logical Equality ''g''(u, v)====
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====
=====Computation of ε''g''=====
=====Computation of ε''f''<sub>14</sub>=====
=====Computation of E''g''=====
=====Computation of E''f''<sub>14</sub>=====
=====Computation of D''g''=====
=====Computation of D''f''<sub>14</sub>=====
=====Computation of d''g''=====
=====Computation of d''f''<sub>14</sub>=====
=====Computation of r''g''=====
=====Computation of r''f''<sub>14</sub>=====
=====Computation Summary for Equality=====
=====Computation Summary for Disjunction=====
<br>
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
{| 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>
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
|
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<math>\begin{array}{c*{8}{l}}
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon g
\boldsymbol\varepsilon f_{14}
& = & u \!\cdot\! v \cdot 1
& = & u \!\cdot\! v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot 1
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
\\[6pt]
\\[6pt]
\mathrm{E}g
\mathrm{E}f_{14}
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}
& = & 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{)}
& + & 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 \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{))}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\\[6pt]
\mathrm{D}g
\mathrm{D}f_{14}
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 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{)}
& + & \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{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
\\[6pt]
\\[6pt]
\mathrm{d}g
\mathrm{d}f_{14}
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& = & u \!\cdot\! v \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \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]
\\[6pt]
\mathrm{r}g
\mathrm{r}f_{14}
& = & u \!\cdot\! v \cdot 0
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)} v \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v
\end{array}</math>
\end{array}</math>
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