Changes

MyWikiBiz, Author Your Legacy — Sunday April 28, 2024
Jump to navigationJump to search

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

Revision as of 18:36, 13 January 2021

144,514 bytes added ,  18:36, 13 January 2021
check
Line 5,609: Line 5,609:  
| align="center" | <math>\text{Map}\!</math>
 
| align="center" | <math>\text{Map}\!</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,627: Line 5,627:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,645: Line 5,645:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,663: Line 5,663:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,681: Line 5,681:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,699: Line 5,699:  
\end{array}\!</math>
 
\end{array}\!</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,717: Line 5,717:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,730: Line 5,730:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,743: Line 5,743:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 5,756: Line 5,756:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,074: Line 6,074:  
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
 
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
 
|-
 
|-
−
| align="center" | <math>\underline\text{Operand}\!</math>
+
| align="center" | <math>\underline{\text{Operand}}\!</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,091: Line 6,091:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,109: Line 6,109:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,127: Line 6,127:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,145: Line 6,145:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,163: Line 6,163:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,181: Line 6,181:  
\end{array}\!</math>
 
\end{array}\!</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,199: Line 6,199:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,212: Line 6,212:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,225: Line 6,225:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 6,238: Line 6,238:  
\end{array}</math>
 
\end{array}</math>
 
|-
 
|-
−
| align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math>
+
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math>
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 7,639: Line 7,639:  
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]
 
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]
 
|}
 
|}
 +
 +
==Appendices==
 +
 +
===Appendix 1. Propositional Forms and Differential Expansions===
 +
 +
====Table A1. Propositional Forms on Two Variables====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
 +
|- style="background:ghostwhite"
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
 +
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>x\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp; || &nbsp; || &nbsp;
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>y\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp; || &nbsp; || &nbsp;
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(~)}
 +
\\
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)~ ~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~ ~(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{,~} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{~~} y \texttt{)}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{false}
 +
\\
 +
\text{neither}~ x ~\text{nor}~ y
 +
\\
 +
y ~\text{without}~ x
 +
\\
 +
\text{not}~ x
 +
\\
 +
x ~\text{without}~ y
 +
\\
 +
\text{not}~ y
 +
\\
 +
x ~\text{not equal to}~ y
 +
\\
 +
\text{not both}~ x ~\text{and}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0
 +
\\
 +
\lnot x \land \lnot y
 +
\\
 +
\lnot x \land y
 +
\\
 +
\lnot x
 +
\\
 +
x \land \lnot y
 +
\\
 +
\lnot y
 +
\\
 +
x \ne y
 +
\\
 +
\lnot x \lor \lnot y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\\
 +
\texttt{~ ~ ~ ~} y \texttt{~~}
 +
\\
 +
\texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{~~} x \texttt{~ ~ ~ ~}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\\
 +
\texttt{((~))}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x ~\text{and}~ y
 +
\\
 +
x ~\text{equal to}~ y
 +
\\
 +
y
 +
\\
 +
\text{not}~ x ~\text{without}~ y
 +
\\
 +
x
 +
\\
 +
\text{not}~ y ~\text{without}~ x
 +
\\
 +
x ~\text{or}~ y
 +
\\
 +
\text{true}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x \land y
 +
\\
 +
x = y
 +
\\
 +
y
 +
\\
 +
x \Rightarrow y
 +
\\
 +
x
 +
\\
 +
x \Leftarrow y
 +
\\
 +
x \lor y
 +
\\
 +
1
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A2. Propositional Forms on Two Variables====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math>
 +
|- style="background:ghostwhite"
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
 +
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
 +
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>x\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp; || &nbsp; || &nbsp;
 +
|- style="background:ghostwhite"
 +
| &nbsp;
 +
| align="right" | <math>y\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp; || &nbsp; || &nbsp;
 +
|-
 +
| <math>f_{0}\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0</math>
 +
| <math>\texttt{(~)}\!</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{1}\\f_{2}\\f_{4}\\f_{8}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ x ~\text{nor}~ y
 +
\\
 +
y ~\text{without}~ x
 +
\\
 +
x ~\text{without}~ y
 +
\\
 +
x ~\text{and}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot x \land \lnot y
 +
\\
 +
\lnot x \land y
 +
\\
 +
x \land \lnot y
 +
\\
 +
x \land y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{3}\\f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}\\f_{1100}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1\\1~1~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ x
 +
\\
 +
x
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot x
 +
\\
 +
x
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{6}\\f_{9}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0110}\\f_{1001}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0\\1~0~0~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x ~\text{not equal to}~ y
 +
\\
 +
x ~\text{equal to}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
x \ne y
 +
\\
 +
x = y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{5}\\f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}\\f_{1010}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1\\1~0~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ y
 +
\\
 +
y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot y
 +
\\
 +
y
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_{7}\\f_{11}\\f_{13}\\f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\
 +
\texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ x ~\text{and}~ y
 +
\\
 +
\text{not}~ x ~\text{without}~ y
 +
\\
 +
\text{not}~ y ~\text{without}~ x
 +
\\
 +
x ~\text{or}~ y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot x \lor \lnot y
 +
\\
 +
x \Rightarrow y
 +
\\
 +
x \Leftarrow y
 +
\\
 +
x \lor y
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A3. E''f'' Expanded Over Differential Features====
 +
 +
<br>
 +
 +
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
 +
|- style="background:ghostwhite"
 +
| style="width:10%; border-bottom:1px solid black" | &nbsp;
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{0}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{1}\\f_{2}\\f_{4}\\f_{8}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{3}\\f_{12}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{6}\\f_{9}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\\
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\\
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{5}\\f_{10}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{7}\\f_{11}\\f_{13}\\f_{14}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\\
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{15}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 +
|- style="background:ghostwhite"
 +
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>4\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>4\!</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>
 +
 +
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
 +
|- style="background:ghostwhite"
 +
| style="width:10%; border-bottom:1px solid black" | &nbsp;
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
 +
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{0}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{1}\\f_{2}\\f_{4}\\f_{8}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\\
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
y
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\\
 +
y
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\\
 +
x
 +
\\
 +
x
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
x
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}1\\1\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}1\\1\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}0\\0\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}1\\1\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}1\\1\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}1\\1\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}1\\1\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\\
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
y
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\\
 +
y
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
x
 +
\\
 +
x
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{15}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A5. E''f'' Expanded Over Ordinary Features====
 +
 +
<br>
 +
 +
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
 +
|- style="background:ghostwhite"
 +
| style="width:10%; border-bottom:1px solid black" | &nbsp;
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
 +
<math>\mathrm{E}f|_{xy}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{0}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{1}\\f_{2}\\f_{4}\\f_{8}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{3}\\f_{12}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{6}\\f_{9}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{5}\\f_{10}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{(} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}y \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" |
 +
<math>\begin{matrix}
 +
f_{7}\\f_{11}\\f_{13}\\f_{14}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
 +
\\
 +
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
 +
\\
 +
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}
 +
\\
 +
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{15}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A6. D''f'' Expanded Over Ordinary Features====
 +
 +
<br>
 +
 +
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
 +
|- style="background:ghostwhite"
 +
| style="width:10%; border-bottom:1px solid black" | &nbsp;
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
 +
<math>\mathrm{D}f|_{xy}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{0}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 +
\\
 +
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{15}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
 +
| 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>
 +
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
===Appendix 2. Differential Forms===
 +
 +
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.
 +
 +
Table A7 expands the differential forms that result over a ''logical basis'':
 +
 +
{| align="center" cellpadding="6" style="text-align:center"
 +
|
 +
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
 +
|}
 +
 +
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse.  Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis.  In this setting it is frequently convenient to use the following abbreviations:
 +
 +
{| align="center" cellpadding="6" style="text-align:center"
 +
|
 +
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math>
 +
|}
 +
 +
Table A8 expands the differential forms that result over an ''algebraic basis'':
 +
 +
{| align="center" cellpadding="6" style="text-align:center"
 +
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math>
 +
|}
 +
 +
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse.  Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.
 +
 +
====Table A7. Differential Forms Expanded on a Logical Basis====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| &nbsp;
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
 +
| <math>\mathrm{d}f~\!</math>
 +
|-
 +
| <math>f_{0}\!</math>
 +
| style="border-right:none" | <math>\texttt{(~)}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
& + &
 +
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
& + &
 +
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
& + &
 +
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
 +
& + &
 +
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)} ~\partial x
 +
& + &
 +
\texttt{(} x \texttt{)} ~\partial y
 +
\\
 +
\texttt{~} y \texttt{~} ~\partial x
 +
& + &
 +
\texttt{(} x \texttt{)} ~\partial y
 +
\\
 +
\texttt{(} y \texttt{)} ~\partial x
 +
& + &
 +
\texttt{~} x \texttt{~} ~\partial y
 +
\\
 +
\texttt{~} y \texttt{~} ~\partial x
 +
& + &
 +
\texttt{~} x \texttt{~} ~\partial y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\partial x
 +
\\
 +
\partial x
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
\\
 +
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\partial x & + & \partial y
 +
\\
 +
\partial x & + & \partial y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\partial y
 +
\\
 +
\partial y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
& + &
 +
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y
 +
& + &
 +
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
& + &
 +
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}
 +
& + &
 +
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y
 +
& + &
 +
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~} ~\partial x
 +
& + &
 +
\texttt{~} x \texttt{~} ~\partial y
 +
\\
 +
\texttt{(} y \texttt{)} ~\partial x
 +
& + &
 +
\texttt{~} x \texttt{~} ~\partial y
 +
\\
 +
\texttt{~} y \texttt{~} ~\partial x
 +
& + &
 +
\texttt{(} x \texttt{)} ~\partial y
 +
\\
 +
\texttt{(} y \texttt{)} ~\partial x
 +
& + &
 +
\texttt{(} x \texttt{)} ~\partial y
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| style="border-right:none" | <math>\texttt{((~))}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A8. Differential Forms Expanded on an Algebraic Basis====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| &nbsp;
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
 +
| <math>\mathrm{d}f~\!</math>
 +
|-
 +
| <math>f_{0}\!</math>
 +
| style="border-right:none" | <math>\texttt{(~)}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}x
 +
\\
 +
\mathrm{d}x
 +
\end{matrix}\!</math>
 +
| <math>\begin{matrix}
 +
\mathrm{d}x
 +
\\
 +
\mathrm{d}x
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\\
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\\
 +
\mathrm{d}x & + & \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}y
 +
\\
 +
\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y
 +
\\
 +
\mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-right:none" |
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| style="border-right:none" | <math>\texttt{((~))}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A9. Tangent Proposition as Pointwise Linear Approximation====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{d}f =
 +
\\[2pt]
 +
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}^2\!f =
 +
\\[2pt]
 +
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\mathrm{d}f|_{x \, y}</math>
 +
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
 +
|-
 +
| style="border-right:none" | <math>f_0\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math>
 +
|-
 +
| style="border-right:none" |
 +
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\end{matrix}\!</math>
 +
| <math>\begin{matrix}
 +
\mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>f_{15}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" |
 +
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math>
 +
|- style="background:ghostwhite; height:40px"
 +
| style="border-right:none" | <math>f\!</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\mathrm{D}f
 +
\\
 +
= & \mathrm{d}f & + & \mathrm{d}^2\!f
 +
\\
 +
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\mathrm{d}f|_{x \, y}</math>
 +
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math>
 +
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math>
 +
|-
 +
| style="border-right:none" | <math>f_0\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-left:4px double black" |
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &
 +
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &
 +
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y
 +
\end{matrix}</math>
 +
|-
 +
| style="border-right:none" | <math>f_{15}\!</math>
 +
| style="border-left:4px double black" | <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A11. Partial Differentials and Relative Differentials====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
 +
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math>
 +
|- style="background:ghostwhite; height:50px"
 +
| &nbsp;
 +
| <math>f\!</math>
 +
| <math>\frac{\partial f}{\partial x}\!</math>
 +
| <math>\frac{\partial f}{\partial y}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}f =
 +
\\[2pt]
 +
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math>
 +
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>\texttt{(~)}\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|-
 +
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~(} y \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~~} y \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}0\\0\end{matrix}</math>
 +
| <math>\begin{matrix}1\\1\end{matrix}</math>
 +
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(~} x \texttt{~~} y \texttt{~)}
 +
\\
 +
\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
 +
\texttt{((} x \texttt{)~} y \texttt{~)}
 +
\\
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\\
 +
\texttt{~} y \texttt{~}
 +
\\
 +
\texttt{(} y \texttt{)}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{~} x \texttt{~}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\\
 +
\texttt{(} x \texttt{)}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y
 +
\\
 +
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\\
 +
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y
 +
\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
| <math>0\!</math>
 +
|}
 +
 +
<br>
 +
 +
====Table A12. Detail of Calculation for the Difference Map====
 +
 +
<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%"
 +
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math>
 +
|- style="background:ghostwhite"
 +
| style="width:6%" | &nbsp;
 +
| style="width:14%; border-left:1px solid black"  | <math>f\!</math>
 +
| style="width:20%; border-left:4px double black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}
 +
\\[4pt]
 +
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}
 +
\end{array}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
 +
\\[4pt]
 +
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}
 +
\end{array}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}
 +
\end{array}</math>
 +
| style="width:20%; border-left:1px solid black" |
 +
<math>\begin{array}{cr}
 +
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
 +
\\[4pt]
 +
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}
 +
\end{array}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{0}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>0 ~+~ 0 ~=~ 0\!</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{1}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{2}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{4}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{8}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{3}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{(} x \texttt{)}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{12}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>x\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} x \texttt{)}
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & x
 +
\\[4pt]
 +
+ & x
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{6}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{9}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{5}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{(} y \texttt{)}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{10}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>y\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{(} y \texttt{)}
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 1
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & y
 +
\\[4pt]
 +
+ & y
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{7}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:4px double black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{11}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{13}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:1px solid black" | <math>f_{14}\!</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}
 +
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}
 +
~ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
+ & \texttt{((} x \texttt{)(} y \texttt{))}
 +
\\[4pt]
 +
= & 0
 +
\end{matrix}</math>
 +
|-
 +
| style="border-top:4px double black" | <math>f_{15}\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1\!</math>
 +
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
| style="border-top:4px double black; border-left:1px solid black"  | <math>1 ~+~ 1 ~=~ 0\!</math>
 +
|}
 +
 +
<br>
 +
 +
===Appendix 3. Computational Details===
 +
 +
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{8}
 +
& = && f_{8}(u, v)
 +
\\[4pt]
 +
& = && uv
 +
\\[4pt]
 +
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
 +
& + &  uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + &  uv \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{8}
 +
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{9}{l}}
 +
\mathrm{E}f_{8}
 +
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)
 +
\\[4pt]
 +
& = & \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{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\\[20pt]
 +
\mathrm{E}f_{8}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
 +
\\[4pt]
 +
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{*{9}{c}}
 +
\mathrm{E}f_{8}
 +
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)
 +
\\[6pt]
 +
& = & u \cdot v
 +
& + & u \cdot \mathrm{d}v
 +
& + & v \cdot \mathrm{d}u
 +
& + & \mathrm{d}u \cdot \mathrm{d}v
 +
\\[6pt]
 +
\mathrm{E}f_{8}
 +
& = & \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{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{8}
 +
& = && \mathrm{E}f_{8}
 +
& + &  \boldsymbol\varepsilon f_{8}
 +
\\[4pt]
 +
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{8}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
& + &  uv
 +
\\[20pt]
 +
\mathrm{D}f_{8}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{~}
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}
 +
\\[20pt]
 +
\mathrm{D}f_{8}
 +
& = && \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{~}
 +
& + &  \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{~}
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{*{9}{l}}
 +
\mathrm{D}f_{8}
 +
& = & \boldsymbol\varepsilon f_{8}
 +
& + & \mathrm{E}f_{8}
 +
\\[6pt]
 +
& = & f_{8}(u, v)
 +
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[6pt]
 +
& = & uv
 +
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
& = & 0
 +
& + & u \cdot \mathrm{d}v
 +
& + & v \cdot \mathrm{d}u
 +
& + & \mathrm{d}u ~ \mathrm{d}v
 +
\\[6pt]
 +
\mathrm{D}f_{8}
 +
& = & 0
 +
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
 +
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math>
 +
|
 +
<math>\begin{array}{c*{9}{l}}
 +
\mathrm{D}f_{8}
 +
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{8}
 +
& = &  u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
 +
& + &  u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v
 +
\\[6pt]
 +
\mathrm{E}f_{8}
 +
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & u ~ \texttt{(} v \texttt{)}  \cdot  \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v
 +
\\[20pt]
 +
\mathrm{D}f_{8}
 +
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}\!</math>
 +
|}
 +
 +
=====Computation of d''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{8}
 +
& = & uv \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 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\\[6pt]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{8}
 +
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>8</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}
 +
\\[20pt]
 +
\mathrm{D}f_{8}
 +
& = & uv \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 \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\\[6pt]
 +
\mathrm{d}f_{8}
 +
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\\[20pt]
 +
\mathrm{r}f_{8}
 +
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Conjunction=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f_{8}
 +
& = & uv \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\\[6pt]
 +
\mathrm{E}f_{8}
 +
& = & uv \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 \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\\[6pt]
 +
\mathrm{D}f_{8}
 +
& = & uv \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 \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\\[6pt]
 +
\mathrm{d}f_{8}
 +
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\\[6pt]
 +
\mathrm{r}f_{8}
 +
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{9}
 +
& = && f_{9}(u, v)
 +
\\[4pt]
 +
& = && \texttt{((} u \texttt{,~} v \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{9}(1, 1)
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)
 +
\\[4pt]
 +
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{9}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{9}
 +
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
 +
\\[4pt]
 +
& = && \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{) }
 +
& + &  \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{))}
 +
\\[20pt]
 +
\mathrm{E}f_{9}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \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{~}
 +
& + &  0
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \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{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{9}
 +
& = && \mathrm{E}f_{9}
 +
& + &  \boldsymbol\varepsilon f_{9}
 +
\\[4pt]
 +
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{9}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}
 +
& + &  \texttt{((} u \texttt{,} v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{9}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{~}
 +
& + &  \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{~}
 +
\\[4pt]
 +
&& + & \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{)}
 +
& + &  \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{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[20pt]
 +
\mathrm{D}f_{9}
 +
& = && \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{)}
 +
& + &  \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{)}
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{*{9}{l}}
 +
\mathrm{D}f_{9}
 +
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
 +
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{9}
 +
& = & \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{)}
 +
& + & \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]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{9}
 +
& = & \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{)}
 +
& + & \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{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>9</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}
 +
\\[20pt]
 +
\mathrm{D}f_{9}
 +
& = & \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{)}
 +
& + & \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]
 +
\mathrm{d}f_{9}
 +
& = & \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{)}
 +
& + & \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{)}
 +
\\[20pt]
 +
\mathrm{r}f_{9}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====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 F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f_{9}
 +
& = & uv \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{9}
 +
& = & uv \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}f_{9}
 +
& = & uv \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}f_{9}
 +
& = & uv \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}f_{9}
 +
& = & uv \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>
 +
 +
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{11}
 +
& = && f_{11}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(} u \texttt{(} v \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{11}(1, 1)
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ }
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ }
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{11}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \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{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  \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{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \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{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  \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{~}
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{11}
 +
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{(}
 +
\\
 +
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\\
 +
&&& \texttt{(}
 +
\\
 +
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\
 +
&&& \texttt{))}
 +
\\[4pt]
 +
& = &&
 +
u 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{))}
 +
\\[4pt]
 +
& = &&
 +
u 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 ~ \mathrm{d}v \texttt{)}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{E}f_{11}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \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{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \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{~}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \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{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{11}
 +
& = && \mathrm{E}f_{11}
 +
& + &  \boldsymbol\varepsilon f_{11}
 +
\\[4pt]
 +
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{11}(u, v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{(} v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
 +
& + &  \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  0
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &  \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{c*{9}{l}}
 +
\mathrm{D}f_{11}
 +
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{11}
 +
& = & u v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{11}
 +
& = &
 +
u 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 ~ \mathrm{d}v \texttt{)}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\cdot
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = &
 +
u 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 ~ \mathrm{d}v \texttt{~}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\cdot
 +
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\\[20pt]
 +
\mathrm{r}f_{11}
 +
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Implication=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f_{11}
 +
& = & u v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{11}
 +
& = & u 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 ~ \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[6pt]
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\\[6pt]
 +
\mathrm{r}f_{11}
 +
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{14}
 +
& = && f_{14}(u, v)
 +
\\[4pt]
 +
& = && \texttt{((} u \texttt{)(} v \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{14}(1, 1)
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ }
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ }
 +
& + &  0
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{14}
 +
& = && \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{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{~}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{~}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{14}
 +
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{((}
 +
\\
 +
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\\
 +
&&& \texttt{)(}
 +
\\
 +
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\
 +
&&& \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })
 +
\\[4pt]
 +
& = && \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{))}
 +
& + &  \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{))}
 +
\\[20pt]
 +
\mathrm{E}f_{14}
 +
& = && \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{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \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{~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \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{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \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{~}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{14}
 +
& = && \mathrm{E}f_{14}
 +
& + &  \boldsymbol\varepsilon f_{14}
 +
\\[4pt]
 +
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{14}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}
 +
& + &  \texttt{((} u \texttt{)(} v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{14}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & 0
 +
& + &  0
 +
& + &  \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{~~}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
 +
\\[4pt]
 +
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  0
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
 +
\\[20pt]
 +
\mathrm{D}f_{14}
 +
& = && uv \!\cdot\! \mathrm{d}u ~ \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{))}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{*{9}{l}}
 +
\mathrm{D}f_{14}
 +
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{14}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \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{))}
 +
\\[6pt]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{14}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>14</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}
 +
\\[20pt]
 +
\mathrm{D}f_{14}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
 +
& + & \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{))}
 +
\\[6pt]
 +
\mathrm{d}f_{14}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot 0
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
\\[20pt]
 +
\mathrm{r}f_{14}
 +
& = & \texttt{ } u \texttt{  } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====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 F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f_{14}
 +
& = & uv \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_{14}
 +
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \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_{14}
 +
& = & uv \cdot \mathrm{d}u ~ \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_{14}
 +
& = & uv \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_{14}
 +
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
===Appendix 4. Source Materials===
 +
 +
===Appendix 5. Various Definitions of the Tangent Vector===
 +
 +
==References==
 +
 +
===Works Cited===
 +
 +
{| cellpadding=3
 +
| valign=top | [AuM]
 +
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963.  Reprinted, Dover, New York, NY, 1977.
 +
|-
 +
| valign=top | [BiG]
 +
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968.  Reprinted, Dover, New York, NY, 1980.
 +
|-
 +
| valign=top | [Boo]
 +
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854.  Reprinted, Dover, New York, NY, 1958.
 +
|-
 +
| valign=top | [BoT]
 +
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.
 +
|-
 +
| valign=top | [dCa]
 +
| do Carmo, M.P., ''Riemannian Geometry''.  Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988.  Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.
 +
|-
 +
| valign=top | [Che46]
 +
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.
 +
|-
 +
| valign=top | [Che56]
 +
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.
 +
|-
 +
| valign=top | [Cho86]
 +
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.
 +
|-
 +
| valign=top | [Cho93]
 +
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.
 +
|-
 +
| valign=top | [DoM]
 +
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.
 +
|-
 +
| valign=top | [Fuji]
 +
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.
 +
|-
 +
| valign=top | [Hic]
 +
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.
 +
|-
 +
| valign=top | [Hir]
 +
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.
 +
|-
 +
| valign=top | [How]
 +
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969.  Reprinted in [SeH, 479-490].
 +
|-
 +
| valign=top | [JGH]
 +
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.
 +
|-
 +
| valign=top | [KoA]
 +
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.
 +
|-
 +
| valign=top | [Koh]
 +
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.
 +
|-
 +
| valign=top | [LaS]
 +
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.
 +
|-
 +
| valign=top | [La83]
 +
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.
 +
|-
 +
| valign=top | [La84]
 +
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.
 +
|-
 +
| valign=top | [La85]
 +
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.
 +
|-
 +
| valign=top | [La93]
 +
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.
 +
|-
 +
| valign=top | [Lie80]
 +
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975.  Original paper 1880.
 +
|-
 +
| valign=top | [Lie84]
 +
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976.  Original paper 1884.
 +
|-
 +
| valign=top | [LoS]
 +
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.
 +
|-
 +
| valign=top | [Mel]
 +
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.
 +
|-
 +
| valign=top | [Men]
 +
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297].  Originally published 1842.
 +
|-
 +
| valign=top | [M&M]
 +
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.
 +
|-
 +
| valign=top | [P1]
 +
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8,  C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960.  Cited as CP [volume].[paragraph].
 +
|-
 +
| valign=top | [P2]
 +
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976.  Cited as NE [volume], [page].
 +
|-
 +
| valign=top | [Rob]
 +
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.
 +
|-
 +
| valign=top | [SeH]
 +
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.
 +
|-
 +
| valign=top | [SpB]
 +
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.
 +
|-
 +
| valign=top | [Sp65]
 +
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.
 +
|-
 +
| valign=top | [Sp79]
 +
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2.  1st edition 1970.  2nd edition, Publish or Perish Inc., Houston, TX, 1979.
 +
|-
 +
| valign=top | [Sty]
 +
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964.  MIT Press, Cambridge, MA, 1969.
 +
|-
 +
| valign=top | [Wie]
 +
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948.  2nd edition, MIT Press, Cambridge, MA, 1961.
 +
|}
 +
 +
===Works Consulted===
 +
 +
{| cellpadding=3
 +
| valign=top | [Ami]
 +
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.
 +
|-
 +
| valign=top | [Ed87]
 +
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.
 +
|-
 +
| valign=top | [Ed88]
 +
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.
 +
|-
 +
| valign=top | [Fla]
 +
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963.  Reprinted, Dover, Mineola, NY, 1989. 
 +
|-
 +
| valign=top | [Has]
 +
| Hassoun, M.H. (ed.), ''Associative Neural Memories :  Theory and Implementation'', Oxford University Press, New York, NY, 1993.
 +
|-
 +
| valign=top | [KoB]
 +
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.
 +
|-
 +
| valign=top | [MaB]
 +
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.
 +
|-
 +
| valign=top | [Mac]
 +
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.
 +
|-
 +
| valign=top | [McC]
 +
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.
 +
|-
 +
| valign=top | [Mc1]
 +
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93.  Reprinted in [McC].
 +
|-
 +
| valign=top | [MiP]
 +
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969.  2nd printing 1972.  Expanded edition 1988.
 +
|-
 +
| valign=top | [Rum]
 +
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.
 +
|}
 +
 +
===Incidental Works===
 +
 +
{| cellpadding=3
 +
| valign=top | [Dew]
 +
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910.  Reprinted, Prometheus Books, Buffalo, NY, 1991.
 +
|-
 +
| valign=top | [Fou]
 +
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972.  Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.
 +
|-
 +
| valign=top | [Hom]
 +
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980.  First printed 1919.
 +
|-
 +
| valign=top | [Jam]
 +
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.
 +
|-
 +
| valign=top | [Ler]
 +
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988.  Originally published in French, 1911.
 +
|-
 +
| valign=top | [Mus]
 +
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979.  English edition first published by Secker and Warburg, 1954.  Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.
 +
|-
 +
| valign=top | [PlaR]
 +
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980.  First printed 1930 & 1935.
 +
|-
 +
| valign=top | [PlaS]
 +
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.
 +
|-
 +
| valign=top | [Qui]
 +
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940.  Revised edition, 1951.  Harvard University Press, Cambridge, MA, 1981.
 +
|-
 +
| valign=top | [SaD]
 +
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977.  1st published 1969.
 +
|-
 +
| valign=top | [Sha]
 +
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.
 +
|-
 +
| valign=top | [Sh1]
 +
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.
 +
|-
 +
| valign=top | [Sh2]
 +
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.
 +
|-
 +
| valign=top | [Sh3]
 +
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.
 +
|-
 +
| valign=top | [Web]
 +
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.
 +
|-
 +
| valign=top | [Whi]
 +
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992.  Originally published in numerous editions, 1855&ndash;1892.
 +
|-
 +
| valign=top | [Wil]
 +
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.
 +
|}
 +
 +
==Document History==
 +
 +
<pre>
 +
Author:  Jon Awbrey
 +
Created:  16 Dec 1993
 +
Relayed:  31 Oct 1994
 +
Revised:  03 Jun 2003
 +
Recoded:  03 Jun 2007
 +
</pre>
 +
 +
[[Category:Adaptive Systems]]
 +
[[Category:Artificial Intelligence]]
 +
[[Category:Boolean Algebra]]
 +
[[Category:Boolean Functions]]
 +
[[Category:Charles Sanders Peirce]]
 +
[[Category:Combinatorics]]
 +
[[Category:Computer Science]]
 +
[[Category:Cybernetics]]
 +
[[Category:Differential Logic]]
 +
[[Category:Discrete Systems]]
 +
[[Category:Dynamical Systems]]
 +
[[Category:Formal Languages]]
 +
[[Category:Formal Sciences]]
 +
[[Category:Formal Systems]]
 +
[[Category:Functional Logic]]
 +
[[Category:Graph Theory]]
 +
[[Category:Group Theory]]
 +
[[Category:Inquiry]]
 +
[[Category:Knowledge Representation]]
 +
[[Category:Linguistics]]
 +
[[Category:Logic]]
 +
[[Category:Logical Graphs]]
 +
[[Category:Mathematics]]
 +
[[Category:Mathematical Systems Theory]]
 +
[[Category:Philosophy]]
 +
[[Category:Science]]
 +
[[Category:Semiotics]]
 +
[[Category:Systems Science]]
 +
[[Category:Visualization]]
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/468105...469880"

Navigation menu