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

Revision as of 18:40, 4 October 2017

40,187 bytes added ,  18:40, 4 October 2017
add § 7 : Appendices
Line 7,639: Line 7,639:  
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 79]
 
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 79]
 
|}
 
|}
 +
 +
===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>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/468106"

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