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

Revision as of 19:03, 16 September 2013

4,337 bytes added ,  19:03, 16 September 2013
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| style="border-right:none" | <math>f\!</math>
 
| style="border-right:none" | <math>f\!</math>
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| style="border-left:4px double black" | <math>\mathrm{D}f</math>
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| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
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| <math>\mathrm{d}f</math>
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| <math>\mathrm{d}f~\!</math>
 
|-
 
|-
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
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====Table A8. Differential Forms Expanded on an Algebraic Basis====
 
====Table A8. Differential Forms Expanded on an Algebraic Basis====
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<br>
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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"
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|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math>
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|- style="background:ghostwhite; height:40px"
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| &nbsp;
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| style="border-right:none" | <math>f\!</math>
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| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math>
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| <math>\mathrm{d}f~\!</math>
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|-
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| <math>f_{0}\!</math>
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| style="border-right:none" | <math>\texttt{(~)}\!</math>
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| style="border-left:4px double black" | <math>0\!</math>
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| <math>0\!</math>
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|-
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| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\
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\texttt{(} x \texttt{)~} y \texttt{~}
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\\
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\
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\texttt{~} x \texttt{~~} y \texttt{~}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
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\\
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
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\\
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
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\\
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
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\\
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(} x \texttt{)}
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\\
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\texttt{~} x \texttt{~}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\mathrm{d}x
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\\
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\mathrm{d}x
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\end{matrix}\!</math>
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| <math>\begin{matrix}
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\mathrm{d}x
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\\
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\mathrm{d}x
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{~(} x \texttt{,~} y \texttt{)~}
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\\
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\texttt{((} x \texttt{,~} y \texttt{))}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\mathrm{d}x & + & \mathrm{d}y
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\\
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\mathrm{d}x & + & \mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\mathrm{d}x & + & \mathrm{d}y
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\\
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\mathrm{d}x & + & \mathrm{d}y
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
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| style="border-right:none" |
 +
<math>\begin{matrix}
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\texttt{(} y \texttt{)}
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\\
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\texttt{~} y \texttt{~}
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\end{matrix}</math>
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| style="border-left:4px double black" |
 +
<math>\begin{matrix}
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\mathrm{d}y
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\\
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\mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\mathrm{d}y
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\\
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\mathrm{d}y
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\end{matrix}</math>
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|-
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| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
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| style="border-right:none" |
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<math>\begin{matrix}
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\texttt{(~} x \texttt{~~} y \texttt{~)}
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\\
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\texttt{(~} x \texttt{~(} y \texttt{))}
 +
\\
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\texttt{((} x \texttt{)~} y \texttt{~)}
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\\
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\texttt{((} x \texttt{)(} y \texttt{))}
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\end{matrix}</math>
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| style="border-left:4px double black" |
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<math>\begin{matrix}
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
 +
\\
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y
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\end{matrix}</math>
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|
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<math>\begin{matrix}
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
 +
\\
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y
 +
\\
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\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
 +
\\
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\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y
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\end{matrix}</math>
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|-
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| <math>f_{15}\!</math>
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| style="border-right:none" | <math>\texttt{((~))}\!</math>
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| style="border-left:4px double black" | <math>0\!</math>
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| <math>0\!</math>
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|}
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<br>
    
====Table A12. Detail of Calculation for the Difference Map====
 
====Table A12. Detail of Calculation for the Difference Map====
Jon Awbrey
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