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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 03:24, 4 September 2013
, 03:24, 4 September 2013update
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"
|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math>
|- style="background:ghostwhite"
|- style="height:40px; background:ghostwhite"
|
|
| align="center" | '''Operator<br>or<br>Operand'''
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math>
| align="center" | '''Proposition<br>or<br>Component'''
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math>
| align="center" | '''Transformation<br>or<br>Mapping'''
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"
|-
|-
| align="center" | <math>\underline\text{Operand}\!</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
F = (F_1, F_2) \\
F = (f, g) : U \!\to\! X
\end{array}</math>
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<math>\begin{array}{l}
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\
F_i : \mathbb{B}^n \!\to\! \mathbb{B}
\end{array}</math>
|
|
<math>\begin{array}{l}
F : [u, v] \!\to\! [x, y] \\
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]
\end{array}</math>
|-
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B'''
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math>
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|
<math>\begin{array}{l}
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)
\end{array}</math>
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
|
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<math>\begin{array}{l}
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}
\end{array}</math>
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|
<math>\begin{array}{l}
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]
\end{array}</math>
|-
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
| align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math>
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<math>\begin{array}{l}
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
\end{array}</math>
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
\end{array}</math>
|-
|-
| Operator
| align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math>
|}
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
\end{array}</math>
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|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
\end{array}</math>
|-
|-
| align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
\end{array}\!</math>
|-
|-
| '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''D'''
| align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math>
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|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
\end{array}</math>
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|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
\end{array}</math>
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → d''X''<sup> •</sup>)
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math>
|
|
<math>\begin{array}{l}
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
\end{array}</math>
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
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<math>\begin{array}{l}
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
\end{array}</math>
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
\end{array}</math>
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|
<math>\begin{array}{l}
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
\end{array}</math>
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
| align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math>
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|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
\end{array}</math>
|-
|-
| Functor
| align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math>
|}
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
\end{array}</math>
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|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
\end{array}</math>
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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
\end{array}</math>
|}
|}
<br>
===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>===
===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>===