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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 04:30, 2 September 2013
, 04:30, 2 September 2013update
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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ '''Table 55. Synopsis of Terminology : Restrictive and Alternative Subtypes'''
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math>
|- style="background:ghostwhite"
|- style="height:40px; background:ghostwhite"
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! Operator
| align="center" | <math>\text{Operator}\!</math>
! Proposition
| align="center" | <math>\text{Proposition}\!</math>
! Map
| align="center" | <math>\text{Map}\!</math>
|-
|-
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\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}
\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>
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<math>\begin{array}{l}
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}
\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}
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]
\end{array}</math>
|-
|-
| 〈''u'', ''v'', d''u'', d''v''〉 → '''B'''
| 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 J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}
\end{array}</math>
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<math>\begin{array}{l}
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]
\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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<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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<math>\begin{array}{l}
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\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}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]
\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}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}
\end{array}</math>
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|
<math>\begin{array}{l}
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]
\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%"
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
|-
|-
| 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}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\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}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]
\end{array}\!</math>
|-
|-
| '''B'''<sup>2</sup> × '''D'''<sup>2</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}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}
\end{array}</math>
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|
<math>\begin{array}{l}
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]
\end{array}</math>
|-
|-
| (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>)
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\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, \eta) \\
\mathsf{e} : (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{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]
\end{array}</math>
|-
|-
| Operator
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
|}
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<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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{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{l}
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]
\end{array}</math>
|-
|-
| Operator
| 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}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]
\end{array}</math>
|-
|-
| 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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<math>\begin{array}{l}
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}
\end{array}</math>
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|
<math>\begin{array}{l}
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]
\end{array}</math>
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
|}
|}