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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 17:36, 31 August 2013
, 17:36, 31 August 2013update
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:70%"
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ '''Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators'''
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math>
|- style="background:ghostwhite"
|- style="height:40px; background:ghostwhite"
! Item
| align="center" | <math>\text{Symbol}\!</math>
! Notation
| align="center" | <math>\text{Notation}\!</math>
! Description
| align="center" | <math>\text{Description}\!</math>
! Type
| align="center" | <math>\text{Type}\!</math>
|-
|-
| ''U''<sup> •</sup>
| align="center" | <math>U^\bullet\!</math>
| = [''u'', ''v'']
| <math>= [u, v]\!</math>
| Source Universe
| <math>\text{Source universe}\!</math>
| ['''B'''<sup>2</sup>]
| <math>[\mathbb{B}^2]\!</math>
|-
|-
| ''X''<sup> •</sup>
| align="center" | <math>X^\bullet~\!</math>
| = [''x'']
| <math>= [x]\!</math>
| Target Universe
| <math>\text{Target universe}\!</math>
| ['''B'''<sup>1</sup>]
| <math>[\mathbb{B}^1]~\!</math>
|-
|-
| E''U''<sup> •</sup>
| align="center" | <math>\mathrm{E}U^\bullet\!</math>
| = [''u'', ''v'', d''u'', d''v'']
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math>
| Extended Source Universe
| <math>\text{Extended source universe}\!</math>
| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>]
| <math>[\mathbb{B}^2 \times \mathbb{D}^2]\!</math>
|-
|-
| E''X''<sup> •</sup>
| align="center" | <math>\mathrm{E}X^\bullet\!</math>
| = [''x'', d''x'']
| <math>= [x, \mathrm{d}x]~\!</math>
| Extended Target Universe
| <math>\text{Extended target universe}\!</math>
| ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>]
| <math>[\mathbb{B}^1 \times \mathbb{D}^1]\!</math>
|-
|-
| ''J''
| align="center" | <math>J\!</math>
| ''J'' : ''U'' → '''B'''
| <math>J : U \to \mathbb{B}\!</math>
| Proposition
| <math>\text{Proposition}\!</math>
| ('''B'''<sup>2</sup> → '''B''') ∈ ['''B'''<sup>2</sup>]
| <math>(\mathbb{B}^2 \to \mathbb{B}) \in [\mathbb{B}^2]\!</math>
|-
|-
| ''J''
| align="center" | <math>J\!</math>
| ''J'' : ''U''<sup> •</sup> → ''X''<sup> •</sup>
| <math>J : U^\bullet \to X^\bullet\!</math>
| Transformation, or Mapping
| <math>\text{Transformation or Map}\!</math>
| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| <math>[\mathbb{B}^2] \to [\mathbb{B}^1]\!</math>
|-
|-
| valign="top" |
| align="center" |
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{matrix}
\boldsymbol\varepsilon
\\
\eta
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
\\
\mathrm{E}
\\
\mathrm{D}
\\
\mathrm{d}
\end{matrix}</math>
|
<math>\begin{array}{l}
\mathrm{W} : U^\bullet \to \mathrm{E}U^\bullet,
\\
\mathrm{W} : X^\bullet \to \mathrm{E}X^\bullet,
\\\\
\mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet)
\\
\text{for}~ \mathrm{W} = \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}
\end{array}</math>
|
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
<math>\begin{array}{ll}
\text{Tacit extension operator} & \boldsymbol\varepsilon
\\
\text{Trope extension operator} & \eta
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"
\\
\text{Enlargement operator} & \mathrm{E}
|-
\\
\text{Difference operator} & \mathrm{D}
\\
\text{Differential operator} & \mathrm{d}
\end{array}</math>
|
<math>\begin{array}{l}
{[\mathbb{B}^2] \to [\mathbb{B}^2 \times \mathbb{D}^2]},
\\
{[\mathbb{B}^1] \to [\mathbb{B}^1 \times \mathbb{D}^1]},
\\\\
([\mathbb{B}^2] \to [\mathbb{B}^1]) \to
\\
([\mathbb{B}^2 \times \mathbb{D}^2] \to [\mathbb{B}^1 \times \mathbb{D}^1])
\end{array}</math>
|-
|-
| align="center" |
<math>\begin{matrix}
\mathsf{e}
\\
\mathsf{E}
\\
\mathsf{D}
\\
\mathsf{T}
\end{matrix}</math>
|
<math>\begin{array}{l}
\mathsf{W} : U^\bullet \to \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,
\\
\mathsf{W} : X^\bullet \to \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,
\\\\
\mathsf{W} : (U^\bullet \to X^\bullet) \to (\mathsf{T}U^\bullet \to \mathsf{T}X^\bullet)
\\
\text{for}~ \mathsf{W} = \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T}
\end{array}</math>
|
|
<math>\begin{array}{llll}
\text{Radius operator} & \mathsf{e} & = & (\boldsymbol\varepsilon, \eta)
\\
\text{Secant operator} & \mathsf{E} & = & (\boldsymbol\varepsilon, \mathrm{E})
\\
\text{Chord operator} & \mathsf{D} & = & (\boldsymbol\varepsilon, \mathrm{D})
\\
\text{Tangent functor} & \mathsf{T} & = & (\boldsymbol\varepsilon, \mathrm{d})
\end{array}</math>
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"
{| 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"
|
|
<math>\begin{array}{l}
{[\mathbb{B}^2] \to [\mathbb{B}^2 \times \mathbb{D}^2]},
\\
{[\mathbb{B}^1] \to [\mathbb{B}^1 \times \mathbb{D}^1]},
\\\\
([\mathbb{B}^2] \to [\mathbb{B}^1]) \to
\\
([\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:60%"
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:70%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"
|+ '''Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
|+ '''Table 55. Synopsis of Terminology : Restrictive and Alternative Subtypes'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
!
!