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Revision as of 17:36, 31 August 2013
, 17:36, 31 August 2013update
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| W+|}+| valign="top" |+| W :+|-+| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,+|-+| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,+|-+| (''U''<sup> •</sup> → ''X''<sup> •</sup>)+|-+| →+|-+| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,+|-+| for each W in the set:+|-+| {<math>\epsilon</math>, <math>\eta</math>, E, D, d}+|}+| valign="top" |+| Operator+|}+| valign="top" |+| +| ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] ,+|-+| ['''B'''<sup>1</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>] ,+|-+| (['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>])+|-+| →+|-+| (['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>])+|-+| +|-+| +|}+{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"+| <math>\epsilon</math>+|-+| <math>\eta</math>+|-+| E+|-+| D+|-+| d
−|}
−| valign="top" |
−| colspan="2" |
−| Tacit Extension Operator || <math>\epsilon</math>
−|-
−| Trope Extension Operator || <math>\eta</math>
−|-
−| Enlargement Operator || E
−|-
−| Difference Operator || D
−|-
−| Differential Operator || d
−|}
−|-
−| valign="top" |
−| <font face=georgia>'''W'''</font>
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
−| <font face=georgia>'''W'''</font> :
−|-
−| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
−|-
−| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
−|-
−| →
−|-
−| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
−|-
−| for each <font face=georgia>'''W'''</font> in the set:
−|-
−| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
−|}
−| valign="top" |
−| Operator
−|}
−| valign="top" |
−|
−|-
−| ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] ,
−|-
−| ['''B'''<sup>1</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>] ,
−|-
−| (['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>])
−|-
−| →
−|-
−| (['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>])
−|-
−|
−|-
−|
−|}
−|-
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"+| <font face=georgia>'''e'''</font>+|-+| <font face=georgia>'''E'''</font>+|-+| <font face=georgia>'''D'''</font>+|-+| <font face=georgia>'''T'''</font>+|}+| valign="top" |
−| colspan="2" |
−| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
−|-
−| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
−|-
−| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
−|-
−| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
−|}
<br>
<br>
−{| 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%"
−|}
|}
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<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"
!
!
