Line 5,499: |
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| | <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> | | | <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> |
| | <math>\text{Extended source universe}\!</math> | | | <math>\text{Extended source universe}\!</math> |
− | | <math>[\mathbb{B}^2 \times \mathbb{D}^2]\!</math> | + | | <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math> |
| |- | | |- |
| | align="center" | <math>\mathrm{E}X^\bullet\!</math> | | | align="center" | <math>\mathrm{E}X^\bullet\!</math> |
| | <math>= [x, \mathrm{d}x]~\!</math> | | | <math>= [x, \mathrm{d}x]~\!</math> |
| | <math>\text{Extended target universe}\!</math> | | | <math>\text{Extended target universe}\!</math> |
− | | <math>[\mathbb{B}^1 \times \mathbb{D}^1]\!</math> | + | | <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math> |
| |- | | |- |
| | align="center" | <math>J\!</math> | | | align="center" | <math>J\!</math> |
Line 5,554: |
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| | | | | |
| <math>\begin{array}{l} | | <math>\begin{array}{l} |
− | {[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \times \mathbb{D}^2]}, | + | {[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]}, |
| \\ | | \\ |
− | {[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1]}, | + | {[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]}, |
| \\\\ | | \\\\ |
| ([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\! | | ([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\! |
| \\ | | \\ |
− | ([\mathbb{B}^2 \times \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1]) | + | ([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]) |
| \end{array}</math> | | \end{array}</math> |
| |- | | |- |
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| | | | | |
| <math>\begin{array}{l} | | <math>\begin{array}{l} |
− | {[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \times \mathbb{D}^2]}, | + | {[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]}, |
| \\ | | \\ |
− | {[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1]}, | + | {[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]}, |
| \\\\ | | \\\\ |
| ([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\! | | ([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\! |
| \\ | | \\ |
− | ([\mathbb{B}^2 \times \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1]) | + | ([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]) |
| \end{array}</math> | | \end{array}</math> |
| |} | | |} |
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| | | |
| <br><font face="courier new"> | | <br><font face="courier new"> |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:90%" |
| | | | | |
| {| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%" | | {| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%" |
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| But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]. | | But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]. |
| | | |
− | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%" | + | <br> |
− | |+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators''' | + | |
− | |- style="background:ghostwhite" | + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
− | ! Item | + | |+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math> |
− | ! Notation | + | |- style="height:40px; background:ghostwhite" |
− | ! Description | + | | align="center" | <math>\text{Symbol}\!</math> |
− | ! Type | + | | align="center" | <math>\text{Notation}\!</math> |
| + | | align="center" | <math>\text{Description}\!</math> |
| + | | align="center" | <math>\text{Type}\!</math> |
| |- | | |- |
− | | valign="top" | ''U''<sup> •</sup> | + | | align="center" | <math>U^\bullet\!</math> |
− | | valign="top" | <font face="courier new">= </font>[''u'', ''v'']
| + | | <math>= [u, v]\!</math> |
− | | valign="top" | Source Universe | + | | <math>\text{Source universe}\!</math> |
− | | valign="top" | ['''B'''<sup>''n''</sup>] | + | | <math>[\mathbb{B}^n]\!</math> |
| |- | | |- |
− | | valign="top" | ''X''<sup> •</sup> | + | | align="center" | <math>X^\bullet~\!</math> |
− | | valign="top" | | + | | <math>\begin{array}{l} |
− | {| align="left" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:100%" | + | = [x, y] \\ |
− | | <font face="courier new">= </font>[''x'', ''y''] | + | = [f, g] |
| + | \end{array}</math> |
| + | | <math>\text{Target universe}\!</math> |
| + | | <math>[\mathbb{B}^k]\!</math> |
| |- | | |- |
− | | <font face="courier new">= </font>[''f'', ''g''] | + | | align="center" | <math>\mathrm{E}U^\bullet\!</math> |
− | |} | + | | <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> |
− | | valign="top" | Target Universe | + | | <math>\text{Extended source universe}\!</math> |
− | | valign="top" | ['''B'''<sup>''k''</sup>]
| + | | <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math> |
| |- | | |- |
− | | valign="top" | E''U''<sup> •</sup> | + | | align="center" | <math>\mathrm{E}X^\bullet\!</math> |
− | | valign="top" | <font face="courier new">= </font>[''u'', ''v'', d''u'', d''v'']
| + | | <math>\begin{array}{l} |
− | | valign="top" | Extended Source Universe
| + | = [x, y, \mathrm{d}x, \mathrm{d}y] \\ |
− | | valign="top" | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] | + | = [f, g, \mathrm{d}f, \mathrm{d}g] |
| + | \end{array}</math> |
| + | | <math>\text{Extended target universe}\!</math> |
| + | | <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math> |
| |- | | |- |
− | | valign="top" | E''X''<sup> •</sup> | + | | align="center" | |
− | | valign="top" |
| + | <math>\begin{matrix} |
− | {| align="left" border="0" cellpadding="0" cellspacing="0" style="text-align:left; width:100%" | + | f \\ g |
− | | <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y''] | + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{array}{ll} |
| + | f : U \!\to\! [x] \cong \mathbb{B} \\ |
| + | g : U \!\to\! [y] \cong \mathbb{B} |
| + | \end{array}</math> |
| + | | <math>\text{Proposition}\!</math> |
| + | | |
| + | <math>\begin{array}{l} |
| + | \mathbb{B}^n \!\to\! \mathbb{B} \\ |
| + | \in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n] |
| + | \end{array}</math> |
| |- | | |- |
− | | <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g''] | + | | align="center" | <math>F\!</math> |
− | |}
| + | | <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math> |
− | | valign="top" | Extended Target Universe
| + | | <math>\text{Transformation of Map}\!</math> |
− | | valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] | + | | <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math> |
| |- | | |- |
− | | ''F'' | + | | align="center" | |
− | | ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
| + | <math>\begin{matrix} |
− | | Transformation, or Mapping | + | \boldsymbol\varepsilon |
− | | ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
| + | \\ |
− | |-
| + | \eta |
− | | valign="top" |
| + | \\ |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | \mathrm{E} |
− | |
| + | \\ |
− | |-
| + | \mathrm{D} |
− | | ''f''
| + | \\ |
− | |-
| + | \mathrm{d} |
− | | ''g''
| + | \end{matrix}</math> |
− | |}
| + | | |
− | | valign="top" |
| + | <math>\begin{array}{l} |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | \mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet, |
− | | ''f'', ''g'' : ''U'' → '''B'''
| + | \\ |
− | |-
| + | \mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet, |
− | | ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
| + | \\ |
− | |-
| + | \mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet) |
− | | ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
| + | \\ |
− | |}
| + | \text{for each}~ \mathrm{W} ~\text{in the set:} |
− | | valign="top" |
| + | \\ |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \} |
− | | Proposition
| + | \end{array}</math> |
− | |}
| + | | |
− | | valign="top" |
| + | <math>\begin{array}{ll} |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100" | + | \text{Tacit extension operator} & \boldsymbol\varepsilon |
− | | '''B'''<sup>''n''</sup> → '''B'''
| + | \\ |
− | |- | + | \text{Trope extension operator} & \eta |
− | | ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
| + | \\ |
− | |-
| + | \text{Enlargement operator} & \mathrm{E} |
− | | = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
| + | \\ |
− | |}
| + | \text{Difference operator} & \mathrm{D} |
− | |-
| + | \\ |
− | | valign="top" |
| + | \text{Differential operator} & \mathrm{d} |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | \end{array}</math> |
− | | W
| + | | |
− | |}
| + | <math>\begin{array}{l} |
− | | valign="top" |
| + | {[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]}, |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | \\ |
− | | W :
| + | {[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]}, |
− | |-
| + | \\\\ |
− | | ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
| + | ([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\! |
− | |-
| + | \\ |
− | | ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
| + | ([\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>) ,
| |
− | |-
| |
− | | for each W in the set:
| |
− | |-
| |
− | | {<math>\epsilon</math>, <math>\eta</math>, E, D, d}
| |
− | |}
| |
− | | valign="top" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | Operator
| |
− | |}
| |
− | | valign="top" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100" | |
− | |
| |
− | |- | |
− | | ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
| |
− | |-
| |
− | | ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
| |
− | |-
| |
− | | (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
| |
− | |-
| |
− | | →
| |
− | |-
| |
− | | (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
| |
− | |-
| |
− | |
| |
− | |-
| |
− | |
| |
− | |}
| |
| |- | | |- |
| + | | 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 each}~ \mathsf{W} ~\text{in the set:} |
| + | \\ |
| + | \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \} |
| + | \end{array}</math> |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
| + | <math>\begin{array}{lll} |
− | | <math>\epsilon</math>
| + | \text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta) |
− | |-
| + | \\ |
− | | <math>\eta</math>
| + | \text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E}) |
− | |-
| + | \\ |
− | | E
| + | \text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D}) |
− | |-
| + | \\ |
− | | D
| + | \text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d}) |
− | |-
| + | \end{array}</math> |
− | | d
| |
− | |}
| |
− | | valign="top" |
| |
− | | colspan="2" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%" | |
− | | Tacit Extension Operator || <math>\epsilon</math>
| |
− | |-
| |
− | | Trope Extension Operator || <math>\eta</math>
| |
− | |-
| |
− | | Enlargement Operator || E
| |
− | |-
| |
− | | Difference Operator || D
| |
− | |-
| |
− | | Differential Operator || d
| |
− | |}
| |
− | |-
| |
− | | valign="top" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | <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" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | |
− | | Operator
| |
− | |}
| |
− | | valign="top" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100" | |
− | |
| |
− | |-
| |
− | | ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
| |
− | |-
| |
− | | ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
| |
− | |-
| |
− | | (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
| |
− | |-
| |
− | | →
| |
− | |-
| |
− | | (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
| |
− | |-
| |
− | |
| |
− | |-
| |
− | |
| |
− | |}
| |
− | |-
| |
| | | | | |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%" | + | <math>\begin{array}{l} |
− | | <font face=georgia>'''e'''</font>
| + | {[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]}, |
− | |-
| + | \\ |
− | | <font face=georgia>'''E'''</font>
| + | {[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]}, |
− | |-
| + | \\\\ |
− | | <font face=georgia>'''D'''</font>
| + | ([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\! |
− | |-
| + | \\ |
− | | <font face=georgia>'''T'''</font>
| + | ([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]) |
| + | \end{array}</math> |
| |} | | |} |
− | | valign="top" |
| |
− | | colspan="2" |
| |
− | {| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"
| |
− | | 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>
| |
| | | |
− | {| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%" | + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:90%" |
| |+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes''' | | |+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |