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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 22:00, 3 September 2013

2,920 bytes removed ,  22:00, 3 September 2013
update
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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>
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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&nbsp;58 and 59 lay the groundwork for discussing a typical map ''F''&nbsp;:&nbsp;['''B'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F''&nbsp;:&nbsp;['''B'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''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&nbsp;58 and 59 lay the groundwork for discussing a typical map ''F''&nbsp;:&nbsp;['''B'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F''&nbsp;:&nbsp;['''B'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''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>&nbsp;&bull;</sup>
+
| align="center" | <math>U^\bullet\!</math>
−
| valign="top" | <font face="courier new">=&nbsp;</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>&nbsp;&bull;</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">=&nbsp;</font>[''x'', ''y'']
+
= [f, g]
 +
\end{array}</math>
 +
| <math>\text{Target universe}\!</math>
 +
| <math>[\mathbb{B}^k]\!</math>
 
|-
 
|-
−
| <font face="courier new">=&nbsp;</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>&nbsp;&bull;</sup>
+
| align="center" | <math>\mathrm{E}X^\bullet\!</math>
−
| valign="top" | <font face="courier new">=&nbsp;</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> &times; '''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>&nbsp;&bull;</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">=&nbsp;</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">=&nbsp;</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> &times; '''D'''<sup>''k''</sup>]
+
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math>
 
|-
 
|-
−
| ''F''
+
| align="center" |
−
| ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>
+
<math>\begin{matrix}
−
| Transformation, or Mapping
+
\boldsymbol\varepsilon
−
| ['''B'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>]
+
\\
−
|-
+
\eta
−
| valign="top" |
+
\\
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
\mathrm{E}
−
| &nbsp;
+
\\
−
|-
+
\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'' &rarr; '''B'''
+
\\
−
|-
+
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,
−
| ''f'' : ''U'' &rarr; [''x''] &sube; ''X''<sup>&nbsp;&bull;</sup>
+
\\
−
|-
+
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
−
| ''g'' : ''U'' &rarr; [''y''] &sube; ''X''<sup>&nbsp;&bull;</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> &rarr; '''B'''
+
\\
−
|-
+
\text{Trope extension operator} & \eta
−
| &isin; ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> &rarr; '''B''')
+
\\
−
|-
+
\text{Enlargement operator}    & \mathrm{E}
−
| = ('''B'''<sup>''n''</sup> +&rarr; '''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&nbsp;:
+
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},
−
|-
+
\\\\
−
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,
+
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!
−
|-
+
\\
−
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])
−
|-
+
\end{array}</math>
−
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
  −
|-
  −
| &rarr;
  −
|-
  −
| (E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>)&nbsp;,
  −
|-
  −
| for each W in the set:
  −
|-
  −
| {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;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"
  −
| &nbsp;
  −
|-
  −
| ['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]&nbsp;,
  −
|-
  −
| ['''B'''<sup>''k''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]&nbsp;,
  −
|-
  −
| (['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup>])
  −
|-
  −
| &rarr;
  −
|-
  −
| (['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>])
  −
|-
  −
| &nbsp;
  −
|-
  −
| &nbsp;
  −
|}
   
|-
 
|-
 +
| 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" | &nbsp;
  −
| 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>&nbsp;:
  −
|-
  −
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,
  −
|-
  −
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
  −
|-
  −
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
  −
|-
  −
| &rarr;
  −
|-
  −
| (<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>)&nbsp;,
  −
|-
  −
| for each <font face=georgia>'''W'''</font> in the set:
  −
|-
  −
| {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<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"
  −
| &nbsp;
  −
|-
  −
| ['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]&nbsp;,
  −
|-
  −
| ['''B'''<sup>''k''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]&nbsp;,
  −
|-
  −
| (['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup>])
  −
|-
  −
| &rarr;
  −
|-
  −
| (['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>])
  −
|-
  −
| &nbsp;
  −
|-
  −
| &nbsp;
  −
|}
  −
|-
   
|
 
|
−
{| 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" | &nbsp;
  −
| colspan="2"  |
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:60%"
  −
| Radius Operator || <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\eta</math>›
  −
|-
  −
| Secant Operator || <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›
  −
|-
  −
| Chord Operator  || <font face=georgia>'''D'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;D›
  −
|-
  −
| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;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"
Jon Awbrey
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