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

Revision as of 03:24, 4 September 2013

5,053 bytes removed ,  03:24, 4 September 2013
update
Line 6,098: Line 6,098:  
<br>
 
<br>
   −
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:90%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"
−
|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''
+
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math>
−
|- style="background:ghostwhite"
+
|- style="height:40px; background:ghostwhite"
 
| &nbsp;
 
| &nbsp;
−
| align="center" | '''Operator<br>or<br>Operand'''
+
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math>
−
| align="center" | '''Proposition<br>or<br>Component'''
+
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math>
−
| align="center" | '''Transformation<br>or<br>Mapping'''
+
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math>
−
|-
  −
| Operand
  −
| valign="top" |
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
  −
|-
  −
| ''F'' = ‹''f'', ''g''› : ''U'' &rarr; ''X''
  −
|}
  −
| valign="top" |
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 &rarr; '''B'''
  −
|-
  −
| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> &rarr; '''B'''
  −
|}
  −
| valign="top" |
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100"
  −
| ''F'' : [''u'', ''v''] &rarr; [''x'', ''y'']
  −
|-
  −
| ''F'' : '''B'''<sup>''n''</sup> &rarr; '''B'''<sup>''k''</sup>
  −
|}
   
|-
 
|-
 +
| align="center" | <math>\underline\text{Operand}\!</math>
 
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| Tacit
+
F = (F_1, F_2) \\
−
|-
+
F = (f, g) : U \!\to\! X
−
| Extension
+
\end{array}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <math>\epsilon</math> :
+
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\
−
|-
+
F_i : \mathbb{B}^n \!\to\! \mathbb{B}
−
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
−
|-
  −
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;&nbsp;(E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <math>\epsilon</math>''F''<sub>''i''</sub> :
+
F : [u, v] \!\to\! [x, y] \\
−
|-
+
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''B'''
+
\end{array}</math>
 
|-
 
|-
−
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''
+
| align="center" | <math>\begin{matrix}\underline\text{Tacit}\\\text{extension}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <math>\epsilon</math>''F'' :
+
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
−
|-
+
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'', ''y'']
+
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)
−
|-
+
\end{array}</math>
−
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>]
  −
|}
  −
|-
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| Trope
  −
|-
  −
| Extension
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <math>\eta</math> :
+
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\
−
|-
+
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
−
|-
  −
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <math>\eta</math>''F''<sub>''i''</sub> :
+
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\
−
|-
+
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
+
\end{array}</math>
 
|-
 
|-
−
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| align="center" | <math>\begin{matrix}\underline\text{Trope}\\\text{extension}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <math>\eta</math>''F'' :
+
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
−
|-
+
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
+
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
−
|-
+
\end{array}</math>
−
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
  −
|}
  −
|-
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| Enlargement
+
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
−
|-
+
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
−
| Operator
+
\end{array}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| E :
+
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
 
|-
 
|-
−
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
+
| align="center" | <math>\begin{matrix}\underline\text{Enlargement}\\\text{operator}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| E''F''<sub>''i''</sub> :
+
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
−
|-
+
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
+
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
−
|-
+
\end{array}</math>
−
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| E''F'' :
+
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
−
|-
+
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
+
\end{array}</math>
−
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
  −
|}
  −
|-
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| Difference
+
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
 +
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
 +
\end{array}</math>
 
|-
 
|-
−
| Operator
+
| align="center" | <math>\begin{matrix}\underline\text{Difference}\\\text{operator}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| D :
+
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
−
|-
+
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
−
|-
+
\end{array}</math>
−
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| D''F''<sub>''i''</sub> :
+
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
−
|-
+
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
+
\end{array}</math>
−
|-
  −
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| D''F'' :
+
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
+
\end{array}</math>
−
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
  −
|}
   
|-
 
|-
 +
| align="center" | <math>\begin{matrix}\underline\text{Differential}\\\text{operator}\end{matrix}</math>
 
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| Differential
+
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
−
|-
+
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
−
| Operator
+
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
−
|}
+
\end{array}</math>
 
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| d :
+
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
−
|-
+
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
−
|-
  −
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| d''F''<sub>''i''</sub> :
+
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
+
\end{array}\!</math>
 
|-
 
|-
−
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
+
| align="center" | <math>\begin{matrix}\underline\text{Remainder}\\\text{operator}\end{matrix}\!</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| d''F'' :
+
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~
−
|-
+
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
+
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)
−
|-
+
\end{array}</math>
−
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
  −
|}
  −
|-
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| Remainder
+
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
−
|-
+
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
−
| Operator
+
\end{array}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| r :
+
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
 
|-
 
|-
−
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
+
| align="center" | <math>\begin{matrix}\underline\text{Radius}\\\text{operator}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| r''F''<sub>''i''</sub> :
+
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\
−
|-
+
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
+
\end{array}</math>
−
|-
  −
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
  −
|}
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| r''F'' :
  −
|-
  −
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
  −
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
  −
|}
  −
|-
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| Radius
  −
|-
  −
| Operator
  −
|}
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
  −
|-
  −
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
  −
|-
  −
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
  −
|}
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
   
| &nbsp;
 
| &nbsp;
−
|-
  −
| &nbsp;
  −
|-
  −
| &nbsp;
  −
|}
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| <font face=georgia>'''e'''</font>''F'' :
  −
|-
  −
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
  −
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
  −
|}
  −
|-
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| Secant
  −
|-
  −
| Operator
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
+
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
 
|-
 
|-
−
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
+
| align="center" | <math>\begin{matrix}\underline\text{Secant}\\\text{operator}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| &nbsp;
+
\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>
 
| &nbsp;
 
| &nbsp;
−
|-
  −
| &nbsp;
  −
|}
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| <font face=georgia>'''E'''</font>''F'' :
  −
|-
  −
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
  −
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
  −
|}
  −
|-
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| Chord
  −
|-
  −
| Operator
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
+
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
 
|-
 
|-
−
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
+
| align="center" | <math>\begin{matrix}\underline\text{Chord}\\\text{operator}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| &nbsp;
+
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\
−
|-
+
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
−
| &nbsp;
+
\end{array}</math>
−
|-
   
| &nbsp;
 
| &nbsp;
−
|}
  −
|
  −
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
  −
| <font face=georgia>'''D'''</font>''F'' :
  −
|-
  −
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
  −
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
  −
|}
  −
|-
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| Tangent
+
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
 +
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
 +
\end{array}</math>
 
|-
 
|-
−
| Functor
+
| align="center" | <math>\begin{matrix}\underline\text{Tangent}\\\text{functor}\end{matrix}</math>
−
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
+
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\
−
|-
+
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
−
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
+
\end{array}</math>
−
|-
  −
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| d''F''<sub>''i''</sub> :
+
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\
−
|-
+
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}
−
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
+
\end{array}</math>
−
|-
  −
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
  −
|}
   
|
 
|
−
{| align="left" border="0" cellpadding="2" cellspacing="0" style="text-align:left; width:100%"
+
<math>\begin{array}{l}
−
| <font face=georgia>'''T'''</font>''F'' :
+
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\
−
|-
+
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]
−
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
+
\end{array}</math>
−
|-
  −
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
   
|}
 
|}
−
|}<br>
+
 
 +
<br>
    
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
Jon Awbrey
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