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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 04:18, 8 June 2008

904 bytes removed ,  04:18, 8 June 2008
→‎Work Area 2: reorder tables
Line 3,611: Line 3,611:     
==Propositional Forms on Two Variables==
 
==Propositional Forms on Two Variables==
−
  −
To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  For future reference, I will set here a few Tables that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.
  −
  −
By way of initial orientation, Table&nbsp;1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
      
===Table 1===
 
===Table 1===
Line 4,190: Line 4,186:     
===Table 3===
 
===Table 3===
−
  −
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
      
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
+
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
−
|- style="background:ghostwhite; height:36px"
+
|- style="background:ghostwhite; height:48px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>\operatorname{E}f|_{xy}</math>
  −
| <math>\operatorname{E}f|_{x(y)}</math>
  −
| <math>\operatorname{E}f|_{(x)y}</math>
  −
| <math>\operatorname{E}f|_{(x)(y)}</math>
  −
|- style="height:36px"
  −
| <math>f_{0}\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{1}\!</math>
+
| <math>\operatorname{T}_{11}f</math>
−
|-
+
|-  
−
| height="36px" | <math>f_{2}\!</math>
+
| <math>\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
−
|-
  −
| height="36px" | <math>f_{4}\!</math>
  −
|-
  −
| height="36px" | <math>f_{8}\!</math>
   
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| <math>\operatorname{T}_{10}f</math>
−
|-
+
|-  
−
| height="36px" | <math>(x)\ y\!</math>
+
| <math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
−
|-
  −
| height="36px" | <math>x\ (y)\!</math>
  −
|-
  −
| height="36px" | <math>x\ y\!</math>
   
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| <math>\operatorname{T}_{01}f</math>
−
|-
+
|-  
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| <math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
−
|-
  −
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
  −
|-
  −
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
   
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| <math>\operatorname{T}_{00}f</math>
−
|-
+
|-  
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| <math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
−
|-
+
|}
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
|- style="height:36px"
 +
| <math>f_{0}\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
  −
|}
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>f_{1}\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
| height="36px" | <math>f_{2}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>f_{4}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>f_{8}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\ y\!</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{3}\!</math>
+
| height="36px" | <math>x\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{12}\!</math>
+
| height="36px" | <math>x\ (y)\!</math>
 +
|-
 +
| height="36px" | <math>(x)\ y\!</math>
 +
|-
 +
| height="36px" | <math>(x)(y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>x\ y\!</math>
−
|}
  −
|
  −
{| align="center"
   
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)(y)\!</math>
−
|}
  −
|
  −
{| align="center"
   
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>x\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)(y)\!</math>
 +
|-
 +
| height="36px" | <math>(x)\ y\!</math>
 +
|-
 +
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\ y\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,323: Line 4,298:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{6}\!</math>
+
| height="36px" | <math>f_{3}\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{9}\!</math>
+
| height="36px" | <math>f_{12}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>(x)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,366: Line 4,341:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{5}\!</math>
+
| height="36px" | <math>f_{6}\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{10}\!</math>
+
| height="36px" | <math>f_{9}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,409: Line 4,384:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{7}\!</math>
+
| height="36px" | <math>f_{5}\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{11}\!</math>
+
| height="36px" | <math>f_{10}\!</math>
 +
|}
 +
|
 +
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{13}\!</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{14}\!</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(y)\!</math>
−
|-
  −
| height="36px" | <math>((x)\ y)\!</math>
  −
|-
  −
| height="36px" | <math>((x)(y))\!</math>
   
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>y\!</math>
−
|-
  −
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
  −
|-
  −
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
   
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(y)\!</math>
 +
|}
 +
|
 +
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 +
|-
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>f_{7}\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>f_{11}\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>f_{13}\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>f_{14}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|}
 
|}
−
|- style="height:36px"
+
|
−
| <math>f_{15}\!</math>
+
{| align="center"
−
| <math>((~))\!</math>
+
|-
−
| <math>((~))\!</math>
+
| height="36px" | <math>((x)(y))\!</math>
−
| <math>((~))\!</math>
+
|-
−
| <math>((~))\!</math>
+
| height="36px" | <math>((x)\ y)\!</math>
−
| <math>((~))\!</math>
+
|-
 +
| height="36px" | <math>(x\ (y))\!</math>
 +
|-
 +
| height="36px" | <math>(x\ y)\!</math>
 
|}
 
|}
−
<br>
+
|
−
 
+
{| align="center"
−
===Table 4===
+
|-
−
 
+
| height="36px" | <math>((x)\ y)\!</math>
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
|-
−
|+ '''Table 4.  <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
+
| height="36px" | <math>((x)(y))\!</math>
−
|- style="background:ghostwhite; height:36px"
+
|-
−
| &nbsp;
+
| height="36px" | <math>(x\ y)\!</math>
−
| <math>f\!</math>
  −
| <math>\operatorname{D}f|_{xy}</math>
  −
| <math>\operatorname{D}f|_{x(y)}</math>
  −
| <math>\operatorname{D}f|_{(x)y}</math>
  −
| <math>\operatorname{D}f|_{(x)(y)}</math>
  −
|- style="height:36px"
  −
| <math>f_{0}\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
  −
| <math>(~)\!</math>
   
|-
 
|-
 +
| height="36px" | <math>(x\ (y))\!</math>
 +
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{1}\!</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{2}\!</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{4}\!</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{8}\!</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|}
 
|}
−
|
+
|- style="height:36px"
−
{| align="center"
+
| <math>f_{15}\!</math>
−
|-
+
| <math>((~))\!</math>
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| <math>((~))\!</math>
−
|-
+
| <math>((~))\!</math>
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| <math>((~))\!</math>
−
|-
+
| <math>((~))\!</math>
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
|- style="background:ghostwhite; height:36px"
−
|-
+
| colspan="2" | Fixed Point Total :
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 
|}
 
|}
 +
<br>
 +
 +
===Table 4===
 +
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 +
|+ '''Table 4.  <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
 +
|- style="background:ghostwhite; height:36px"
 +
| &nbsp;
 +
| <math>f\!</math>
 +
| <math>\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
 +
| <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
 +
| <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
 +
| <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
 +
|- style="height:36px"
 +
| <math>f_{0}\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|-
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>f_{1}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>f_{2}\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>f_{4}\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>f_{8}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\ y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{3}\!</math>
+
| height="36px" | <math>(y)\!</math>
 +
|-
 +
| height="36px" | <math>y\!</math>
 +
|-
 +
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{12}\!</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 +
|-
 +
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
 
| height="36px" | <math>(x)\!</math>
 
| height="36px" | <math>(x)\!</math>
 +
|-
 +
| height="36px" | <math>x\!</math>
 
|-
 
|-
 
| height="36px" | <math>x\!</math>
 
| height="36px" | <math>x\!</math>
Line 4,585: Line 4,583:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(~)\!</math>
 +
|-
 +
| height="36px" | <math>(~)\!</math>
 +
|-
 +
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 +
|-
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>f_{3}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>f_{12}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{6}\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{9}\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 +
|-
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>f_{6}\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>f_{9}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{5}\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{10}\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 +
|-
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>f_{5}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>f_{10}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
−
|-
   
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>f_{7}\!</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>f_{11}\!</math>
+
| height="36px" | <math>(~)\!</math>
−
|-
  −
| height="36px" | <math>f_{13}\!</math>
  −
|-
  −
| height="36px" | <math>f_{14}\!</math>
   
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>((~))\!</math>
 +
|}
 +
|
 +
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 +
|-
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>f_{7}\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>f_{11}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>f_{13}\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>f_{14}\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(y)\!</math>
 
|}
 
|}
−
|- style="height:36px"
+
|
−
| <math>f_{15}\!</math>
+
{| align="center"
−
| <math>((~))\!</math>
+
|-
−
| <math>(~)\!</math>
+
| height="36px" | <math>x\!</math>
−
| <math>(~)\!</math>
+
|-
−
| <math>(~)\!</math>
+
| height="36px" | <math>x\!</math>
 +
|-
 +
| height="36px" | <math>(x)\!</math>
 +
|-
 +
| height="36px" | <math>(x)\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(~)\!</math>
 +
|-
 +
| height="36px" | <math>(~)\!</math>
 +
|-
 +
| height="36px" | <math>(~)\!</math>
 +
|-
 +
| height="36px" | <math>(~)\!</math>
 +
|}
 +
|- style="height:36px"
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 
| <math>(~)\!</math>
 
| <math>(~)\!</math>
 
|}
 
|}
Line 4,776: Line 4,800:     
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 5.  <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
+
|+ '''Table 5.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
−
|- style="background:ghostwhite; height:48px"
+
|- style="background:ghostwhite; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
|
+
| <math>\operatorname{E}f|_{xy}</math>
−
{| align="center"
+
| <math>\operatorname{E}f|_{x(y)}</math>
−
|-
+
| <math>\operatorname{E}f|_{(x)y}</math>
−
| <math>\operatorname{T}_{11}f</math>
+
| <math>\operatorname{E}f|_{(x)(y)}</math>
−
|-
  −
| <math>\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
  −
|}
  −
|
  −
{| align="center"
  −
|-
  −
| <math>\operatorname{T}_{10}f</math>
  −
|-
  −
| <math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
  −
|}
  −
|
  −
{| align="center"
  −
|-
  −
| <math>\operatorname{T}_{01}f</math>
  −
|-
  −
| <math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
  −
|}
  −
|
  −
{| align="center"
  −
|-
  −
| <math>\operatorname{T}_{00}f</math>
  −
|-
  −
| <math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
  −
|}
   
|- style="height:36px"
 
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
Line 4,841: Line 4,841:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 4,900: Line 4,900:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|-
 
|-
Line 4,943: Line 4,943:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|}
 
|}
 
|-
 
|-
Line 4,986: Line 4,986:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 5,037: Line 5,037:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|}
 
|}
 
|- style="height:36px"
 
|- style="height:36px"
Line 5,085: Line 5,085:  
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
−
|- style="background:ghostwhite; height:36px"
+
|}
−
| colspan="2" | Fixed Point Total :
  −
| <math>4\!</math>
  −
| <math>4\!</math>
  −
| <math>4\!</math>
  −
| <math>16\!</math>
  −
|}
   
<br>
 
<br>
   Line 5,097: Line 5,091:     
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 6.  <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
+
|+ '''Table 6.  <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 
|- style="background:ghostwhite; height:36px"
 
|- style="background:ghostwhite; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
+
| <math>\operatorname{D}f|_{xy}</math>
−
| <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
+
| <math>\operatorname{D}f|_{x(y)}</math>
−
| <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
+
| <math>\operatorname{D}f|_{(x)y}</math>
−
| <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
+
| <math>\operatorname{D}f|_{(x)(y)}</math>
 
|- style="height:36px"
 
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
Line 5,138: Line 5,132:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 5,197: Line 5,191:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|-
 
|-
Line 5,240: Line 5,234:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|-
 
|-
Line 5,283: Line 5,277:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 5,334: Line 5,328:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|}
 
|}
 
|- style="height:36px"
 
|- style="height:36px"
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/67729"

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