12,080
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 15:10, 19 May 2008
, 15:10, 19 May 2008→Formal development: add tables
==Formal development==
==Formal development==
…
Table 4
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 4. Propositional Calculus : Basic Notation'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| <math>\mathcal{A}</math>
| <math>\{ a_1, \ldots, a_n \}</math>
| Alphabet
| <math>[n] = \mathbf{n}</math>
|-
| <math>A_i\!</math>
| <math>\{ \overline{a_i}, a_i \}\!</math>
| Dimension <math>i\!</math>
| <math>\mathbb{B}</math>
|-
| <math>A\!</math>
| <math>\langle \mathcal{A} \rangle</math><br>
<math>\langle a_1, \ldots, a_n \rangle</math><br>
<math>\{ (a_1, \ldots, a_n) \}\!</math>
<math>A_1 \times \ldots \times A_n</math><br>
<math>\textstyle \prod_i A_i\!</math>
| Set of cells,<br>
coordinate tuples,<br>
points, or vectors<br>
in the universe<br>
of discourse
| <math>\mathbb{B}^n</math>
|-
| <math>A^*\!</math>
| <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| Linear functions
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
|-
| <math>A^\uparrow</math>
| <math>(A \to \mathbb{B})</math>
| Boolean functions
| <math>\mathbb{B}^n \to \mathbb{B}</math>
|-
| <math>A^\circ</math>
| <math>[ \mathcal{A} ]</math><br>
<math>(A, A^\uparrow)</math><br>
<math>(A\ +\!\to \mathbb{B})</math><br>
<math>(A, (A \to \mathbb{B}))</math><br>
<math>[ a_1, \ldots, a_n ]</math>
| Universe of discourse<br>
based on the features<br>
<math>\{ a_1, \ldots, a_n \}</math>
| <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{B}^n]</math>
|}
<br>
Table 5
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 5. Differential Extension : Basic Notation'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| <math>\operatorname{d}\mathcal{A}</math>
| <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| Alphabet of<br>
differential<br>
features
| <math>[n] = \mathbf{n}</math>
|-
| <math>\operatorname{d}A_i</math>
| <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}</math>
| Differential<br>
dimension <math>i\!</math>
| <math>\mathbb{D}</math>
|-
| <math>\operatorname{d}A</math>
| <math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>
<math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>
<math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>
<math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>
<math>\textstyle \prod_i \operatorname{d}A_i</math>
| Tangent space<br>
at a point:<br>
Set of changes,<br>
motions, steps,<br>
tangent vectors<br>
at a point
| <math>\mathbb{D}^n</math>
|-
| <math>\operatorname{d}A^*</math>
| <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>
| Linear functions<br>
on <math>\operatorname{d}A</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
|-
| <math>\operatorname{d}A^\uparrow</math>
| <math>(\operatorname{d}A \to \mathbb{B})</math>
| Boolean functions<br>
on <math>\operatorname{d}A</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
|-
| <math>\operatorname{d}A^\circ</math>
| <math>[\operatorname{d}\mathcal{A}]</math><br>
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
| Tangent universe<br>
at a point of <math>A^\circ,</math><br>
based on the<br>
tangent features<br>
<math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{D}^n]</math>
|}
<br>
'''…'''
==Expository examples==
==Expository examples==