definition of ascending sequence of faces of affine simplex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of affine simplex.
- The reader knows a definition of face of affine simplex.
Target Context
- The reader will have a definition of ascending sequence of faces of affine simplex.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( [p_0, ..., p_n]\): \(= \text{ the affine simplex }\)
\( \sigma\): \( \in \{\text{ the permutations of } (0, ..., n)\}\)
\(*S\): \(= ([p_{\sigma_0}], [p_{\sigma_0}, p_{\sigma_1}], ..., [p_{\sigma_0}, ..., p_{\sigma_n}])\)
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Conditions:
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2: Natural Language Description
For any real vectors space, \(V\), any affine-independent set of base points on \(V\), \(\{p_0, ..., p_n\} \subseteq V\), and the affine simplex, \([p_0, ..., p_n]\), for any permutation, \(\sigma\), of \((0, ..., n)\), \(S := ([p_{\sigma_0}], [p_{\sigma_0}, p_{\sigma_1}], ..., [p_{\sigma_0}, ..., p_{\sigma_n}])\)
3: Note
Each element of \(S\) is a face of \([p_0, ..., p_n]\) and \([p_{\sigma_0}] \subset [p_{\sigma_0}, p_{\sigma_1}] \subset ... \subset [p_{\sigma_0}, ..., p_{\sigma_n}]\), which is the reason why \(S\) is called ascending sequence of faces of \([p_0, ..., p_n]\).
There are \((n + 1)!\) ascending sequences of faces of \([p_0, ..., p_n]\), because for example, for the affine simplex, \([p_0, p_1]\), the permutations of \((0, 1)\) are \((0, 1)\) and \((1, 0)\), and \(([p_0], [p_0, p_1])\) and \(([p_1], [p_0, p_1])\) are the ascending sequences of faces of \([p_0, p_1]\).
