563: Ascending Sequence of Faces of Affine Simplex

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definition of ascending sequence of faces of affine simplex

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of affine simplex.
• The reader knows a definition of face of affine simplex.

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Target Context

• The reader will have a definition of ascending sequence of faces of affine simplex.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$[p_0,...,p_n]$: $=\text{ the affine simplex }$
$\sigma$: $\in \{\text{ the permutations of }(0,...,n)\}$
$\ast S$: $=([p_{{\sigma }_0}],[p_{{\sigma }_0},p_{{\sigma }_1}],...,[p_{{\sigma }_0},...,p_{{\sigma }_n}])$
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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 $(0,...,n)$, $S:=([p_{{\sigma }_0}],[p_{{\sigma }_0},p_{{\sigma }_1}],...,[p_{{\sigma }_0},...,p_{{\sigma }_n}])$

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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 $[p_0,p_1]$.

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References

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