definition of slicing-and-halving map on Euclidean set
Topics
About: set
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 Euclidean set.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of slicing-and-halving map on Euclidean set.
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:
\( \mathbb{R}^{d'}\): \(= \text{ the Euclidean set }\)
\( J\): \(\subseteq \{1, ..., d'\}\)
\( k\): \(\in J\)
\( r'\): \(\in \mathbb{R}^{d'}\)
\(*\lambda_{J, r', k}\): \(: Pow (\mathbb{R}^{d'}) \to Pow (\mathbb{R}^{d'}), S \mapsto \{s \in S \vert \forall j \in \{1, ..., d'\} \setminus J (s^j = r'^j) \land r'^k \le s^k\}\)
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Conditions:
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2: Natural Language Description
For the Euclidean set, \(\mathbb{R}^{d'}\), any subset, \(J \subseteq \{1, ..., d'\}\), any \(k \in J\), and any point, \(r' \in \mathbb{R}^{d'}\), the map, \(\lambda_{J, r', k}: Pow (\mathbb{R}^{d'}) \to Pow (\mathbb{R}^{d'}), S \mapsto \{s \in S \vert \forall j \in \{1, ..., d'\} \setminus J (s^j = r'^j) \land r'^k \le s^k\}\)
3: Note
The reason why we do as \(\forall j \in \{1, ..., d'\} \setminus J (s^j = r'^j)\) instead of \(\forall j \in J (s^j = r'^j)\) is that the remained \(J\) components are usually more important than the fixed \(\{1, ..., d'\} \setminus J\) components. In fact, in many cases, we do the projection that takes the \(J\) components after the slicing map, getting the subset of \(\mathbb{R}^{\vert J \vert}\).
We do not use any name like "half-slicing", because that seems like slicing only half way instead of slicing completely or slicing into halves. This concept is not like those but slicing it completely and halving the slice after that.
We are saying "Euclidean set", because this concept does not require any topology, any vectors space structure, or something, although \(\mathbb{R}^{d'}\) is typically a Euclidean topological space or something.
