device-56.home 2026-3-30:2:37:13
This commit is contained in:
oskar
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---
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excalidraw-plugin: parsed
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tags: [excalidraw]
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---
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==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
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# Excalidraw Data
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## Text Elements
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## Embedded Files
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0c2ad4dae5ed514b2fd2099d68d0760c69fbce4d: $$[1^{1}]$$
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f9c478059e1cce4e96a58e13baac6bfbb4f93eff: $$[1^{1}X^{1}$$
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214cc7890168fcc3649c7581bd637e267a6de7bf: $$[1^{1}2^{2}$$
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84fa363149df51fc341b5998c6b4e67ed3dbdb8e: $$[1^{3}$$
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f49bf0b1178f855291cee484fe1921917621c6c0: $$[1^{1}3^{2}$$
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c6ae105adef2251de3956c588d95fb1ed29d572b: $$[1^{2}2^{2}$$
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f7ab87077d1c10f7309f5bbce5c714d5f25bef0f: $$[1^{2}2^{3}$$
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539b82e2e9bc43b1e2665050071a453e72116434: $$[2^{1}X^{2}$$
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082b26651855e6e47f735b3481c3b45c1678ff69: $$2^{1}X^{\neq 2}$$
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76483ecab4b1eb14f16fcd978d387e8ac0f6344a: $$[2^{3}$$
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32a8965b6d15d90d3898edfdb7969dbd9a92f3c8: $$[3^{1}X^{3}$$
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cc5e37cfba585550da4471dd85852d6a38354787: $$[3^{1}X^{\leq 2}$$
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b752624b6df8d46597c3242c793dafa003ae9422: $$[3^{2}X^{3}$$
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848fe42b60268d03c05e522d05a9dd988119d57b: $$[3^{2}X^{\leq 2}$$
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98ffd4aa77ffc7c0ea17203a781a2cd621186d8e: $$[n^{1}$$
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8aef2acc0c00b47668a81c160807e1ab6d4b03ce: $$[1^{2}X^{1}$$
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%%
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## Drawing
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```compressed-json
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||||
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||||
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||||
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|
||||
```
|
||||
%%
|
||||
File diff suppressed because one or more lines are too long
|
Before Width: | Height: | Size: 80 KiB |
183
Excalidraw/demo_théorème_du_début_2.excalidraw.md
Normal file
183
Excalidraw/demo_théorème_du_début_2.excalidraw.md
Normal file
@@ -0,0 +1,183 @@
|
||||
---
|
||||
|
||||
excalidraw-plugin: parsed
|
||||
tags: [excalidraw]
|
||||
|
||||
---
|
||||
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
|
||||
|
||||
|
||||
# Excalidraw Data
|
||||
|
||||
## Text Elements
|
||||
## Embedded Files
|
||||
0c2ad4dae5ed514b2fd2099d68d0760c69fbce4d: $$[1^{1}]$$
|
||||
|
||||
f9c478059e1cce4e96a58e13baac6bfbb4f93eff: $$[1^{1}X^{1}$$
|
||||
|
||||
214cc7890168fcc3649c7581bd637e267a6de7bf: $$[1^{1}2^{2}$$
|
||||
|
||||
84fa363149df51fc341b5998c6b4e67ed3dbdb8e: $$[1^{3}$$
|
||||
|
||||
f49bf0b1178f855291cee484fe1921917621c6c0: $$[1^{1}3^{2}$$
|
||||
|
||||
539b82e2e9bc43b1e2665050071a453e72116434: $$[2^{1}X^{2}$$
|
||||
|
||||
082b26651855e6e47f735b3481c3b45c1678ff69: $$2^{1}X^{\neq 2}$$
|
||||
|
||||
76483ecab4b1eb14f16fcd978d387e8ac0f6344a: $$[2^{3}$$
|
||||
|
||||
32a8965b6d15d90d3898edfdb7969dbd9a92f3c8: $$[3^{1}X^{3}$$
|
||||
|
||||
cc5e37cfba585550da4471dd85852d6a38354787: $$[3^{1}X^{\leq 2}$$
|
||||
|
||||
b752624b6df8d46597c3242c793dafa003ae9422: $$[3^{2}X^{3}$$
|
||||
|
||||
848fe42b60268d03c05e522d05a9dd988119d57b: $$[3^{2}X^{\leq 2}$$
|
||||
|
||||
98ffd4aa77ffc7c0ea17203a781a2cd621186d8e: $$[n^{1}$$
|
||||
|
||||
8aef2acc0c00b47668a81c160807e1ab6d4b03ce: $$[1^{2}X^{1}$$
|
||||
|
||||
f297bb8c586c081e6495ec2b90fc2134c5756fa0: $$[1^{2}X^{\neq 1}$$
|
||||
|
||||
%%
|
||||
## Drawing
|
||||
```compressed-json
|
||||
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|
||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
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||||
|
||||
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||||
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||||
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|
||||
|
||||
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|
||||
|
||||
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||||
|
||||
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||||
|
||||
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||||
|
||||
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||||
|
||||
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||||
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||||
|
||||
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||||
|
||||
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|
||||
|
||||
AB8gBQ0xQQBwAH2Hk1wgJSD/AfwIAA==
|
||||
```
|
||||
%%
|
||||
2
Excalidraw/demo_théorème_du_début_2.excalidraw.svg
Normal file
2
Excalidraw/demo_théorème_du_début_2.excalidraw.svg
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File diff suppressed because one or more lines are too long
|
After Width: | Height: | Size: 76 KiB |
@@ -114,5 +114,28 @@ Le début arrivera toujours sur l'un de ces cycles :
|
||||
### Théorème du début — Démonstration
|
||||
|
||||
3) Simplifier les cas (en assimiler certains)
|
||||
![[attachments/Pasted image 20260330021624.png]] <!-- element class="fragment" -->
|
||||
La version de Conway : <!-- element class="fragment" -->
|
||||
|
||||
![[attachments/Capture d’écran 2026-03-29 à 22.50.04.png]]
|
||||
![[attachments/Capture d’écran 2026-03-29 à 22.50.04.png]] <!-- element class="fragment" -->
|
||||
|
||||
|
||||
---
|
||||
## Théorème du découpage
|
||||
|
||||
> [!definition] Découpage
|
||||
> Quand les descendants de $L$ et $R$ n'interfèrent jamais dans $LR$, c'est-à-dire :
|
||||
> $\forall n,\quad (LR)_{n} = L_{n}R_{n}$
|
||||
>
|
||||
> On dit que $LR$ se **découpe** en $L \cdot R$
|
||||
|
||||
|
||||
--
|
||||
## Théorème du découpage
|
||||
|
||||
|
||||
> [!definition] Découpage trivial
|
||||
> $[\;\;]\cdot R$ ou $L \cdot [\;\;]$
|
||||
|
||||
> [!definition] Atome
|
||||
> - **atome** : morceau sans découpage non trivial
|
||||
|
||||
BIN
attachments/Pasted image 20260330021624.png
Normal file
BIN
attachments/Pasted image 20260330021624.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 20 KiB |
@@ -60,10 +60,12 @@ header-auto-numbering:
|
||||
> - i Il est évident que cela arrive lorsque le dernier chiffre de $L_{n}$ est toujours différent du premier chiffre de $R_{n}$ (ou bien quand l'une des deux est vide)
|
||||
> ---
|
||||
> - def On appelle **trivial** un découpage du type $[\;]\cdot L$ ou $L\cdot [\;]$
|
||||
^def-decoupage
|
||||
|
||||
> [!definition] Atome
|
||||
> Les **atomes** (ou *éléments*) sont les chaînes qui ne possèdent pas de découpage non trivial.
|
||||
> - source:: [[sources/1 - articles/Open problems in communication and computation (Cover, T. M., 1938-, Gopinath, B) (z-library.sk, 1lib.sk, z-lib.sk).pdf#page=181&selection=221,11,241,7&color=note|(John Horton Conway, 1987)]]
|
||||
^def-atome
|
||||
|
||||
- i toute chaîne est **composée** d'un certain nombre d'éléments. On dit que cette chaîne **comprends** lesdits éléments.
|
||||
- source:: [[sources/1 - articles/Open problems in communication and computation (Cover, T. M., 1938-, Gopinath, B) (z-library.sk, 1lib.sk, z-lib.sk).pdf#page=181&selection=243,5,260,9&color=note|(John Horton Conway, 1987)]]
|
||||
@@ -211,7 +213,7 @@ header-auto-numbering:
|
||||
> > - en assimilant $[3^{1}X^{1}$ et $[3^{1}(\leq 2)^{2}$ aux deux cas $[3^{1}X^{3}$, $[3^{1}X^{\leq 2}$
|
||||
> > - en assimilant $[3^{2}(\leq 2)^{\leq 3}$ aux cas $[3^{2}X^{3}$, $[3^{2}X^{\leq 2}$
|
||||
> > - en assimilant $[1^{3}2^{2}$ et $[1^{3}X^{1}$ au seul cas $[1^{3}$
|
||||
> > - en séparant $[1^{2}2^{\leq 3}$ en $[1^{2}X^{1}$ (qui est déjà listé), $[1^{2}2^{2}$, $[1^{2}2^{3}$
|
||||
> > - en assimilant $[1^{2}2^{\leq 3}$ à $[1^{2}X^{1}$ (qui est déjà listé) et $[1^{2}X^{\neq 1}$
|
||||
> > - en séparant $[2^{1} X^{\leq 2}$ en $[2^{1}X^{2}$ et $[2^{1}X^{\neq 2}$
|
||||
> > - en ajoutant $[1^{1}3^{2}$
|
||||
> > Cela nous donne la liste suivante :
|
||||
@@ -220,8 +222,7 @@ header-auto-numbering:
|
||||
> > - $[1^{2}X^{1}$
|
||||
> > - $[1^{1}2^{2}$
|
||||
> > - $[1^{1}3^{2}$
|
||||
> > - $[1^{2}2^{2}$
|
||||
> > - $[1^{2}2^{3}$
|
||||
> > - $[1^{2}X^{\neq 1}$
|
||||
> > - $[1^{3}$
|
||||
> > - $[2^{1}X^{2}$
|
||||
> > - $[2^{1}X^{\neq 2}$
|
||||
@@ -234,7 +235,21 @@ header-auto-numbering:
|
||||
> >
|
||||
> > Cela nous permet d'atteindre le schéma original de Conway :
|
||||
> > ![[sources/1 - articles/Open problems in communication and computation (Cover, T. M., 1938-, Gopinath, B) (z-library.sk, 1lib.sk, z-lib.sk).pdf#page=186&rect=12,345,377,408|schéma original de Conway p.186]]
|
||||
^theoreme-debut
|
||||
|
||||
> [!proposition]+ théorème du découpage
|
||||
> Un chaîne de $\geq 2$ jour $LR$ se découpe en $L \cdot R$ seulement dans ces cas :
|
||||
>
|
||||
> | L | R |
|
||||
> | --------- | --------------------------------------------------------------------------------------------- |
|
||||
> | $n]$ | $[m$ |
|
||||
> | $2]$ | $[1^1X^1$ ou $[1^{3}$ ou $[3^{1}X^{\neq 3}$ ou $[n^{1}$ |
|
||||
> | $\neq 2]$ | $[2^{2} 1^{1}X^{1}$ ou $[2^{2}1^{3}$ ou $[2^{2}3^{1}X\neq 3$ ou $[2^{2}n^{(0 \text{ ou } 1)}$ |
|
||||
> avec $n \geq 4$ et $m \leq 3$
|
||||
> ou bien quand l'un des deux est vide ($L = [\;\;]$ ou $R = [\;\;]$)
|
||||
> > [!démonstration]- Démonstration
|
||||
> > Cela suit directement du [[désintégration audioactive#^theoreme-debut|téorème du début]] appliqué à $R$, et du fait que le dernier chiffre de $L$ est constant
|
||||
^theoreme-decoupage
|
||||
|
||||
|
||||
## Tableau des éléments
|
||||
|
||||
Reference in New Issue
Block a user