Mes motivations sont de définir mathématiquement des concepts philosophiques bien choisis d'existence, d'essence et de puissance d'interaction.
Il se peut que la formalisation mathématique de certaines notions ou de certains concepts philosophiques ait peu d'intérêt sur le plan mathématique :
Cela ne veut pas, nécessairement, dire qu'il faille se tourner vers d'autres notions ou concepts, certes plus riches, mais polysémiques, fourre-tout ou plus flous (Par exemple comme le concept "amour" qui a plusieurs sens, qu'il conviendrait de distinguer).
Introduction et notion de dimension
modifier
Nous nous plaçons dans un cadre proche de celui de la mécanique newtonienne, où
T
o
u
t
{\displaystyle Tout}
T
=
R
{\displaystyle T=\mathbb {R} }
T
{\displaystyle T}
R
{\displaystyle \mathbb {R} }
T
o
u
t
{\displaystyle Tout}
T
=
⨆
t
∈
T
T
(
t
)
{\displaystyle \displaystyle {{\cal {T}}=\bigsqcup _{t\in T}T(t)}}
où
∀
t
∈
T
,
T
(
t
)
{\displaystyle \forall t\in T,\,\,T(t)}
T
o
u
t
{\displaystyle Tout}
t
{\displaystyle t}
T
{\displaystyle {\cal {T}}}
t
{\displaystyle t}
T
(
t
)
≠
∅
{\displaystyle T(t)\neq \emptyset }
c
a
r
d
Q
(
{
T
(
t
)
|
t
∈
T
}
)
=
c
a
r
d
Q
(
T
)
{\displaystyle {card}_{Q}(\{T(t)|t\in T\})={card}_{Q}(T)}
On suppose de plus que
∀
t
∈
T
,
∃
n
t
∈
N
:
T
(
t
)
∈
P
(
R
n
t
(
t
)
)
{\displaystyle \forall t\in T,\exists n_{t}\in \mathbb {N} \,\,:\,\,T(t)\in {\mathcal {P}}{\Big (}\mathbb {R} ^{n_{t}}(t){\Big )}}
T
∈
P
(
⨆
t
∈
T
R
n
t
(
t
)
)
{\displaystyle \displaystyle {{\cal {T}}\in {\mathcal {P}}{\Big (}\bigsqcup _{t\in T}\mathbb {R} ^{n_{t}}(t){\Big )}}}
Remarque :
∀
t
∈
T
,
T
(
t
)
peut être bornée
{\displaystyle \forall t\in T,T(t)\,\,{\mbox{peut être bornée}}}
R
n
t
(
t
)
{\displaystyle \mathbb {R} ^{n_{t}}(t)}
Remarque :
∀
t
∈
T
,
A
(
t
)
∈
P
(
T
(
t
)
)
,
∅
(
t
)
⊂
A
(
t
)
⊂
T
(
t
)
e
t
∀
t
1
,
t
2
∈
T
,
t
1
≠
t
2
,
∅
(
t
1
)
≠
∅
,
∅
(
t
2
)
≠
∅
,
{\displaystyle \displaystyle {\forall t\in T,\,\,A(t)\in {\cal {P}}{\Big (}T(t){\Big )},\emptyset (t)\subset A(t)\subset T(t)\,\,et\,\,\forall t_{1},t_{2}\in T,\,\,t_{1}\neq t_{2},\,\,\emptyset (t_{1})\neq \emptyset ,\,\,\emptyset (t_{2})\neq \emptyset ,}}
∅
(
t
1
)
≠
∅
(
t
2
)
et
∅
(
t
1
)
⋂
∅
(
t
2
)
=
∅
{\displaystyle \displaystyle {\emptyset (t_{1})\neq \emptyset (t_{2})\,\,{\mbox{et}}\,\,\emptyset (t_{1})\bigcap \emptyset (t_{2})=\emptyset }}
Remarque importante : J'hésite à employer la notation
∀
t
∈
T
,
∅
(
t
)
{\displaystyle \forall t\in T,\,\,\emptyset (t)}
Par définition
et
∅
(
t
)
=
{
}
T
(
t
)
{\displaystyle \emptyset (t)={\{\}}_{T(t)}}
∀
t
1
,
t
2
∈
T
:
t
1
≠
t
2
,
∅
(
t
1
)
=
{
}
T
(
t
1
)
≠
{
}
T
(
t
2
)
=
∅
(
t
2
)
{\displaystyle \forall t_{1},t_{2}\in T\colon t_{1}\neq t_{2},\,\,\emptyset (t_{1})={\{\}}_{T(t_{1})}\neq {\{\}}_{T(t_{2})}=\emptyset (t_{2})}
∀
t
1
,
t
2
∈
T
:
t
1
≠
t
2
,
∅
(
t
1
)
≠
∅
,
∅
(
t
2
)
≠
∅
{\displaystyle \forall t_{1},t_{2}\in T\colon t_{1}\neq t_{2},\,\,\emptyset (t_{1})\neq \emptyset ,\,\,\emptyset (t_{2})\neq \emptyset }
∅
(
t
1
)
≠
∅
(
t
2
)
et
∅
(
t
1
)
⋂
∅
(
t
2
)
=
∅
{\displaystyle \displaystyle {\emptyset (t_{1})\neq \emptyset (t_{2})\,\,{\mbox{et}}\,\,\emptyset (t_{1})\bigcap \emptyset (t_{2})=\emptyset }}
Dans cette optique
∀
t
∈
T
∅
(
t
)
≠
∅
{\displaystyle \forall t\in T\,\,\emptyset (t)\neq \emptyset }
∀
t
∈
T
c
a
r
d
Q
o
u
E
(
∅
(
t
)
)
=
c
a
r
d
Q
o
u
E
(
∅
)
=
0
{\displaystyle \forall t\in T\,\,{card}_{Q\,\,ou\,\,E}{\Big (}\emptyset (t){\Big )}={card}_{Q\,\,ou\,\,E}(\emptyset )=0}
Si cette notation pose vraiment problème, on la remplacera par et on reviendra à la notation
∅
{\displaystyle \emptyset }
(
R
″
=
−
∞
F
(
R
)
⨆
R
⨆
+
∞
F
(
R
)
{\displaystyle {\Big (}\mathbb {R} ''=-\infty _{{\cal {F}}(\mathbb {R} )}\bigsqcup \mathbb {R} \bigsqcup +\infty _{{\cal {F}}(\mathbb {R} )}}
On a
R
⊊
R
′
⊊
R
″
)
{\displaystyle \mathbb {R} \subsetneq \mathbb {R} '\subsetneq \mathbb {R} ''{\Big )}}
(Pour les notations nouvelles, se reporter à la version la plus récente du document : "La définition du cardinal quantitatif sur
R
n
{\displaystyle {\mathbb {R} }^{n}}
R
″
n
{\displaystyle {\mathbb {R} ''}^{n}}
)
Soit
A
⊂
T
{\displaystyle {\cal {A}}\subset {\cal {T}}}
On se place dans un repère orthonormé direct
R
{\displaystyle {\cal {R}}}
On désigne par
c
a
r
d
E
(
A
)
{\displaystyle {card}_{E}({\cal {A}})}
A
{\displaystyle {\cal {A}}}
c
a
r
d
Q
(
A
)
=
c
a
r
d
Q
,
R
(
A
)
{\displaystyle {card}_{Q}({\cal {A}})={card}_{Q,{\cal {R}}}({\cal {A}})}
A
{\displaystyle {\cal {A}}}
R
{\displaystyle {\cal {R}}}
"La définition du cardinal quantitatif sur
R
n
{\displaystyle {\mathbb {R} }^{n}}
R
″
n
{\displaystyle {\mathbb {R} ''}^{n}}
) qui se veut plus fine que celle de cardinal équipotentiel et qui se veut être la notion optimale de nombre d'éléments, dans le système
Z
F
C
{\displaystyle ZFC}
D
dimension de
T
⟺
d
e
´
f
D
∈
P
(
(
⨆
t
∈
T
|
T
(
t
)
≠
∅
(
t
)
T
(
t
)
)
⨆
(
⨆
t
∈
T
|
T
(
t
)
=
∅
(
t
)
{
∅
(
t
)
}
)
)
{\displaystyle \displaystyle {{\cal {D}}\,\,{\mbox{dimension de}}\,\,{\cal {T}}\,\,\Longleftrightarrow _{d{\acute {e}}f}\,\,{\cal {D}}\in {\cal {P}}{\bigg (}{\Big (}\bigsqcup _{t\in T|T(t)\neq \emptyset (t)}T(t){\Big )}\bigsqcup {\Big (}\bigsqcup _{t\in T|T(t)=\emptyset (t)}\{\emptyset (t)\}{\Big )}{\bigg )}}}
D
0
{\displaystyle {\cal {D}}_{0}}
T
{\displaystyle {\cal {T}}}
⟺
d
e
´
f
{\displaystyle \Longleftrightarrow _{d{\acute {e}}f}}
D
0
{\displaystyle {\cal {D}}_{0}}
T
{\displaystyle {\cal {T}}}
et
∃
x
:
T
⟶
T
⨆
P
(
T
)
:
t
⟼
x
(
t
)
∈
{
∅
(
t
)
}
si
T
(
t
)
=
∅
(
t
)
et
x
(
t
)
∈
T
(
t
)
⨆
{
∅
(
t
)
}
si
T
(
t
)
≠
∅
(
t
)
{\displaystyle \displaystyle {{\mbox{et}}\,\,\exists x\,\,:\,\,T\longrightarrow {\cal {T}}\bigsqcup {\cal {P}}({\cal {T}})\,\,:\,\,t\longmapsto x(t)\in \{\emptyset (t)\}\,\,{\mbox{si}}\,\,T(t)=\emptyset (t)\,\,{\mbox{et}}\,\,x(t)\in T(t)\bigsqcup \{\emptyset (t)\}\,\,{\mbox{si}}\,\,T(t)\neq \emptyset (t)}}
et
D
0
=
⨆
t
∈
T
{
x
(
t
)
}
=
{
x
(
t
)
|
t
∈
T
}
{\displaystyle \displaystyle {{\mbox{et}}\,\,{\cal {D}}_{0}=\bigsqcup _{t\in T}\{x(t)\}}=\{x(t)|t\in T\}}
et
c
a
r
d
Q
(
{
x
(
t
)
|
t
∈
T
}
)
=
c
a
r
d
Q
(
{
T
(
t
)
|
t
∈
T
}
)
[
=
c
a
r
d
Q
(
T
)
]
{\displaystyle {card}_{Q}(\{x(t)|t\in T\})={card}_{Q}(\{T(t)|t\in T\})[={card}_{Q}(T)]}
c
a
r
d
E
(
{
x
(
t
)
|
t
∈
T
}
)
=
c
a
r
d
E
(
{
T
(
t
)
|
t
∈
T
}
)
[
=
c
a
r
d
E
(
T
)
]
{\displaystyle {card}_{E}(\{x(t)|t\in T\})={card}_{E}(\{T(t)|t\in T\})[={card}_{E}(T)]}
D
{\displaystyle {\cal {D}}}
T
{\displaystyle {\cal {T}}}
⟺
d
e
´
f
{\displaystyle \Longleftrightarrow _{d{\acute {e}}f}}
D
dimension de
T
et
∃
D
0
dimension temporelle (éternelle) minimale, représentative de
T
:
D
0
dimension de
D
{\displaystyle \displaystyle {{\cal {D}}\,\,{\mbox{dimension de}}\,\,{\cal {T}}\,\,{\mbox{et}}\,\,\exists {\cal {D}}_{0}\,\,{\mbox{dimension temporelle (éternelle) minimale, représentative de}}\,\,{\cal {T}}\,\,:\,\,{\cal {D}}_{0}\,\,{\mbox{dimension de}}\,\,{\cal {D}}}}
(
⨆
t
∈
T
|
T
(
t
)
≠
∅
(
t
)
T
(
t
)
)
⨆
(
⨆
t
∈
T
|
T
(
t
)
=
∅
(
t
)
{
∅
(
t
)
}
)
{\displaystyle \displaystyle {{\Big (}\bigsqcup _{t\in T|T(t)\neq \emptyset (t)}T(t){\Big )}\bigsqcup {\Big (}\bigsqcup _{t\in T|T(t)=\emptyset (t)}\{\emptyset (t)\}{\Big )}}}
⨆
t
∈
T
{
∅
(
t
)
}
{\displaystyle \displaystyle {\bigsqcup _{t\in T}\{\emptyset (t)\}}}
T
)
{\displaystyle {\cal {T}})}
(
⨆
t
∈
T
|
T
(
t
)
≠
∅
(
t
)
T
(
t
)
)
⨆
(
⨆
t
∈
T
|
T
(
t
)
=
∅
(
t
)
{
∅
(
t
)
}
)
{\displaystyle \displaystyle {{\Big (}\bigsqcup _{t\in T|T(t)\neq \emptyset (t)}T(t){\Big )}\bigsqcup {\Big (}\bigsqcup _{t\in T|T(t)=\emptyset (t)}\{\emptyset (t)\}{\Big )}}}
⨆
t
∈
T
{
∅
(
t
)
}
{\displaystyle \displaystyle {\bigsqcup _{t\in T}\{\emptyset (t)\}}}
Soit
D
une dimension de
T
:
D
=
⨆
t
∈
T
D
D
(
t
)
où
∀
t
∈
T
D
,
D
(
t
)
∈
P
(
T
)
et
T
D
∈
P
(
T
)
{\displaystyle \displaystyle {{\mbox{Soit}}\,\,{\cal {D}}\,\,{\mbox{une dimension de}}\,\,{\cal {T}}\,\,:\,\,{\cal {D}}=\bigsqcup _{t\in T_{\cal {D}}}D(t)\,\,{\mbox{où}}\,\,\forall t\in T_{\cal {D}},\,\,D(t)\in {\cal {P}}({\cal {T}})\,\,{\mbox{et}}\,\,T_{\cal {D}}\in {\cal {P}}(T)}}
D
0
{\displaystyle {\cal {D}}_{0}}
D
{\displaystyle {\cal {D}}}
⟺
d
e
´
f
{\displaystyle \Longleftrightarrow _{d{\acute {e}}f}}
D
0
{\displaystyle {\cal {D}}_{0}}
T
{\displaystyle {\cal {T}}}
et
∃
x
:
T
D
⟶
T
⨆
P
(
T
)
:
t
⟼
x
(
t
)
∈
{
∅
(
t
)
}
si
T
(
t
)
=
∅
(
t
)
et
x
(
t
)
∈
T
(
t
)
⨆
{
∅
(
t
)
}
si
T
(
t
)
≠
∅
(
t
)
{\displaystyle \displaystyle {{\mbox{et}}\,\,\exists x\,\,:\,\,T_{\cal {D}}\longrightarrow {\cal {T}}\bigsqcup {\cal {P}}({\cal {T}})\,\,:\,\,t\longmapsto x(t)\in \{\emptyset (t)\}\,\,{\mbox{si}}\,\,T(t)=\emptyset (t)\,\,{\mbox{et}}\,\,x(t)\in T(t)\bigsqcup \{\emptyset (t)\}\,\,{\mbox{si}}\,\,T(t)\neq \emptyset (t)}}
et
D
0
=
⨆
t
∈
T
D
{
x
(
t
)
}
=
{
x
(
t
)
|
t
∈
T
D
}
{\displaystyle \displaystyle {{\mbox{et}}\,\,{\cal {D}}_{0}=\bigsqcup _{t\in T_{\cal {D}}}\{x(t)\}}=\{x(t)|t\in T_{\cal {D}}\}}
et
c
a
r
d
Q
(
{
x
(
t
)
|
t
∈
T
D
}
)
=
c
a
r
d
Q
(
{
D
(
t
)
|
t
∈
T
D
}
)
[
=
c
a
r
d
Q
(
T
D
)
]
{\displaystyle \displaystyle {{\mbox{et}}\,\,{card}_{Q}(\{x(t)|t\in T_{\cal {D}}\})={card}_{Q}(\{D(t)|t\in T_{\cal {D}}\})[={card}_{Q}(T_{\cal {D}})]}}
On reprend les notations de la définition précédente.
Si, de plus,
T
D
=
]
−
∞
,
b
0
)
⨆
(
⨆
i
∈
N
n
−
1
∗
(
a
i
,
b
i
)
)
⨆
(
a
n
,
+
∞
[
avec
∀
i
∈
N
n
∗
,
a
i
∈
T
,
∀
i
∈
N
n
−
1
,
b
i
∈
T
,
T
=
R
{\displaystyle \displaystyle {{\mbox{Si, de plus,}}\,\,T_{\cal {D}}=]-\infty ,b_{0})\bigsqcup {\Big (}\bigsqcup _{i\in \mathbb {N} _{n-1}^{*}}(a_{i},b_{i}){\Big )}\bigsqcup (a_{n},+\infty [\,\,{\mbox{avec}}\,\,\forall i\in \mathbb {N} _{n}^{*},\,\,a_{i}\in T,\,\,\forall i\in \mathbb {N} _{n-1},\,\,b_{i}\in T,\,\,T=\mathbb {R} }}
alors
D
{\displaystyle {\cal {D}}}
D
0
{\displaystyle {\cal {D}}_{0}}
D
{\displaystyle {\cal {D}}}
Si, de plus
T
D
=
R
{\displaystyle T_{\cal {D}}=\mathbb {R} }
alors
D
{\displaystyle {\cal {D}}}
D
0
{\displaystyle {\cal {D}}_{0}}
D
{\displaystyle {\cal {D}}}
D
1
{\displaystyle {\cal {D}}_{1}}
D
2
{\displaystyle {\cal {D}}_{2}}
⟺
d
e
´
f
{\displaystyle \Longleftrightarrow _{d{\acute {e}}f}}
D
1
dimension de
D
2
et
∃
D
0
dimension temporelle minimale, représentative de
D
2
:
D
0
dimension de
D
1
{\displaystyle \displaystyle {{\cal {D}}_{1}\,\,{\mbox{dimension de}}\,\,{\cal {D}}_{2}\,\,{\mbox{et}}\,\,\exists {\cal {D}}_{0}\,\,{\mbox{dimension temporelle minimale, représentative de}}\,\,{\cal {D}}_{2}\,\,:\,\,{\cal {D}}_{0}\,\,{\mbox{dimension de}}\,\,{\cal {D}}_{1}}}
Remarque : Si, de plus
D
0
{\displaystyle {\cal {D}}_{0}}
D
2
{\displaystyle {\cal {D}}_{2}}
D
1
{\displaystyle {\cal {D}}_{1}}
D
2
{\displaystyle {\cal {D}}_{2}}
Au lieu de considérer un espace
T
×
T
{\displaystyle {\cal {T}}\times T}
T
o
u
t
{\displaystyle Tout}
∀
t
∈
T
{\displaystyle \forall t\in T}
T
×
{
t
}
{\displaystyle {\cal {T}}\times \{t\}}
Nous considérons
∀
t
∈
T
{\displaystyle \forall t\in T}
T
(
t
)
{\displaystyle T(t)}
⨆
t
∈
T
T
(
t
)
=
T
{\displaystyle \displaystyle {\bigsqcup _{t\in T}T(t)={\cal {T}}}}
Remarque : On ne prend pas nécessairement
T
=
E
×
T
{\displaystyle {\cal {T}}=E\times T}
∀
t
∈
T
,
T
(
t
)
=
E
×
{
t
}
{\displaystyle \forall t\in T,\,\,T(t)=E\times \{t\}}
∀
t
∈
T
,
T
(
t
)
est une tranche variable de
T
,
{\displaystyle \forall t\in T,\,\,T(t)\,\,{\mbox{est une tranche variable de}}\,\,{\cal {T}},}
en tout cas beaucoup plus variable qu'une tranche
E
×
{
t
}
{\displaystyle E\times \{t\}}
E
×
T
{\displaystyle E\times T}
On peut d'ailleurs considérer que
∀
t
∈
T
,
T
(
t
)
{\displaystyle \forall t\in T,\,\,T(t)}
t
{\displaystyle t}
On suppose donc
∀
t
1
,
t
2
∈
T
,
t
1
≠
t
2
,
T
(
t
1
)
⋂
T
(
t
2
)
=
∅
{\displaystyle \displaystyle {\forall t_{1},t_{2}\in T,\,\,t_{1}\neq t_{2},\,\,T(t_{1})\bigcap T(t_{2})=\emptyset }}
Mais rien ne nous dit qu'à plus grande échelle cette propriété est vérifiée :
Il est fort probable que plus la partie de
T
{\displaystyle {\cal {T}}}
(
⨆
t
∈
T
|
T
(
t
)
≠
∅
(
t
)
T
(
t
)
)
⨆
(
⨆
t
∈
T
|
T
(
t
)
=
∅
(
t
)
{
∅
(
t
)
}
)
{\displaystyle \displaystyle {{\Big (}\bigsqcup _{t\in T|T(t)\neq \emptyset (t)}T(t){\Big )}\bigsqcup {\Big (}\bigsqcup _{t\in T|T(t)=\emptyset (t)}\{\emptyset (t)\}{\Big )}}}
(
{\displaystyle {\Big (}}
T
{\displaystyle {\cal {T}}}
En effet,
T
=
⨆
t
∈
T
T
(
t
)
{\displaystyle \displaystyle {{\cal {T}}=\bigsqcup _{t\in T}T(t)}}
∃
t
∈
T
,
T
(
t
)
=
∅
(
t
)
{\displaystyle \exists t\in T,\,\,T(t)=\emptyset (t)}
∄
x
t
∈
∅
(
t
)
{\displaystyle \not \exists x_{t}\in \emptyset (t)}
∀
t
∈
T
{\displaystyle \forall t\in T}
∅
(
t
)
{\displaystyle \emptyset (t)}
C'est le singleton constitué de l'élément
∅
(
t
)
{\displaystyle \emptyset (t)}
{
∅
(
t
)
}
{\displaystyle \{\emptyset (t)\}}
)
{\displaystyle {\Big )}}
T
o
u
t
{\displaystyle Tout}
Je pensais le contraire, mais je pense aujourd'hui que la question de l'existence de Dieu est un indécidable irréductible, du moins, dans l'état de nos connaissances actuelles.
Déjà, le monde microscopique quantique avec la logique qui lui est associée, est une réalité :
On pourrait aussi envisager que Tout corresponde à un enchevêtrement de mondes ayant chacun sa propre logique.
De fait, toute démonstration utilisant la logique classique, avec son principe du tiers exclus, est inappropriée lorsqu'on étudie Tout, et en particulier Dieu.
Bien que nous ayons une connaissance et une appréhension de certaines des propriétés de Tout :
Comme nous n'aurons toujours qu'une connaissance locale et relative de ce dernier, la logique qui lui est associée, nous sera à jamais inaccessible.
La logique quantique est différente de la logique classique, la notion d'inclusion quantique de parties quantiques, est différente de celle d'inclusion classique de parties classiques :
Dans le cadre de la mécanique quantique qui est différent et plus général que celui de la mécanique newtonienne, les parties considérées ne se ramènent pas nécessairement à une tribu de parties, classique :
En effet à l'échelle quantique, il peut y avoir superposition de 2 états (ou évènements) :
Et comme toute théorie physique est relative et révolutionnable et dépend de logiques plus ou moins complexes et abstraites, qui lui sont propres, et de notion d'inclusions plus ou moins complexes et abstraites, qui lui sont propres, et que Tout est un indéfinissable, malgré la connaissance que nous avons de certaines de ses propriétés :
Jamais nous n'atteindrons ou n'accèderont à des notions de logique, d'inclusion et même d'existence universelles :
Les notions de logique, d'inclusion et d'existence auxquelles nous aurons accès, dépendront, toujours, des cadres relatifs dans lesquels nous nous placerons, et celles du cadre universel, nous resteront, à jamais, inaccessibles :
Ce n'est nullement du galimacia à la Jean-Paul Sartre sur l'existence, là elle est dépouillée de toute ambiguïté :
Il ne s'agit pas du fait d'exister à travers soi et les autres, qui est plus relatif à la puissance de nos interactions avec soi et les autres, de plus, la notion de puissance, peut être subjective et relative, à chacun.
Il est fort probable qu'il n'existe pas d'état premier de Tout et que Tout soit incréé, et puis supposons que cet état premier ait existé, à cet état premier, Tout s'est réduit au pire à l'Ensemble vide, donc, dans ce cas, Tout existe, et existera toujours, et sinon Tout s'est réduit ou se réduira au pire à l'Ensemble vide, donc Tout a toujours existé, existe, et existera toujours, pas nécessairement par rapport à l'Espace-Temps, mais par rapport à quelque chose d'éternel, l'Ensemble vide, le complémentaire de Tout dans lui-même, qui peut s'identifier parfois à Tout, dans son état minimal.
Il est possible que Tout ne s'est jamais contracté et réduit à l'Ensemble vide :
De toute façon qu'il se soit réduit ou pas, qu'il se réduise un jour, ou ne se réduise jamais à l'Ensemble vide, Tout est Eternel.
Soient
D
,
O
∈
P
(
T
)
{\displaystyle \displaystyle {{\cal {D}},{\cal {O}}\in {\cal {P}}({\cal {T}})}}
L'application existence, pour les corps ou les processus, est définie par :
e
x
i
s
t
e
n
c
e
:
P
(
T
)
⟶
P
(
T
)
:
O
⟼
e
x
i
s
t
e
n
c
e
(
O
)
=
O
{\displaystyle \displaystyle {{existence}\,\,:\,\,{\cal {P}}({\cal {T}})\longrightarrow {\cal {P}}({\cal {T}})\,\,:\,\,{\cal {O}}\,\,\longmapsto \,\,{existence}({\cal {O}})={\cal {O}}}}
O
=
O
(
t
0
)
∈
P
(
T
(
t
0
)
)
{\displaystyle {\cal {O}}=O(t_{0})\in {\cal {P}}{\Big (}T(t_{0}){\Big )}}
t
0
∈
T
{\displaystyle t_{0}\in T}
e
x
i
s
t
e
n
c
e
(
O
(
t
0
)
)
=
O
(
t
0
)
{\displaystyle {existence}{\Big (}O(t_{0}){\Big )}=O(t_{0})}
modifier
∃
x
∈
E
′
⟺
E
′
≠
∅
⟺
c
a
r
d
E
(
E
′
)
≥
1
⟺
c
a
r
d
Q
(
E
′
)
≥
1
{\displaystyle \displaystyle {\exists x\in E'\,\,\Longleftrightarrow \,\,E'\neq \emptyset \,\,\Longleftrightarrow \,\,{card}_{E}(E')\geq 1\,\,\Longleftrightarrow \,\,{card}_{Q}(E')\geq 1}}
On définit l'application existence commune à tout corps et au corps
D
{\displaystyle {\cal {D}}}
(ou dans un certain sens, l'application existence par rapport au corps
D
{\displaystyle {\cal {D}}}
e
x
i
s
t
e
n
c
e
(
.
|
D
)
:
P
(
T
)
⟶
P
(
T
)
:
O
⟼
e
x
i
s
t
e
n
c
e
(
O
|
D
)
=
O
∩
D
{\displaystyle \displaystyle {{existence}(.|{\cal {D}})\,\,:\,\,{\cal {P}}({\cal {T}})\,\,\longrightarrow \,\,{\cal {P}}({\cal {T}})\,\,:\,\,{\cal {O}}\,\,\longmapsto \,\,{existence}({\cal {O}}|{\cal {D}})={\cal {O}}\cap {\cal {D}}}}
e
x
i
s
t
e
n
c
e
(
O
|
D
)
{\displaystyle existence({\cal {O}}|{\cal {D}})}
O
{\displaystyle {\cal {O}}}
D
{\displaystyle {\cal {D}}}
(ou dans un certain sens, l'existence du corps
O
{\displaystyle {\cal {O}}}
D
{\displaystyle {\cal {D}}}
O
=
O
(
t
0
)
,
D
=
D
(
t
0
)
∈
P
(
T
(
t
0
)
)
{\displaystyle {\cal {O}}=O(t_{0}),{\cal {D}}=D(t_{0})\in {\cal {P}}{\Big (}T(t_{0}){\Big )}}
t
0
∈
T
{\displaystyle t_{0}\in T}
e
x
i
s
t
e
n
c
e
(
O
(
t
0
)
|
D
(
t
0
)
)
=
O
(
t
0
)
∩
D
(
t
0
)
{\displaystyle {existence}{\Big (}O(t_{0}){\Big |}D(t_{0}){\Big )}=O(t_{0})\cap D(t_{0})}
1)
e
x
i
s
t
e
n
c
e
(
O
|
D
)
=
O
∩
D
=
D
∩
O
=
e
x
i
s
t
e
n
c
e
(
D
|
O
)
{\displaystyle {existence}({\cal {O}}|{\cal {D}})={\cal {O}}\cap {\cal {D}}={\cal {D}}\cap {\cal {O}}={existence}({\cal {D}}|{\cal {O}})}
2)
e
x
i
s
t
e
n
c
e
(
D
|
D
)
=
D
∩
D
=
D
=
e
x
i
s
t
e
n
c
e
(
D
)
{\displaystyle {existence}({\cal {D}}|{\cal {D}})={\cal {D}}\cap {\cal {D}}={\cal {D}}={existence}({\cal {D}})}
3) Si
D
⊂
O
{\displaystyle {\cal {D}}\subset {\cal {O}}}
e
x
i
s
t
e
n
c
e
(
O
|
D
)
=
O
∩
D
=
D
=
e
x
i
s
t
e
n
c
e
(
D
)
{\displaystyle {existence}({\cal {O}}|{\cal {D}})={\cal {O}}\cap {\cal {D}}={\cal {D}}={existence}({\cal {D}})}
Attention, ici, essence a le même sens que celui de nature.
O
,
O
1
,
O
2
∈
P
(
T
)
{\displaystyle \displaystyle {{\cal {O}},{\cal {O}}_{1},{\cal {O}}_{2}\in {\cal {P}}({\cal {T}})}}
On pose
B
i
j
(
O
1
,
O
2
)
=
{
b
:
O
1
⟶
O
2
|
b
b
i
j
e
c
t
i
o
n
}
{\displaystyle \displaystyle {{\cal {B}}ij({\cal {O}}_{1},{\cal {O}}_{2})={\Big \{}b\,\,:\,\,{\cal {O}}_{1}\longrightarrow {\cal {O}}_{2}{\Big |}b\,\,bijection{\Big \}}}}
Si
∃
b
1
,
2
∈
B
i
j
(
O
1
,
O
2
)
{\displaystyle \exists b_{1,2}\in {\cal {B}}ij({\cal {O}}_{1},{\cal {O}}_{2})}
On dit que
O
2
{\displaystyle {\cal {O}}_{2}}
O
1
{\displaystyle {\cal {O}}_{1}}
b
1
,
2
{\displaystyle b_{1,2}}
et on note
O
2
=
e
s
s
e
n
c
e
(
O
1
|
b
1
,
2
)
{\displaystyle {\cal {O}}_{2}={essence}{\Big (}{\cal {O}}_{1}{\Big |}b_{1,2}{\Big )}}
Si l'on note
b
2
,
1
=
(
b
1
,
2
)
−
1
{\displaystyle b_{2,1}={(b_{1,2})}^{-1}}
b
1
,
2
{\displaystyle b_{1,2}}
On a
O
1
=
e
s
s
e
n
c
e
(
O
2
|
b
2
,
1
)
{\displaystyle {\cal {O}}_{1}={essence}{\Big (}{\cal {O}}_{2}{\Big |}b_{2,1}{\Big )}}
O
2
=
O
1
{\displaystyle {\cal {O}}_{2}={\cal {O}}_{1}}
et dans ce cas,
∃
b
1
,
1
∈
B
i
j
(
O
1
,
O
1
)
{\displaystyle \exists b_{1,1}\in {\cal {B}}ij({\cal {O}}_{1},{\cal {O}}_{1})}
O
1
{\displaystyle {\cal {O}}_{1}}
i
d
O
1
∈
B
i
j
(
O
1
,
O
1
)
{\displaystyle {id}_{{\cal {O}}_{1}}\in {\cal {B}}ij({\cal {O}}_{1},{\cal {O}}_{1})}
et
e
x
i
s
t
e
n
c
e
(
O
1
)
=
O
1
=
e
s
s
e
n
c
e
(
O
1
|
b
1
,
1
)
{\displaystyle {existence}({\cal {O}}_{1})={\cal {O}}_{1}={essence}({\cal {O}}_{1}|b_{1,1})}
O
1
=
O
1
(
t
1
)
∈
P
(
T
(
t
1
)
)
e
t
O
2
=
O
2
(
t
2
)
∈
P
(
T
(
t
2
)
)
{\displaystyle {\cal {O}}_{1}=O_{1}(t_{1})\in {\cal {P}}{\Big (}T(t_{1}){\Big )}\,\,et\,\,{\cal {O}}_{2}=O_{2}(t_{2})\in {\cal {P}}{\Big (}T(t_{2}){\Big )}}
t
1
,
t
2
∈
T
{\displaystyle t_{1},t_{2}\in T}
On pose
B
i
j
(
O
,
.
)
=
{
b
:
O
⟶
D
O
|
D
O
∈
P
(
T
)
e
t
b
b
i
j
e
c
t
i
o
n
}
{\displaystyle \displaystyle {{\cal {B}}ij({\cal {O}},.)={\Big \{}b\,\,:\,\,{\cal {O}}\,\,\longrightarrow \,\,{\cal {D}}_{\cal {O}}{\Big |}{\cal {D}}_{\cal {O}}\in {\cal {P}}({\cal {T}})\,\,et\,\,b\,\,bijection{\Big \}}}}
L'application essence est définie par :
e
s
s
e
n
c
e
(
O
)
:
B
i
j
(
O
,
.
)
⟶
P
(
T
)
:
b
⟼
e
s
s
e
n
c
e
(
O
|
b
)
{\displaystyle \displaystyle {{essence}({\cal {O}})\,\,:\,\,{\cal {B}}ij({\cal {O}},.)\longrightarrow {\cal {P}}({\cal {T}})\,\,:\,\,b\,\,\longmapsto \,\,{essence}({\cal {O}}|b)}}
O
=
O
(
t
0
)
∈
P
(
T
(
t
0
)
)
{\displaystyle {\cal {O}}=O(t_{0})\in {\cal {P}}{\Big (}T(t_{0}){\Big )}}
t
0
∈
T
{\displaystyle t_{0}\in T}
e
s
s
e
n
c
e
(
O
)
{\displaystyle {essence}({\cal {O}})}
{
e
s
s
e
n
c
e
(
O
|
b
)
|
b
∈
B
i
j
(
O
,
.
)
}
{\displaystyle \{{essence}({\cal {O}}|b)|b\in {\cal {B}}ij({\cal {O}},.)\}}
Soit
O
2
∈
P
(
T
)
{\displaystyle \displaystyle {{\cal {O}}_{2}\in {\cal {P}}({\cal {T}})}}
t
1
,
t
2
∈
T
:
t
1
<
t
2
{\displaystyle t_{1},t_{2}\in T\,\,:\,\,t_{1}<t_{2}}
O
1
,
0
(
t
1
)
=
e
s
s
e
n
c
e
s
p
i
n
o
z
i
s
t
e
(
O
2
(
t
2
)
)
{\displaystyle O_{1,0}(t_{1})={essence\,\,spinoziste}{\Big (}O_{2}(t_{2}){\Big )}}
⟺
d
e
´
f
{\displaystyle \Longleftrightarrow _{d{\acute {e}}f}}
O
1
,
0
∈
P
(
T
)
:
O
1
,
0
(
t
1
)
⟹
O
2
(
t
2
)
{\displaystyle \displaystyle {{\cal {O}}_{1,0}\in {\cal {P}}({\cal {T}})\,\,:\,\,O_{1,0}(t_{1})\,\,\Longrightarrow \,\,O_{2}(t_{2})}}
et
(
O
1
∈
P
(
T
)
:
O
1
(
t
1
)
⟹
O
2
(
t
2
)
)
⟹
O
1
,
0
(
t
1
)
⊂
O
1
(
t
1
)
{\displaystyle \displaystyle {{\bigg (}{\cal {O}}_{1}\in {\cal {P}}({\cal {T}})\,\,:\,\,O_{1}(t_{1})\,\,\Longrightarrow \,\,O_{2}(t_{2}){\bigg )}\,\,\Longrightarrow \,\,O_{1,0}(t_{1})\subset O_{1}(t_{1})}}
∀
O
∈
P
(
T
)
:
O
=
⨆
t
∈
T
O
(
t
)
{\displaystyle \displaystyle {\forall {\cal {O}}\in {\cal {P}}({\cal {T}})\,\,:\,\,{\cal {O}}=\bigsqcup _{t\in T}O(t)}}
On pose
∀
T
0
∈
P
(
T
)
,
O
|
T
0
=
⨆
t
∈
T
0
O
(
t
)
{\displaystyle \displaystyle {\forall T_{0}\in {\cal {P}}(T),\,\,{\cal {O}}_{|T_{0}}=\bigsqcup _{t\in T_{0}}O(t)}}
On pourrait pour faire preuve de plus d'abstraction, chercher des définitions qui ne s'appliquent pas, nécessairement, à notre monde, mais qui pourraient s'appliquer à d'autres.
Dans la suite, les cas
T
1
⋂
T
2
≠
∅
{\displaystyle T_{1}\bigcap T_{2}\neq \emptyset }
T
1
>
T
2
{\displaystyle T_{1}>T_{2}}
Soit
O
2
∈
P
(
T
)
{\displaystyle \displaystyle {{\cal {O}}_{2}\in {\cal {P}}({\cal {T}})}}
T
1
,
T
2
∈
P
(
T
)
:
T
1
<
T
2
{\displaystyle T_{1},T_{2}\in {\cal {P}}(T)\,\,:\,\,T_{1}<T_{2}}
O
1
,
0
|
T
1
=
e
s
s
e
n
c
e
s
p
i
n
o
z
i
s
t
e
(
O
2
|
T
2
)
{\displaystyle {{\cal {O}}_{1,0}}_{|T_{1}}={essence\,\,spinoziste}{\Big (}{{\cal {O}}_{2}}_{|T_{2}}{\Big )}}
⟺
d
e
´
f
{\displaystyle \Longleftrightarrow _{d{\acute {e}}f}}
O
1
,
0
∈
P
(
T
)
:
O
1
,
0
|
T
1
⟹
O
2
|
T
2
{\displaystyle \displaystyle {{\cal {O}}_{1,0}\in {\cal {P}}({\cal {T}})\,\,:\,\,{{\cal {O}}_{1,0}}_{|T_{1}}\,\,\Longrightarrow \,\,{{\cal {O}}_{2}}_{|T_{2}}}}
et
(
O
1
∈
P
(
T
)
:
O
1
|
T
1
⟹
O
2
|
T
2
)
⟹
O
1
,
0
|
T
1
⊂
O
1
|
T
1
{\displaystyle \displaystyle {{\bigg (}{\cal {O}}_{1}\in {\cal {P}}({\cal {T}})\,\,:\,\,{{\cal {O}}_{1}}_{|T_{1}}\,\,\Longrightarrow \,\,{{\cal {O}}_{2}}_{|T_{2}}{\bigg )}\,\,\Longrightarrow \,\,{{\cal {O}}_{1,0}}_{|T_{1}}\subset {{\cal {O}}_{1}}_{|T_{1}}}}
Ensemble de relations entre 2 corps
modifier
Soient
O
1
,
O
2
∈
P
(
T
)
{\displaystyle \displaystyle {{\cal {O}}_{1},{\cal {O}}_{2}\in {\cal {P}}({\cal {T}})}}
r
e
l
a
t
i
o
n
s
(
O
1
,
O
2
)
=
d
e
´
f
P
(
O
1
×
O
2
)
{\displaystyle {relations}({\cal {O}}_{1},{\cal {O}}_{2})=_{d{\acute {e}}f}{\cal {P}}({\cal {O}}_{1}\times {\cal {O}}_{2})}
Soient
O
1
,
O
2
∈
P
(
T
)
{\displaystyle \displaystyle {{\cal {O}}_{1},{\cal {O}}_{2}\in {\cal {P}}({\cal {T}})}}
On peut avoir
e
x
i
s
t
e
n
c
e
(
O
1
|
O
2
)
=
∅
{\displaystyle {existence}({\cal {O}}_{1}|{\cal {O}}_{2})=\emptyset }
mais si
O
1
,
O
2
≠
∅
{\displaystyle {\cal {O}}_{1},{\cal {O}}_{2}\neq \emptyset }
r
e
l
a
t
i
o
n
s
(
O
1
,
O
2
)
≠
∅
{\displaystyle {relations}({\cal {O}}_{1},{\cal {O}}_{2})\neq \emptyset }
Remarque :
Pour ce qui concerne la puissance d'interaction totale d'un corps principal :
Pour la puissance externe de ce corps :
Il faut considérer une partition dynamique, dans le temps, composée de ce corps principal et des autres (ou d'autres) corps, interagissant, avec lui
(
1
)
{\displaystyle (1)}
Pour la puissance interne de ce corps principal :
Il faut considérer une partition dynamique, dans le temps, de ce corps principal
(
2
)
{\displaystyle (2)}
Pour couvrir, entièrement, les réseaux formés des corps de chaque partition, on considère des maillages triangulaires de corps :
A chaque maillon du maillage triangulaire de corps, on considère la puissance élémentaire des interactions mutuelles des
3
{\displaystyle 3}
(
1
)
{\displaystyle (1)}
t
{\displaystyle t}
t
{\displaystyle t}
(
2
)
{\displaystyle (2)}
t
{\displaystyle t}
t
{\displaystyle t}
t
{\displaystyle t}
Dans la définition de la puissance élémentaire entre 3 corps :
J'ai tenté de donner des conditions d'intégrabilité, afin que mes intégrales soient parfaitement définies.
Tant qu'à faire, j'ai généralisé la définition de puissance externe, interne et totale, d'interaction, à tout type d'intégrale et de mesure, en particulier à la mesure de comptage sur
N
n
∗
{\displaystyle \mathbb {N} _{n}^{*}}
Comme vous pouvez le constater, les idées mises en oeuvre, sont simples.
Remarque importante : On se place, ici, dans le cadre de la mécanique newtonienne : L'espace
E
×
T
{\displaystyle E\times T}
E
×
T
⊂
R
n
×
R
{\displaystyle E\times T\subset \mathbb {R} ^{n}\times \mathbb {R} }
n
∈
N
∗
{\displaystyle n\in \mathbb {N} ^{*}}
Remarque : La plupart des ensembles considérés reposent, essentiellement, sur des partitions.
I
∈
P
(
E
×
T
)
{\displaystyle I\in {\cal {P}}(E\times T)}
La donnée de cet ensemble induit une application
I
^
:
T
⟶
P
(
E
×
T
)
:
t
⟼
I
^
(
t
)
=
I
∩
(
E
×
{
t
}
)
{\displaystyle {\widehat {I}}\,\,:\,\,T\,\,\longrightarrow \,\,{\cal {P}}(E\times T)\,\,:\,\,t\,\,\longmapsto \,\,{\widehat {I}}(t)=I\cap (E\times \{t\})}
C'est la "tranche" de
I
{\displaystyle I}
t
{\displaystyle t}
On a
I
=
⨆
i
∈
T
I
^
(
t
)
∈
P
(
E
×
T
)
{\displaystyle \displaystyle {I=\bigsqcup _{i\in T}{\widehat {I}}(t)\in {\cal {P}}(E\times T)}}
Par construction,
∀
t
∈
T
,
I
^
(
t
)
∈
P
(
E
×
{
t
}
)
{\displaystyle \forall t\in T,\,\,{\widehat {I}}(t)\in {\cal {P}}(E\times \{t\})}
I
^
:
T
⟶
P
(
E
×
T
)
:
t
⟼
I
^
(
t
)
∈
P
(
E
×
{
t
}
)
{\displaystyle {\widehat {I}}\,\,:\,\,T\longrightarrow {\cal {P}}(E\times T)\,\,:\,\,t\,\,\longmapsto \,\,{\widehat {I}}(t)\in {\cal {P}}(E\times \{t\})}
où
I
=
⨆
t
∈
T
I
^
(
t
)
{\displaystyle \displaystyle {I=\bigsqcup _{t\in T}{\widehat {I}}(t)}}
t
∈
T
e
t
(
i
,
t
)
∈
I
^
(
t
)
{\displaystyle t\in T\,\,et\,\,(i,t)\in {\widehat {I}}(t)}
⟺
t
∈
T
e
t
(
i
,
t
)
∈
I
∩
(
E
×
{
t
}
)
{\displaystyle \iff t\in T\,\,et\,\,(i,t)\in I\cap (E\times \{t\})}
⟺
t
∈
T
e
t
(
i
,
t
)
∈
I
e
t
(
i
,
t
)
∈
E
×
{
t
}
{\displaystyle \iff t\in T\,\,et\,\,(i,t)\in I\,\,et\,\,(i,t)\in E\times \{t\}}
⟺
t
∈
T
e
t
(
i
,
t
)
∈
I
e
t
i
∈
E
e
t
t
=
t
{\displaystyle \iff t\in T\,\,et\,\,(i,t)\in I\,\,et\,\,i\in E\,\,et\,\,t=t}
⟺
(
i
,
t
)
∈
I
e
t
(
i
,
t
)
∈
E
×
T
{\displaystyle \iff (i,t)\in I\,\,et\,\,(i,t)\in E\times T}
⟺
(
i
,
t
)
∈
I
{\displaystyle \iff (i,t)\in I}
I
∈
P
(
E
×
T
)
{\displaystyle I\in {\cal {P}}(E\times T)}
I
∈
P
(
E
×
T
)
{\displaystyle I\in {\cal {P}}(E\times T)}
On pose
A
^
(
I
,
E
,
T
)
{\displaystyle \displaystyle {{\widehat {\cal {A}}}(I,E,T)}}
=
{
I
^
|
I
^
:
T
⟶
P
(
E
×
T
)
:
t
⟼
I
^
(
t
)
=
I
∩
(
E
×
{
t
}
)
}
{\displaystyle \displaystyle {={\Big \{}{\widehat {I}}{\Big |}{\widehat {I}}\,\,:\,\,T\longrightarrow {\cal {P}}(E\times T)\,\,:\,\,t\,\,\longmapsto \,\,{\widehat {I}}(t)=I\cap (E\times \{t\}){\Big \}}}}
=
{
I
^
|
I
^
:
T
⟶
P
(
E
×
T
)
:
t
⟼
I
^
(
t
)
∈
P
(
E
×
{
t
}
)
e
t
I
=
⨆
t
∈
T
I
^
(
t
)
}
{\displaystyle \displaystyle {={\Big \{}{\widehat {I}}{\Big |}{\widehat {I}}\,\,:\,\,T\longrightarrow {\cal {P}}(E\times T)\,\,:\,\,t\,\,\longmapsto \,\,{\widehat {I}}(t)\in {\cal {P}}(E\times \{t\})\,\,et\,\,I=\bigsqcup _{t\in T}{\widehat {I}}(t){\Big \}}}}
A
^
(
E
,
T
)
=
{
I
^
∈
A
^
(
I
,
E
,
T
)
|
I
∈
P
(
E
×
T
)
}
{\displaystyle \displaystyle {{\widehat {\cal {A}}}(E,T)={\Big \{}{\widehat {I}}\in {\widehat {\cal {A}}}(I,E,T){\Big |}I\in {\cal {P}}(E\times T){\Big \}}}}
J
∈
P
(
E
×
T
)
{\displaystyle J\in {\cal {P}}(E\times T)}
On pose
A
^
(
J
,
I
,
E
,
T
)
{\displaystyle \displaystyle {{\widehat {\cal {A}}}(J,I,E,T)}}
=
{
I
^
(
J
)
∈
A
^
(
I
,
E
,
T
)
|
J
^
∈
A
^
(
J
,
E
,
T
)
,
∀
t
∈
T
,
J
^
(
t
)
∈
P
(
I
^
(
J
)
(
t
)
)
e
t
∀
(
j
,
t
)
∈
J
,
∃
I
^
(
J
)
(
j
,
t
)
∈
P
(
I
^
(
t
)
)
:
{\displaystyle \displaystyle {={\Big \{}{\widehat {I}}(J)\in {\widehat {\cal {A}}}(I,E,T){\Big |}{\widehat {J}}\in {\widehat {\cal {A}}}(J,E,T),\,\,\forall t\in T,\,\,{\widehat {J}}(t)\in {\mathcal {P}}{\Big (}{\widehat {I}}(J)(t){\Big )}\,\,et\,\,\forall (j,t)\in J,\,\,\exists {\widehat {I}}(J)(j,t)\in {\cal {P}}{\Big (}{\widehat {I}}(t){\Big )}\,\,:}}
∀
t
∈
T
,
⨆
(
j
,
t
)
∈
J
^
(
t
)
I
^
(
J
)
(
j
,
t
)
=
I
^
(
J
)
(
t
)
}
{\displaystyle \displaystyle {\forall t\in T,\,\,\bigsqcup _{(j,t)\in {\widehat {J}}(t)}{\widehat {I}}(J)(j,t)={\widehat {I}}(J)(t){\Big \}}}}
A
^
(
J
,
.
,
E
,
T
)
=
{
I
^
(
J
)
∈
A
^
(
J
,
I
,
E
,
T
)
|
I
∈
P
(
E
×
T
)
}
{\displaystyle \displaystyle {{\widehat {\cal {A}}}(J,.,E,T)={\Big \{}{\widehat {I}}(J)\in {\widehat {\cal {A}}}(J,I,E,T){\Big |}I\in {\cal {P}}(E\times T){\Big \}}}}
P
^
^
(
I
)
:
I
⟶
P
(
E
×
T
)
:
(
i
,
t
)
⟼
P
^
^
(
i
,
t
)
∈
P
(
E
×
{
t
}
)
{\displaystyle {\widehat {\widehat {P}}}(I)\,\,:\,\,I\longrightarrow {\cal {P}}(E\times T)\,\,:\,\,(i,t)\,\,\longmapsto \,\,{\widehat {\widehat {P}}}(i,t)\in {\cal {P}}(E\times \{t\})}
∀
(
i
1
,
t
1
)
,
(
i
2
,
t
2
)
∈
I
,
(
i
1
,
t
1
)
≠
(
i
2
,
t
2
)
,
P
^
^
(
I
)
(
i
1
,
t
1
)
⋂
P
^
^
(
I
)
(
i
2
,
t
2
)
=
∅
{\displaystyle \displaystyle {\forall (i_{1},t_{1}),(i_{2},t_{2})\in I,\,\,(i_{1},t_{1})\neq (i_{2},t_{2}),\,\,{\widehat {\widehat {P}}}(I)(i_{1},t_{1})\bigcap {\widehat {\widehat {P}}}(I)(i_{2},t_{2})=\emptyset }}
c'est-à-dire
(
P
^
^
(
I
)
(
i
,
t
)
)
(
i
,
t
)
∈
I
p
a
r
t
i
t
i
o
n
{\displaystyle \displaystyle {{{\Big (}{\widehat {\widehat {P}}}(I)(i,t){\Big )}}_{(i,t)\in I}\,\,partition}}
A
^
^
(
I
,
E
,
T
)
{\displaystyle \displaystyle {{\widehat {\widehat {\cal {A}}}}(I,E,T)}}
=
{
P
^
^
(
I
)
|
P
^
^
(
I
)
:
I
⟶
P
(
E
×
T
)
:
(
i
,
t
)
⟼
P
^
^
(
I
)
(
i
,
t
)
∈
P
(
E
×
{
t
}
)
{\displaystyle \displaystyle {={\Big \{}{\widehat {\widehat {P}}}(I){\Big |}{\widehat {\widehat {P}}}(I)\,\,:\,\,I\longrightarrow {\cal {P}}(E\times T)\,\,:\,\,(i,t)\longmapsto {\widehat {\widehat {P}}}(I)(i,t)\in {\cal {P}}(E\times \{t\})}}
t
e
l
l
e
q
u
e
∀
(
i
1
,
t
1
)
,
(
i
2
,
t
2
)
∈
I
,
(
i
1
,
t
1
)
≠
(
i
2
,
t
2
)
,
P
^
^
(
I
)
(
i
1
,
t
1
)
⋂
P
^
^
(
I
)
(
i
2
,
t
2
)
=
∅
}
{\displaystyle \displaystyle {telle\,\,que\,\,\forall (i_{1},t_{1}),(i_{2},t_{2})\in I,\,\,(i_{1},t_{1})\neq (i_{2},t_{2}),\,\,{\widehat {\widehat {P}}}(I)(i_{1},t_{1})\bigcap {\widehat {\widehat {P}}}(I)(i_{2},t_{2})=\emptyset }{\Big \}}}
=
{
P
^
^
(
I
)
|
P
^
^
(
I
)
:
I
⟶
P
(
E
×
T
)
:
(
i
,
t
)
⟼
P
^
^
(
I
)
(
i
,
t
)
∈
P
(
E
×
{
t
}
)
t
e
l
l
e
q
u
e
(
P
^
^
(
I
)
(
i
,
t
)
)
(
i
,
t
)
∈
I
p
a
r
t
i
t
i
o
n
}
{\displaystyle \displaystyle {={\Big \{}{\widehat {\widehat {P}}}(I){\Big |}{\widehat {\widehat {P}}}(I)\,\,:\,\,I\longrightarrow {\cal {P}}(E\times T)\,\,:\,\,(i,t)\longmapsto {\widehat {\widehat {P}}}(I)(i,t)\in {\cal {P}}(E\times \{t\})\,\,telle\,\,que\,\,{{\Big (}{\widehat {\widehat {P}}}(I)(i,t){\Big )}}_{(i,t)\in I}\,\,partition}{\Big \}}}
A
^
^
(
E
,
T
)
=
{
P
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
|
I
∈
P
(
E
×
T
)
}
{\displaystyle \displaystyle {{\widehat {\widehat {\cal {A}}}}(E,T)={\Big \{}{\widehat {\widehat {P}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T){\Big |}I\in {\cal {P}}(E\times T){\Big \}}}}
P
^
^
(
I
)
{\displaystyle {\widehat {\widehat {P}}}(I)}
{
P
^
^
(
I
)
(
i
,
t
)
|
(
i
,
t
)
∈
I
}
=
{
P
^
^
(
I
)
(
i
,
t
)
|
t
∈
T
e
t
(
i
,
t
)
∈
I
^
(
t
)
}
{\displaystyle \displaystyle {\{{\widehat {\widehat {P}}}(I)(i,t)|(i,t)\in I\}=\{{\widehat {\widehat {P}}}(I)(i,t)|t\in T\,\,et\,\,(i,t)\in {\widehat {I}}(t)\}}}
P
(
I
)
=
⨆
t
∈
T
P
^
^
(
I
)
(
t
)
=
⨆
t
∈
T
⨆
(
i
,
t
)
∈
I
^
(
t
)
P
^
^
(
I
)
(
i
,
t
)
=
⨆
(
i
,
t
)
∈
I
P
^
^
(
I
)
(
i
,
t
)
o
u
`
P
^
^
(
I
)
(
t
)
=
⨆
i
∈
I
^
(
t
)
P
^
^
(
I
)
(
i
,
t
)
{\displaystyle \displaystyle {P(I)=\bigsqcup _{t\in T}{\widehat {\widehat {P}}}(I)(t)=\bigsqcup _{t\in T}\bigsqcup _{(i,t)\in {\widehat {I}}(t)}{\widehat {\widehat {P}}}(I)(i,t)=\bigsqcup _{(i,t)\in I}{\widehat {\widehat {P}}}(I)(i,t)\,\,o{\grave {u}}\,\,{\widehat {\widehat {P}}}(I)(t)=\bigsqcup _{i\in {\widehat {I}}(t)}{\widehat {\widehat {P}}}(I)(i,t)}}
on a
∀
t
1
,
t
2
∈
T
,
t
1
≠
t
2
,
P
^
^
(
I
)
(
t
1
)
⋂
P
^
^
(
I
)
(
t
2
)
=
∅
{\displaystyle \displaystyle {\forall t_{1},t_{2}\in T,\,\,t_{1}\neq t_{2},\,\,{\widehat {\widehat {P}}}(I)(t_{1})\bigcap {\widehat {\widehat {P}}}(I)(t_{2})=\emptyset }}
c'est-à-dire
∀
(
i
1
,
t
1
)
,
(
i
2
,
t
2
)
∈
I
,
t
1
≠
t
2
,
(
i
1
,
t
1
)
≠
(
i
2
,
t
2
)
,
P
^
^
(
I
)
(
i
1
,
t
1
)
⋂
P
^
^
(
I
)
(
i
2
,
t
2
)
=
∅
{\displaystyle \displaystyle {\forall (i_{1},t_{1}),(i_{2},t_{2})\in I,\,\,t_{1}\neq t_{2},\,\,(i_{1},t_{1})\neq (i_{2},t_{2}),\,\,{\widehat {\widehat {P}}}(I)(i_{1},t_{1})\bigcap {\widehat {\widehat {P}}}(I)(i_{2},t_{2})=\emptyset }}
I
,
Q
∈
P
(
E
×
T
)
{\displaystyle I,Q\in {\cal {P}}(E\times T)}
Soit
P
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
{\displaystyle {\widehat {\widehat {P}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T)}
On pose
A
^
0
(
I
,
Q
,
P
^
^
(
I
)
,
E
,
T
)
=
{
Q
^
∈
A
^
(
Q
,
E
,
T
)
|
∀
(
i
,
t
)
∈
I
,
Q
^
(
t
)
⋂
P
^
^
(
I
)
(
i
,
t
)
=
∅
}
{\displaystyle \displaystyle {{\widehat {\cal {A}}}_{0}{\Big (}I,Q,{\widehat {\widehat {P}}}(I),E,T{\Big )}=\{{\widehat {Q}}\in {\widehat {\cal {A}}}(Q,E,T)|\forall (i,t)\in I,\,\,{\widehat {Q}}(t)\bigcap {\widehat {\widehat {P}}}(I)(i,t)=\emptyset \}}}
A
^
0
(
I
,
.
,
P
^
^
(
I
)
,
E
,
T
)
=
{
Q
^
∈
A
^
0
(
I
,
Q
,
P
^
^
(
I
)
,
E
,
T
)
|
Q
∈
P
(
E
×
T
)
}
{\displaystyle \displaystyle {{\widehat {\cal {A}}}_{0}{\Big (}I,.,{\widehat {\widehat {P}}}(I),E,T{\Big )}={\Big \{}{\widehat {Q}}\in {\widehat {\cal {A}}}_{0}{\Big (}I,Q,{\widehat {\widehat {P}}}(I),E,T{\Big )}{\Big |}Q\in {\cal {P}}(E\times T){\Big \}}}}
I
∈
P
(
E
×
T
)
{\displaystyle I\in {\cal {P}}(E\times T)}
Soit
I
^
∈
A
(
I
,
E
,
T
)
{\displaystyle {\widehat {I}}\in {\cal {A}}(I,E,T)}
Soit
t
∈
T
{\displaystyle t\in T}
Soit
E
(
t
)
{\displaystyle {\cal {E}}(t)}
I
^
(
t
)
{\displaystyle {\widehat {I}}(t)}
Soit
μ
(
t
)
:
E
(
t
)
⟶
R
¯
{\displaystyle \mu (t)\,\,:\,\,{\cal {E}}(t)\,\,\longrightarrow \,\,{\overline {\mathbb {R} }}}
Soit
E
{\displaystyle {\cal {E}}}
I
{\displaystyle I}
Soit
μ
:
E
⟶
R
¯
{\displaystyle \mu \,\,:\,\,{\cal {E}}\,\,\longrightarrow \,\,{\overline {\mathbb {R} }}}
P
u
i
s
s
a
n
c
e
(
μ
)
:
P
3
(
E
×
T
)
⟶
R
¯
:
(
A
,
B
,
C
)
⟼
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
,
B
,
C
)
{\displaystyle \displaystyle {{{\cal {P}}uissance}(\mu )\,\,:\,\,{\cal {P}}^{3}(E\times T)\,\,\longrightarrow \,\,{\overline {\mathbb {R} }}\,\,:\,\,(A,B,C)\,\,\longmapsto \,\,{{\cal {P}}uissance}(\mu )(A,B,C)}}
∀
(
A
,
B
1
,
B
2
)
∈
P
3
(
E
×
T
)
,
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
,
B
1
,
B
2
)
=
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
,
B
2
,
B
1
)
{\displaystyle \forall (A,B_{1},B_{2})\in {\cal {P}}^{3}(E\times T),\,\,{{\cal {P}}uissance}(\mu )(A,B_{1},B_{2})={{\cal {P}}uissance}(\mu )(A,B_{2},B_{1})}
∀
(
A
,
B
,
C
)
∈
P
3
(
E
×
T
)
{\displaystyle \forall (A,B,C)\in {\cal {P}}^{3}(E\times T)}
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
,
B
,
C
)
=
P
u
i
s
s
a
n
c
e
(
μ
)
(
B
,
A
,
C
)
=
P
u
i
s
s
a
n
c
e
(
μ
)
(
C
,
A
,
B
)
{\displaystyle \displaystyle {{{\cal {P}}uissance}(\mu )(A,B,C)={{\cal {P}}uissance}(\mu )(B,A,C)={{\cal {P}}uissance}(\mu )(C,A,B)}}
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
,
B
,
C
)
{\displaystyle {{\cal {P}}uissance}(\mu )(A,B,C)}
s'appelle la puissance des interactions de
A
{\displaystyle A}
en interaction avec les interactions de
B
{\displaystyle B}
C
{\displaystyle C}
par rapport à la mesure
μ
{\displaystyle \mu }
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
,
B
,
B
)
{\displaystyle {{\cal {P}}uissance}(\mu )(A,B,B)}
A
{\displaystyle A}
B
{\displaystyle B}
B
{\displaystyle B}
A
{\displaystyle A}
B
{\displaystyle B}
(Dans la définition de la puissance élémentaire entre 3 corps :
J'ai tenté de donner des conditions d'intégrabilité, afin que mes intégrales soient parfaitement définies.)
Notations :
f
:
E
×
F
⟶
H
{\displaystyle f\,\,:\,\,E\times F\longrightarrow H}
E
0
×
F
0
∈
P
(
E
×
F
)
{\displaystyle E_{0}\times F_{0}\in {\cal {P}}(E\times F)}
Soit
h
:
H
⟶
K
{\displaystyle h\,\,:\,\,H\,\,\longrightarrow \,\,K}
Alors
f
|
E
0
×
F
0
(
.
,
.
)
=
f
|
E
0
×
F
0
{\displaystyle f{|}_{E_{0}\times F_{0}}(.,.)=f{|}_{E_{0}\times F_{0}}}
et
h
(
f
(
.
,
.
)
)
|
E
0
×
F
0
=
(
h
∘
f
)
|
E
0
×
F
0
(
.
,
.
)
=
(
h
∘
f
)
|
E
0
×
F
0
{\displaystyle h{\Big (}f(.,.){\Big )}{\Big |}_{E_{0}\times F_{0}}=(h\circ f){|}_{E_{0}\times F_{0}}(.,.)=(h\circ f){|}_{E_{0}\times F_{0}}}
Soient
g
:
E
×
F
×
G
⟶
H
{\displaystyle g\,\,:\,\,E\times F\times G\longrightarrow H}
E
0
×
F
0
×
G
0
∈
P
(
E
×
F
×
G
)
{\displaystyle E_{0}\times F_{0}\times G_{0}\in {\cal {P}}(E\times F\times G)}
Soit
k
:
H
⟶
K
{\displaystyle k\,\,:\,\,H\,\,\longrightarrow \,\,K}
Alors
g
|
E
0
×
F
0
×
G
0
(
.
,
.
,
.
)
=
g
|
E
0
×
F
0
×
G
0
{\displaystyle g{|}_{E_{0}\times F_{0}\times G_{0}}(.,.,.)=g{|}_{E_{0}\times F_{0}\times G_{0}}}
et
k
(
g
(
.
,
.
,
.
)
)
|
E
0
×
F
0
×
G
0
=
(
k
∘
g
)
|
E
0
×
F
0
×
G
0
(
.
,
.
,
.
)
=
(
k
∘
g
)
|
E
0
×
F
0
×
G
0
{\displaystyle k{\Big (}g(.,.,.){\Big )}{\Big |}_{E_{0}\times F_{0}\times G_{0}}=(k\circ g){|}_{E_{0}\times F_{0}\times G_{0}}(.,.,.)=(k\circ g){|}_{E_{0}\times F_{0}\times G_{0}}}
t
∈
T
{\displaystyle t\in T}
Soit
A
∈
P
(
E
×
T
)
{\displaystyle A\in {\cal {P}}(E\times T)}
∀
A
^
∈
A
^
(
A
,
E
,
T
)
,
∀
B
^
^
(
I
)
,
C
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
,
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
^
(
t
)
,
B
^
^
(
I
)
(
.
,
t
)
,
C
^
^
(
I
)
(
.
,
t
)
)
|
{
i
∈
E
|
(
i
,
t
)
∈
I
}
2
{\displaystyle \forall {\widehat {A}}\in {\widehat {\cal {A}}}(A,E,T),\forall {\widehat {\widehat {B}}}(I),{\widehat {\widehat {C}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T),\,\,{{\cal {P}}uissance}(\mu ){\Big (}{\widehat {A}}(t),{\widehat {\widehat {B}}}(I)(.,t),{\widehat {\widehat {C}}}(I)(.,t){\Big )}{|}_{\{i\in E|(i,t)\in I\}^{2}}}
∈
L
1
(
{
i
∈
E
|
(
i
,
t
)
∈
I
}
2
,
?
,
?
)
{\displaystyle \in L^{1}{\Big (}\{i\in E|(i,t)\in I\}^{2},\,\,?,\,\,?{\Big )}}
∀
A
^
∈
A
^
(
A
,
E
,
T
)
,
∀
B
^
^
(
I
)
,
C
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
,
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
^
(
.
)
,
B
^
^
(
I
)
(
.
,
.
)
,
C
^
^
(
I
)
(
.
,
.
)
)
|
T
×
I
2
{\displaystyle \forall {\widehat {A}}\in {\widehat {\cal {A}}}(A,E,T),\forall {\widehat {\widehat {B}}}(I),{\widehat {\widehat {C}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T),\,\,{{\cal {P}}uissance}(\mu ){\Big (}{\widehat {A}}(.),{\widehat {\widehat {B}}}(I)(.,.),{\widehat {\widehat {C}}}(I)(.,.){\Big )}{|}_{T\times {I}^{2}}}
∈
L
1
(
T
×
I
2
,
B
(
T
)
⊗
E
⊗
2
,
λ
⊗
μ
⊗
2
)
{\displaystyle \in L^{1}{\Big (}T\times {I}^{2},\,\,{\cal {B}}(T)\otimes {\cal {E}}^{\otimes 2},\,\,\lambda \otimes \mu ^{\otimes 2}{\Big )}}
∀
t
∈
T
,
{\displaystyle \forall t\in T,}
∀
A
^
∈
A
^
(
I
,
A
,
E
,
T
)
,
∀
B
^
^
(
I
)
,
C
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
,
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
^
(
I
)
(
.
,
t
)
,
B
^
^
(
I
)
(
.
,
t
)
,
C
^
^
(
I
)
(
.
,
t
)
)
|
{
i
∈
E
|
(
i
,
t
)
∈
I
}
3
{\displaystyle \forall {\widehat {A}}\in {\widehat {\cal {A}}}(I,A,E,T),\forall {\widehat {\widehat {B}}}(I),{\widehat {\widehat {C}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T),\,\,{{\cal {P}}uissance}(\mu ){\Big (}{\widehat {A}}(I)(.,t),{\widehat {\widehat {B}}}(I)(.,t),{\widehat {\widehat {C}}}(I)(.,t){\Big )}{|}_{\{i\in E|(i,t)\in I\}^{3}}}
∈
L
1
(
{
i
∈
E
|
(
i
,
t
)
∈
I
}
3
,
?
,
?
)
{\displaystyle \in L^{1}{\Big (}\{i\in E|(i,t)\in I\}^{3},\,\,?,\,\,?{\Big )}}
∀
A
^
∈
A
^
(
I
,
A
,
E
,
T
)
,
B
^
^
(
I
)
,
C
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
,
P
u
i
s
s
a
n
c
e
(
μ
)
(
A
^
(
I
)
(
.
,
.
)
,
B
^
^
(
I
)
(
.
,
.
)
,
C
^
^
(
I
)
(
.
,
.
)
)
|
I
3
{\displaystyle \forall {\widehat {A}}\in {\widehat {\cal {A}}}(I,A,E,T),{\widehat {\widehat {B}}}(I),{\widehat {\widehat {C}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T),\,\,{{\cal {P}}uissance}(\mu ){\Big (}{\widehat {A}}(I)(.,.),{\widehat {\widehat {B}}}(I)(.,.),{\widehat {\widehat {C}}}(I)(.,.){\Big )}{|}_{{I}^{3}}}
∈
L
1
(
I
3
,
E
⊗
3
,
μ
⊗
3
)
{\displaystyle \in L^{1}{\Big (}{I}^{3},\,\,{\cal {E}}^{\otimes 3},\,\,\mu ^{\otimes 3}{\Big )}}
En tenant compte des notations de remarque et de ce qui précède et en supprimant et en ne prenant pas en compte les doublons de
P
u
i
s
s
a
n
c
e
(
μ
)
(
O
^
(
t
)
,
P
^
^
(
I
)
(
i
1
,
t
)
,
P
^
^
(
I
)
(
i
2
,
t
)
)
{\displaystyle {{\cal {P}}uissance}(\mu ){\Big (}{\widehat {O}}(t),{\widehat {\widehat {P}}}(I)(i_{1},t),{\widehat {\widehat {P}}}(I)(i_{2},t){\Big )}}
dans les expressions suivantes, on obtient :
Soit
I
∈
P
(
E
×
T
)
{\displaystyle I\in {\cal {P}}(E\times T)}
Soit
t
∈
T
{\displaystyle t\in T}
Soit
E
(
t
)
{\displaystyle {\cal {E}}(t)}
I
^
(
t
)
{\displaystyle {\widehat {I}}(t)}
Soit
μ
(
t
)
:
E
(
t
)
⟶
R
¯
{\displaystyle \mu (t)\,\,:\,\,{\cal {E}}(t)\,\,\longrightarrow \,\,{\overline {\mathbb {R} }}}
Soit
P
^
^
(
I
)
∈
A
^
^
(
I
,
E
,
T
)
{\displaystyle {\widehat {\widehat {P}}}(I)\in {\widehat {\widehat {\cal {A}}}}(I,E,T)}
P
u
i
s
s
a
n
c
e
e
x
t
e
r
n
e
(
I
,
μ
,
P
u
i
s
s
a
n
c
e
(
μ
)
)
(
t
,
.
,
P
^
^
(
I
)
)
{\displaystyle {{\cal {P}}uissance}_{externe}{\Big (}I,\mu ,{{\cal {P}}uissance}(\mu ){\Big )}{\Big (}t,.,{\widehat {\widehat {P}}}(I){\Big )}}
P
u
i
s
s
a
n
c
e
e
x
t
e
r
n
e
(
I
,
μ
,
P
u
i
s
s
a
n
c
e
(
μ
)
)
(
t
,
.
,
P
^
^
(
I
)
)
:
A
^
0
(
I
,
.
,
P
^
^
(
I
)
,
E
,
T
)
⟶
R
¯
:
O
^
⟼
P
u
i
s
s
a
n
c
e
e
x
t
e
r
n
e
(
I
,
μ
,
P
u
i
s
s
a
n
c
e
(
μ
)
)
(
t
,
O
^
,
P
^
^
(
I
)
)
{\displaystyle \displaystyle {{{\cal {P}}uissance}_{externe}{\Big (}I,\mu ,{{\cal {P}}uissance}(\mu ){\Big )}{\Big (}t,.,{\widehat {\widehat {P}}}(I){\Big )}\,\,:\,\,{\widehat {\cal {A}}}_{0}(I,.,{\widehat {\widehat {P}}}(I),E,T)\,\,\longrightarrow \,\,{\overline {\mathbb {R} }}\,\,:\,\,{\widehat {O}}\longmapsto {{\cal {P}}uissance}_{externe}{\Big (}I,\mu ,{{\cal {P}}uissance}(\mu ){\Big )}{\Big (}t,{\widehat {O}},{\widehat {\widehat {P}}}(I){\Big )}}}
où
P
u
i
s
s
a
n
c
e
e
x
t
e
r
n
e
(
I
,
μ
,
P
u
i
s
s
a
n
c
e
(
μ
)
)
(
t
,
O
^
,
P
^
^
(
I
)
)
{\displaystyle {{\cal {P}}uissance}_{externe}{\Big (}I,\mu ,{{\cal {P}}uissance}(\mu ){\Big )}{\Big (}t,{\widehat {O}},{\widehat {\widehat {P}}}(I){\Big )}}
=
1
2
∫
I
^
(
t
)
∫
I
^
(
t
)
I
(
i
1
≠
i
2
)
(
i
1
,
i
2
)
P
u
i
s
s
a
n
c
e
(
μ
)
(
O
^
(
t
)
,
P
^
^
(
I
)
(
i
1
,
t
)
,
P
^
^
(
I
)
(
i
2
,
t
)
)
d
μ
(
t
)
(
i
1
,
t
)
d
μ
(
t
)
(
i
2
,
t
)
{\displaystyle \displaystyle {={\frac {1}{2}}\int _{{\widehat {I}}(t)}\int _{{\widehat {I}}(t)}{\mathbb {I} }_{(i_{1}\neq i_{2})}(i_{1},i_{2}){{\cal {P}}uissance}(\mu ){\Big (}{\widehat {O}}(t),{\widehat {\widehat {P}}}(I)(i_{1},t),{\widehat {\widehat {P}}}(I)(i_{2},t){\Big )}\,\,d\mu (t)(i_{1},t)\,\,d\mu (t)(i_{2},t)}}
+
∫
I
^
(
t
)
P
u
i
s
s
a
n
c
e
(
μ
)
(
O
^
(
t
)
,
P
^
^
(
I
)
(
i
1
,
t
)
,
P
^
^
(
I
)
(
i
1
,
t
)
)
d
μ
(
t
)
(
i
1
,
t
)
{\displaystyle \displaystyle {+\int _{{\widehat {I}}(t)}{{\cal {P}}uissance}(\mu ){\Big (}{\widehat {O}}(t),{\widehat {\widehat {P}}}(I)(i_{1},t),{\widehat {\widehat {P}}}(I)(i_{1},t){\Big )}\,\,d\mu (t)(i_{1},t)}}
∀
t
≥
o
u
>
sup
(
T
)
{\displaystyle \forall t\geq \,\,ou\,\,>\sup(T)}
P
u
i
s
s
a
n
c
e
e
x
t
e
r
n
e
(
I
,
μ
,
P
u
i
s
s
a
n
c
e
(
μ
)
)
(
t
,
O
^
,
P
^
^
(
I
)
)
{\displaystyle {{\cal {P}}uissance}_{externe}{\Big (}I,\mu ,{{\cal {P}}uissance}(\mu ){\Big )}{\Big (}t,{\widehat {O}},{\widehat {\widehat {P}}}(I){\Big )}}
1
2
∫
I
^
(
t
)
∫
I
^
(
t
)
I
(
i
1
≠
i
2
)
(
i
1
,
i
2
)
P
u
i
s
s
a
n
c
e
(
μ
)
(
O
^
,
P
^
^
(
I
)
(
i
1
,
t
)
,
P
^
^
(
I
)
(
i
2
,
t
)
)
d
μ
(
t
)
(
i
1
,
t
)
d
μ
(
t
)
(
i
2
,
t
)
{\displaystyle \displaystyle {{\frac {1}{2}}\int _{{\widehat {I}}(t)}\int _{{\widehat {I}}(t)}{\mathbb {I} }_{(i_{1}\neq i_{2})}(i_{1},i_{2}){{\cal {P}}uissance}(\mu ){\Big (}{\widehat {O}},{\widehat {\widehat {P}}}(I)(i_{1},t),{\widehat {\widehat {P}}}(I)(i_{2},t){\Big )}\,\,d\mu (t)(i_{1},t)\,\,d\mu (t)(i_{2},t)}}
+
∫
I
^
(
t
)
P
u
i
s
s
a
n
c
e
(
μ
)
(
O
^
,
P
^
^
(
I
)
(
i
1
,
t
)
,
P
^
^
(
I
)
(
i
1
,
t
)
)
d
μ
(
t
)
(
i
1
,
t
)
{\displaystyle \displaystyle {+\int _{{\widehat {I}}(t)}{{\cal {P}}uissance}(\mu ){\Big (}{\widehat {O}},{\widehat {\widehat {P}}}(I)(i_{1},t),{\widehat {\widehat {P}}}(I)(i_{1},t){\Big )}\,\,d\mu (t)(i_{1},t)}}
Soit
t
∈
T
{\displaystyle t\in T}
Soit
O
^
∈
A
^
(
J
,
.
,
E
,
T
)
{\displaystyle {\widehat {O}}\in {\widehat {\cal {A}}}(J,.,E,T)}
P
u
i
s
s
a
n
c
e
i
n
t
e
r
n
e
(
ν
,
P
u
i
s
s
a
n
c
e
(
ν
)
)
(
t
,
O
^
)
=
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
t
)
,
O
^
(
t
)
,
O
^
(
t
)
)
I
R
+
(
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
t
)
,
O
^
(
t
)
,
O
^
(
t
)
)
)
{\displaystyle {{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}(t,{\widehat {O}})={{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(t),{\widehat {O}}(t),{\widehat {O}}(t){\Big )}\,\,{\mathbb {I} }_{\mathbb {R} _{+}}{\bigg (}{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(t),{\widehat {O}}(t),{\widehat {O}}(t){\Big )}{\bigg )}}
J
∈
P
(
E
×
T
)
{\displaystyle J\in {\cal {P}}(E\times T)}
Soit
t
∈
T
{\displaystyle t\in T}
Soit
F
(
t
)
{\displaystyle {\cal {F}}(t)}
J
^
(
t
)
{\displaystyle {\widehat {J}}(t)}
Soit
ν
(
t
)
:
F
(
t
)
⟶
R
¯
{\displaystyle \nu (t)\,\,:\,\,{\cal {F}}(t)\longrightarrow {\overline {\mathbb {R} }}}
P
u
i
s
s
a
n
c
e
i
n
t
e
r
n
e
(
ν
,
P
u
i
s
s
a
n
c
e
(
ν
)
)
(
t
,
.
)
{\displaystyle {{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}(t,.)}
P
u
i
s
s
a
n
c
e
i
n
t
e
r
n
e
(
ν
,
P
u
i
s
s
a
n
c
e
(
ν
)
)
(
t
,
.
)
:
A
^
(
J
,
.
,
E
,
T
)
⟶
R
+
¯
:
O
^
⟼
P
u
i
s
s
a
n
c
e
i
n
t
e
r
n
e
(
ν
,
P
u
i
s
s
a
n
c
e
(
ν
)
)
(
t
,
O
^
)
=
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
t
)
,
O
^
(
t
)
,
O
^
(
t
)
)
I
R
+
(
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
t
)
,
O
^
(
t
)
,
O
^
(
t
)
)
)
{\displaystyle \displaystyle {{{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}(t,.)\,\,:\,\,{\widehat {\cal {A}}}(J,.,E,T)\longrightarrow {\overline {\mathbb {R} _{+}}}\,\,:\,\,{\widehat {O}}\,\,\longmapsto \,\,{{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}(t,{\widehat {O}})={{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(t),{\widehat {O}}(t),{\widehat {O}}(t){\Big )}\,\,{\mathbb {I} }_{\mathbb {R} _{+}}{\bigg (}{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(t),{\widehat {O}}(t),{\widehat {O}}(t){\Big )}{\bigg )}}}
En supprimant et en ne prenant pas en compte les doublons, la relation :
Si
t
∈
T
{\displaystyle t\in T}
P
u
i
s
s
a
n
c
e
i
n
t
e
r
n
e
(
ν
,
P
u
i
s
s
a
n
c
e
(
ν
)
)
(
t
,
O
^
)
{\displaystyle {{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}(t,{\widehat {O}})}
=
(
1
6
∫
J
^
(
t
)
∫
J
^
(
t
)
∫
J
^
(
t
)
I
(
j
≠
j
1
,
j
≠
j
2
,
j
1
≠
j
2
)
(
j
1
,
j
2
,
j
)
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
J
)
(
j
,
t
)
,
O
^
(
J
)
(
j
1
,
t
)
,
O
^
(
J
)
(
j
2
,
t
)
)
d
ν
(
t
)
(
j
1
,
t
)
d
ν
(
t
)
(
j
2
,
t
)
d
ν
(
t
)
(
j
,
t
)
{\displaystyle \displaystyle {={\bigg (}{\frac {1}{6}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j\neq j_{1},j\neq j_{2},j_{1}\neq j_{2})}(j_{1},j_{2},j){{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t)}}
+
1
2
∫
J
^
(
t
)
∫
J
^
(
t
)
∫
J
^
(
t
)
I
(
j
≠
j
1
,
j
≠
j
2
,
j
1
=
j
2
)
(
j
1
,
j
2
,
j
)
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
J
)
(
j
,
t
)
,
O
^
(
J
)
(
j
1
,
t
)
,
O
^
(
J
)
(
j
2
,
t
)
)
d
ν
(
t
)
(
j
1
,
t
)
d
ν
(
t
)
(
j
2
,
t
)
d
ν
(
t
)
(
j
,
t
)
{\displaystyle \displaystyle {+{\frac {1}{2}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j\neq j_{1},j\neq j_{2},j_{1}=j_{2})}(j_{1},j_{2},j){{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t)}}
+
∫
J
^
(
t
)
∫
J
^
(
t
)
∫
J
^
(
t
)
I
(
j
=
j
1
,
j
=
j
2
,
j
1
=
j
2
)
P
u
i
s
s
a
n
c
e
(
ν
)
(
O
^
(
J
)
(
j
,
t
)
,
O
^
(
J
)
(
j
1
,
t
)
,
O
^
(
J
)
(
j
2
,
t
)
)
d
ν
(
t
)
(
j
1
,
t
)
d
ν
(
t
)
(
j
2
,
t
)
d
ν
(
t
)
(
j
,
t
)
)
{\displaystyle \displaystyle {+\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j=j_{1},j=j_{2},j_{1}=j_{2})}{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t){\bigg )}}}
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{\displaystyle \displaystyle {{\mathbb {I} }_{\overline {\mathbb {R} _{+}}}{\bigg (}{\frac {1}{6}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j\neq j_{1},j\neq j_{2},j_{1}\neq j_{2})}(j_{1},j_{2},j){{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+{\frac {1}{2}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j\neq j_{1},j\neq j_{2},j_{1}=j_{2})}(j_{1},j_{2},j){{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j=j_{1},j=j_{2},j_{1}=j_{2})}{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t){\bigg )}}}
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{\displaystyle \displaystyle {={\bigg (}{\frac {1}{6}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j\neq j_{1},j\neq j_{2},j_{1}\neq j_{2})}(j_{1},j_{2},j){{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+{\frac {1}{2}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{1},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+\int _{{\widehat {J}}(t)}{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j,t){\Big )}\,\,d\nu (t)(j,t){\bigg )}}}
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{\displaystyle \displaystyle {{\mathbb {I} }_{\overline {\mathbb {R} _{+}}}{\bigg (}{\frac {1}{6}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{\mathbb {I} }_{(j\neq j_{1},j\neq j_{2},j_{1}\neq j_{2})}(j_{1},j_{2},j){{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{2},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j_{2},t)\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+{\frac {1}{2}}\int _{{\widehat {J}}(t)}\int _{{\widehat {J}}(t)}\,\,{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j_{1},t),{\widehat {O}}(J)(j_{1},t){\Big )}\,\,d\nu (t)(j_{1},t)\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+\int _{{\widehat {J}}(t)}{{\cal {P}}uissance}(\nu ){\Big (}{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j,t),{\widehat {O}}(J)(j,t){\Big )}\,\,d\nu (t)(j,t){\bigg )}}}
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{\displaystyle \displaystyle {={\Bigg (}{\frac {1}{3}}\int _{{\widehat {J}}(t)}\,\,{{\cal {P}}uissance}_{externe}{\Big (}J,\nu ,{{\cal {P}}uissance}(\nu ){\Big )}{\Big (}t,{\widehat {O}}(J)(j,.),{\widehat {O}}(J)\setminus {\widehat {O}}(J)(j,.){\Big )}\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+\int _{{\widehat {J}}(t)}{{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}{\Big (}t,{\widehat {O}}(J)(j,.){\Big )}\,\,d\nu (t)(j,t){\Bigg )}}}
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{\displaystyle \displaystyle {{\mathbb {I} }_{\overline {\mathbb {R} _{+}}}{\Bigg (}{\frac {1}{3}}\int _{{\widehat {J}}(t)}\,\,{{\cal {P}}uissance}_{externe}{\Big (}J,\nu ,{{\cal {P}}uissance}(\nu ){\Big )}{\Big (}t,{\widehat {O}}(J)(j,.),{\widehat {O}}(J)\setminus {\widehat {O}}(J)(j,.){\Big )}\,\,d\nu (t)(j,t)}}
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{\displaystyle \displaystyle {+\int _{{\widehat {J}}(t)}{{\cal {P}}uissance}_{interne}{\Big (}\nu ,{{\cal {P}}uissance}(\nu ){\Big )}{\Big (}t,{\widehat {O}}(J)(j,.){\Big )}\,\,d\nu (t)(j,t){\Bigg )}}}
modifier
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{\displaystyle {{\cal {P}}uissance}{\Big (}I,\nu ,\mu ,{{\cal {P}}uissance}(\nu ),{{\cal {P}}uissance}(\mu ){\Big )}(t,O)}
=
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{\displaystyle ={{\cal {P}}uissance}{\Big (}I,\nu ,\mu ,{{\cal {P}}uissance}(\nu ),{{\cal {P}}uissance}(\mu ){\Big )}(t,{\widehat {O}})}
![{\displaystyle {card}_{Q}(\{x(t)|t\in T\})={card}_{Q}(\{T(t)|t\in T\})[={card}_{Q}(T)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4010c04ccd86b63d3a5c1dc7f011647d2a4032da)
![{\displaystyle {card}_{E}(\{x(t)|t\in T\})={card}_{E}(\{T(t)|t\in T\})[={card}_{E}(T)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e63bd309e6bbf5166310ac1a6610f01de3ebcdd2)
![{\displaystyle \displaystyle {{\mbox{et}}\,\,{card}_{Q}(\{x(t)|t\in T_{\cal {D}}\})={card}_{Q}(\{D(t)|t\in T_{\cal {D}}\})[={card}_{Q}(T_{\cal {D}})]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45394f881aa2dfc26c518c0e1253ced5dfc3f930)
![{\displaystyle \displaystyle {{\mbox{Si, de plus,}}\,\,T_{\cal {D}}=]-\infty ,b_{0})\bigsqcup {\Big (}\bigsqcup _{i\in \mathbb {N} _{n-1}^{*}}(a_{i},b_{i}){\Big )}\bigsqcup (a_{n},+\infty [\,\,{\mbox{avec}}\,\,\forall i\in \mathbb {N} _{n}^{*},\,\,a_{i}\in T,\,\,\forall i\in \mathbb {N} _{n-1},\,\,b_{i}\in T,\,\,T=\mathbb {R} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea2ed056c17ea004ed7836efdaa0d1b8ef5c62a)
