Ley:
Sea m·d_{tt}^{2}[z] = pE_{e}(z,q) ==>
Si q = 0 ==> p = m
Ley:
Sea m·d_{tt}^{2}[z] = pE_{g}(z,q) ==>
Si q = 0 ==> p = m
Electro-débil de leptones orbitales:
Ley:
F(t)·G(t) = e^{(1/m)·(q+(-W))}·e^{(1/m)·(W+(-q))}·f(t)·g(t)
d_{t}[F(t)]·d_{t}[G(t)] = ...
... d_{t}[f(t)]·d_{t}[g(t)]+(1/m)^{2}·d_{t}[q+(-W)]·d_{t}[W+(-q)]·f(t)·g(t)
Ley:
Sea A(x,y) = (1/m)·< x,y > ==>
F(x,y)·G(x,y) = e^{ Anti-Potencial[ A(x,y)·a^{2}·< q+(-W),W+(-q) > ] }·f(x,y)·g(x,y)
d_{y}[F(x,y)]·d_{x}[G(x,y)] = ...
... d_{y}[f(x,y)]·d_{x}[g(x,y)]+( A_{x}·A_{y} )·a^{4}·(q+(-W))·(W+(-q))·f(x,y)·g(x,y)
Deducción:
F(x,y) = e^{ int[ A_{x}·a^{2}·(q+(-W)) ]d[y] }·f(x,y)
G(x,y) = e^{ int[ A_{y}·a^{2}·(W+(-q)) ]d[x] }·g(x,y)
d_{y}[F(x,y)]·d_{x}[G(x,y)] = 0 <==> ...
f(x,y) = e^{ int[ ia^{2}·A_{x}·(q+(-W)) ]d[y] }
g(x,y) = e^{ int[ ia^{2}·A_{y}·(W+(-q)) ]d[x] }
Ley:
Sea A(y,x) = (1/m)·< y,x > ==>
F(x,y)·G(x,y) = e^{ Potencial[ A(y,x)·a^{2}·< q+(-W),W+(-q) > ] }·f(x,y)·g(x,y)
d_{x}[F(x,y)]·d_{y}[G(x,y)] = ...
... d_{x}[f(x,y)]·d_{y}[g(x,y)]+( A_{y}·A_{x} )·a^{4}·(q+(-W))·(W+(-q))·f(x,y)·g(x,y)
Deducción:
F(x,y) = e^{ int[ A_{y}·a^{2}·(q+(-W)) ]d[x] }·f(x,y)
G(x,y) = e^{ int[ A_{x}·a^{2}·(W+(-q)) ]d[y] }·g(x,y)
Gravito-débil de leptones orbitales:
Ley:
F(t)·G(t) = e^{(1/m)·(p+(-Z))}·e^{(1/m)·(Z+(-p))}·f(t)·g(t)
d_{t}[F(t)]·d_{t}[G(t)] = ...
... d_{t}[f(t)]·d_{t}[g(t)]+(1/m)^{2}·d_{t}[p+(-Z)]·d_{t}[Z+(-p)]·f(t)·g(t)
Ley:
Sea A(x,y) = (1/m)·< x,y > ==>
F(x,y)·G(x,y) = e^{ Anti-Potencial[ A(x,y)·a^{2}·< p+(-Z),Z+(-p) > ] }·f(x,y)·g(x,y)
d_{y}[F(x,y)]·d_{x}[G(x,y)] = ...
... d_{y}[f(x,y)]·d_{x}[g(x,y)]+( A_{x}·A_{y} )·a^{4}·(p+(-Z))·(Z+(-p))·f(x,y)·g(x,y)
Ley:
Sea A(y,x) = (1/m)·< y,x > ==>
F(x,y)·G(x,y) = e^{ Potencial[ A(y,x)·a^{2}·< p+(-Z),Z+(-p) > ] }·f(x,y)·g(x,y)
d_{x}[F(x,y)]·d_{y}[G(x,y)] = ...
... d_{x}[f(x,y)]·d_{y}[g(x,y)]+( A_{y}·A_{x} )·a^{4}·(p+(-Z))·(Z+(-p))·f(x,y)·g(x,y)
Desintegración alfa:
Ley:
F(t)·G(t) = e^{(1/m)·(n·(q+(-q))+W+(-q))}·e^{(1/m)·(q+(-W))}·f(t)·g(t)
d_{t}[F(t)]·d_{t}[G(t)] = ...
... d_{t}[f(t)]·d_{t}[g(t)]+(1/m)^{2}·d_{t}[n·(q+(-q))+W+(-q)]·d_{t}[q+(-W)]·f(t)·g(t)
Sea A(x,y) = (1/m)·< x,y > ==>
F(x,y)·G(x,y) = e^{ Anti-Potencial[ A(x,y)·a^{2}·< n·(q+(-q))+W+(-q),q+(-W) > ] }·f(x,y)·g(x,y)
d_{y}[F(x,y)]·d_{x}[G(x,y)] = ...
... d_{y}[f(x,y)]·d_{x}[g(x,y)]+( A_{x}·A_{y} )·a^{4}·(n·(q+(-q))+W+(-q))·(q+(-W))·f(x,y)·g(x,y)
Desintegración beta:
Ley:
F(t)·G(t) = e^{(1/m)·(n·(q+(-q))+q+(-W))}·e^{(1/m)·(W+(-q))}·f(t)·g(t)
d_{t}[F(t)]·d_{t}[G(t)] = ...
... d_{t}[f(t)]·d_{t}[g(t)]+(1/m)^{2}·d_{t}[n·(q+(-q))+q+(-W)]·d_{t}[W+(-q)]·f(t)·g(t)
Sea A(x,y) = (1/m)·< x,y > ==>
F(x,y)·G(x,y) = e^{ Anti-Potencial[ A(x,y)·a^{2}·< n·(q+(-q))+q+(-W),W+(-q) > ] }·f(x,y)·g(x,y)
d_{y}[F(x,y)]·d_{x}[G(x,y)] = ...
... d_{y}[f(x,y)]·d_{x}[g(x,y)]+( A_{x}·A_{y} )·a^{4}·(n·(q+(-q))+q+(-W))·(W+(-q))·f(x,y)·g(x,y)
Desintegración gamma:
Ley:
F(t)·G(t) = e^{(1/m)·n·(q+(-q))}·e^{(1/m)·(W+(-W))}·f(t)·g(t)
d_{t}[F(t)]·d_{t}[G(t)] = ...
... d_{t}[f(t)]·d_{t}[g(t)]+(1/m)^{2}·d_{t}[n·(q+(-q))]·d_{t}[W+(-W)]·f(t)·g(t)
Sea A(x,y) = (1/m)·< x,y > ==>
F(x,y)·G(x,y) = e^{ Anti-Potencial[ A(x,y)·a^{2}·< n·(q+(-q)),W+(-W) > ] }·f(x,y)·g(x,y)
d_{y}[F(x,y)]·d_{x}[G(x,y)] = ...
... d_{y}[f(x,y)]·d_{x}[g(x,y)]+( A_{x}·A_{y} )·a^{4}·n·(q+(-q))·(W+(-W))·f(x,y)·g(x,y)
Teorema:
x^{4}+ax^{2}+bx+c = 0 es resoluble
Demostración:
Sea x = u+iv ==>
(u+iv)^{4}+a·(u+iv)^{2}+b·(u+iv)+c = 0
(-6)·(uv)^{2}+2ai·(uv)+c = 0
uv = (1/(6i))·( (-a)+( a^{2}+(-1)·6c )^{(1/2)} ) ...
... || ...
uv = (1/(6i))·( (-a)+(-1)·( a^{2}+(-1)·6c )^{(1/2)} )
4i·(uv)·( u^{2}+(-1)·v^{2} ) = w·( u^{2}+(-1)·v^{2} )
w = (2/3)·( (-a)+( a^{2}+(-1)·6c )^{(1/2)} )
... || ...
w = (2/3)·( (-a)+(-1)·( a^{2}+(-1)·6c )^{(1/2)} )
u^{4}+(a+w)·u^{2}+bu = 0
u^{3}+(a+w)·u+b = 0
Teorema:
x^{5}+ax^{3}+bx^{2}+cx+d = 0 es resoluble
Demostración:
Sea x = u+iv ==>
(u+iv)^{5}+a·(u+iv)^{3}+b·(u+iv)^{2}+c·(u+iv)+d = 0
2bi·(uv) = d
uv = (d/(2bi))
El polinomio tiene 1 punto fijo,
y el coeficiente de Galois es n+2 = 3 y es resoluble
[Ah][ h es solución de uv ]
3a·(uv)·(u+iv)+10·(uv)^{2}·(u+iv) = w·(u+iv)
w = 3a·(d/(2bi))+10·(d/(2bi))^{2}
5·(uv)·(u^{3}+(-i)·v^{3}) = k·(u^{3}+(-i)·v^{3})
u^{5}+(a+k)·u^{3}+bu^{2}+(c+w)·u = 0
u^{4}+(a+k)·u^{2}+bu+(c+w) = 0
Teorema:
x^{6}+ax^{4}+bx^{3}+cx^{2}+dx+p = 0 es irresoluble
Demostración:
(-20)·i·(uv)^{3}+(-6)·a·(uv)^{2}+2ic·(uv)+p·(uv)^{0} = 0
F(uv) = vu = uv
El polinomio tiene 3 puntos fijos,
y el coeficiente de Galois es n+2 = 5 y es irresoluble
[Eh][ h no es solución de uv ]
uv = (z+(-1)·(1/10i)·a)
h^{3}+ph+q = 0
h | 1 | h | p+h^{2} | q+ph+h^{3} = 0
(z+(-h))·( z^{2}+hz+(p+h^{2}) ) = 0
uv = (1/10i)·a+( (1/2)·( (-h)+( h^{2}+(-4)·(h^{2}+p) )^{(1/2)} )
uv = (1/10i)·a+( (1/2)·( (-h)+(-1)·( h^{2}+(-4)·(h^{2}+p) )^{(1/2)} )
Teorema:
x^{7}+ax^{5}+bx^{4}+cx^{3}+dx^{2}+px+q = 0 es resoluble
Demostración:
(-6)·b·(uv)^{2}+2id·(uv)+q·(uv)^{0} = 0
F(uv) = vu = uv
El polinomio tiene 2 puntos fijos,
y el coeficiente de Galois es n+2 = 4 y es resoluble
[Ah][ h es solución de uv ]
Definición: [ de Grupo Galois ]
F(uv) = vu = uv
F(uv·ab) = F(uv)·ba
F(ab·uv) = ba·F(uv)
Teorema:
F((uv·ab)·pq) = F(uv·(ab·pq))
Demostración:
F((uv·ab)·pq) = F(uv·ab)·qp = ( F(uv)·ba )·qp = (vu·ba)·qp = vu·(ba·qp) = ...
... vu·( ba·F(pq) ) = vu·F(ab·pq) = F(uv·(ab·pq))
Teorema:
F(uv·ab) = F(uv)·ba = vu·ba = ba·vu = ba·F(uv) = F(ab·uv)
F(uv·ab) = vu·F(ab) = vu·ba = ba·vu = F(ab)·vu = F(ab·uv)
Definición: [ de coeficiente de Galois de un polinomio ]
Sea P(x) = P_{2n}(u+iv) ==>
Gal(P(x)) = Grado( Q_{n}(uv) )+2 = n+2
Sea P(x) = P_{2n+1}(u+iv) ==>
Gal(P(x)) = Grado( Q_{n+(-1)}(uv) )+2 = n+1
Teorema fundamental del Álgebra:
P_{n+1}(x) = P_{n}(x)·(x+(-1)·a_{n+1}) = (x+(-1)·a_{1})...(n)...(x+(-1)·a_{n})·(x+(-1)·a_{n+1})
Definición:
P(x) es resoluble <==> Grado[P(x)]+(-1)·Gal(P(x)) =[2]= Grado[P(x)]
P(x) es irresoluble <==> ¬( Grado[P(x)]+(-1)·Gal(P(x)) =[2]= Grado[P(x)] )
Teorema:
Sea P(x) = P_{2n}(u+iv) ==>
Si Gal(P(x)) = 2k+1 >] 5 ==> P(x) es irresoluble
Si Gal(P(x)) = 2k >] 5 ==> P(x) es resoluble
Demostración:
Por el teorema fundamental del Álgebra:
P_{2n}(u+iv) tiene 2n raíces
Por Cardano:
Q_{n}(uv) tiene n raíces
Sea Gal(P(x)) = n+2 = 2k+1 ==>
2n+(-1)·(2k+1) = 2·(n+(-k))+1 = 2p+1 =[2]= 1 & ¬( 1 =[2]= 2n )
P(x) es irresoluble
Sea Gal(P(x)) = n+2 = 2k ==>
2n+(-1)·2k = 2·(n+(-k)) = 2p =[2]= 0 & 0 =[2]= 2n
P(x) es resoluble
Teorema:
Sea P(x) = P_{2n+1}(u+iv) ==>
Si Gal(P(x)) = 2k+1 >] 5 ==> P(x) es irresoluble
Si Gal(P(x)) = 2k >] 5 ==> P(x) es resoluble
Demostración:
Por el teorema fundamental del Álgebra:
P_{2n+1}(u+iv) tiene 2n+1 raíces
Por Cardano:
Q_{n+(-1)}(uv) tiene n+(-1) raíces
Gal(P(x)) = n+1
Si n = 2k ==>
2n+1+(-1)·(2k+1) =[2]= 0 & ¬( 0 =[2]= 2n+1 )
P(x) es irresoluble
Si n = 2k+1 ==>
2n+1+(-1)·(2k+2) =[2]= (-1) =[2]= 1 & ( 1 =[2]= 2n+1 )
P(x) es resoluble
Teorema:
Sea f(x) continua ==>
Si [Ax][ x >] 0 ==> f(x) >] x ] ==> [Ec][ f(c) = 0 ]
Sea f(x) continua ==>
Si [Ax][ x [< 0 ==> f(x) [< x ] ==> [Ec][ f(c) = 0 ]
Demostración:
Sea u >] 0 ==>
f(u) >] u >] 0
(-1)·f(-u) [< (-u) [< 0
Teorema:
Sea a [< b ==>
Si f(x) = 2x+(-1)·(a+b) ==> [Ec][ f(c) = 0 ]
Sea a >] b ==>
Si f(x) = 2x+(-1)·(a+b) ==> [Ec][ f(c) = 0 ]
Demostración:
f(b) = b+(-a) >] 0
f(a) = a+(-b) [< 0
Teorema:
Sea f(x) = x^{2n+1}+(-a) ==> [E!c][ f(c) = 0 ]
Demostración:
Se define c = a^{( 1/(2n+1) )}
d_{x}[f(x)] = (2n+1)·x^{2n} >] 0
f(x) es creciente
Sea s >] 0 ==>
f(c+s) = (c+s)^{2n+1}+(-a) >] c^{2n+1}+(-a) = 0
f(c+(-s)) = (c+(-s))^{2n+1}+(-a) [< c^{2n+1}+(-a) = 0
Teorema:
Sea f(x) = x^{2n+2}+(-x) ] ==> [E!c][ d_{x}[f(c)] = 0 ]
Demostración:
d_{x}[f(x)] = (2n+2)·x^{2n+1}+(-1)
Se define c = ( 1/(2n+2) )^{( 1/(2n+1) )}
d_{xx}^{2}[f(x)] = (2n+2)·(2n+1)·x^{2n} >] 0
d_{x}[f(x)] es creciente
Sea s >] 0 ==>
d_{x}[f(c+s)] = (2n+2)·(c+s)^{2n+1}+(-1) >] (2n+2)·c^{2n+1}+(-1) = 0
d_{x}[f(c+(-s))] = (2n+2)·(c+(-s))^{2n+1}+(-1) [< (2n+2)·c^{2n+1}+(-1) = 0
Problema:
Demostrad:
Sea f(x) = x^{[2n+1:b]}+(-a) ==> [E!c][ f(c) = 0 ]
Arte:
[Ef(x)][ Si ( F(x) = int[ f(x) ]d[x] & lim[x = 0][ F(x) ] = ( 1 || (-1) ) ) ==> ...
... int[x = (-2)]-[2][ f(x) ]d[x] = 0 ]
Exposición:
f(x) = 0·(1/x)
F(x) = x^{0}
int[x = (-2)]-[2][ f(x) ]d[x] = 2^{0}+(-1)·(-2)^{0} = 1+(-1) = 0
Destructor:
int[x = (-2)]-[2][ f(x) ]d[x] = F(2)+(-1)·F(-2) = F(1+1)+F((-1)+(-1)) = F(1+(-1))+(-1)·F((-1)+1) = ...
... F(0)+(-1)·F(0) = 0·F(0) = 0
Arte:
[Ef(x)][ Si ( F(x) = int[ f(x) ]d[x] & lim[x = 1][ F(x) ] = 2n ) ==> ...
... int[x = (1/(2n))]-[(1/n)][ f(2nx) ]d[x] = 0 ]
Exposición:
f(x) = 2n·0·(1/x)
F(x) = 2nx^{0}
int[x = (1/(2n))]-[(1/n)][ f(2nx) ]d[x] = (1/(2n))·( 2n2^{0}+(-1)·2n1^{0} ) = 0
Destructor:
int[x = (1/(2n))]-[(1/n)][ f(2nx) ]d[x] = (1/(2n))·( F(2)+(-1)·F(1) ) = ...
... (1/(2n))·( F( (3/2)+(1/2) )+(-1)·F(1) = (1/(2n))·( F( (3/2)+(-1)·(1/2) )+(-1)·F(1) ) = ...
... (1/(2n))·( F(1)+(-1)·F(1) ) = (1/(2n))·2n·0 = 0
Teorema:
Sea F(x) = int[ f(x) ]d[x] ==>
Si lim[y = oo][ F(y) ] = c ==> lim[y = oo][ int[x = a]-[b][ f(x+y) ]d[x] ] = 0c
Demostración:
lim[y = oo][ int[x = a]-[b][ f(x+y) ]d[x] ] = lim[y = oo][ F(b+y)+(-1)·F(a+y) ] = ...
... F(b+oo)+(-1)·F(a+oo) = F(oo)+(-1)·F(oo) = 0c
Teorema:
lim[y = (1/k)][ int[x = (-1)]-[1][ (1/2)·(2n+1)·y·(xy)^{2n} ]d[x] ] ] = (1/k)^{2n+1}
Dual:
No estaba buena de cuerpo y cara ni tenía un cuerpo atlético.
Estaba buena de cuerpo y cara o tenía un cuerpo atlético.
Dual:
No estaba buena de cuerpo y cara y era fea.
Estaba buena de cuerpo y cara o era guapa.
Generador de destructor:
Estoy en un lugar haciendo esto,
no haciendo esto,
estoy haciendo esto.
Estoy en un lugar no haciendo esto,
haciendo esto,
no estoy haciendo esto.
Definición: [ de tensor de curvatura de Cristofel ]
d_{tt}^{2}[x_{s}]+R_{ijk}^{s}·d_{t}[x_{i}]·d_{t}[x_{j}]·d_{tt}^{2}[x_{k}] = 0
Teorema:
R_{kkk}^{k} = kt ==> x_{k}(t) = i·(1/k)^{(1/2)}·( t /o(t)o/ (1/2)·t^{2} )^{[o(t)o] (1/2)}
R_{ijk}^{s} = (ij)^{(1/2)}·t·(k/s)^{(1/2)}
Demostración:
(-1)·( 1/( d_{t}[x_{k}]^{2}·d_{tt}^{2}[x_{k}] ) )·d_{tt}^{2}[x_{k}] = R_{kkk}^{k} = kt
(-1)·( t /o(t)o/ ( x_{k} )^{[o(t)o] 2} ) = k·(1/2)·t^{2}
x_{k}(t) = i·(1/k)^{(1/2)}·( t /o(t)o/ (1/2)·t^{2} )^{[o(t)o] (1/2)}
Teorema:
R_{kkk}^{k} = e^{kt} ==> x_{k}(t) = ik·( t /o(t)o/ e^{kt} )^{[o(t)o] (1/2)}
R_{ijk}^{s} = e^{(1/2)·(i+j)·t}·(s/k)·e^{(1/2)·(k+(-s))·t}
Demostración:
(-1)·( 1/( d_{t}[x_{k}]^{2}·d_{tt}^{2}[x_{k}] ) )·d_{tt}^{2}[x_{k}] = R_{kkk}^{k} = kt
(-1)·( t /o(t)o/ ( x_{k} )^{[o(t)o] 2} ) = (1/k)·e^{kt}
x_{k}(t) = ik·( t /o(t)o/ e^{kt} )^{[o(t)o] (1/2)}
Homologías de Jûanagoras-Schoze:
Arte:
Sea h_{n}: S_{n} ---> S_{n+1} ==>
[En][Ef(x)][ f: S_{1} ---> S_{n} & f(x) es biyectiva ]
Exposición:
n = 1
Se define f(x) = x
h(n) = 1
Arte:
Sea h_{n}: S_{n} ---> S_{n+1} ==>
[En][Eg(x)][ g: S_{1} ---> S_{n+1} & g(x) es biyectiva ]
Exposición:
n = 0
Se define g(x) = x
h(n) = 0
Arte:
Sea h_{n}: P_{n}(A) ---> P_{n+1}(A) ==>
[En][Ef(x)][ f: A ---> P_{n}(A) & f(x) es biyectiva ]
Exposición:
n = 1
Se define f(x) = {x}
h(n) = 1
Arte:
Sea h_{n}: P_{n}(A) ---> P_{n+1}(A) ==>
[En][Eg(x)][ g: A ---> P_{n+1}(A) & g(x) es biyectiva ]
Exposición:
n = 0
Se define g(x) = {x}
h(n) = 0
Teorema:
Sea h_{n}: ( Z/[n]_{m} ) ---> ( Z/[n+1]_{m} ) ==>
[Ef(x)][ f: [0,m+(-1)]_{N} ---> ( Z/[n]_{m} ) & f(x) es biyectiva ]
Demostración:
Sea n = mk+r ==>
Se define f(r) = [r]_{m}
Teorema:
Sea h_{n}: ( Z/[n]_{m} ) ---> ( Z/[n+1]_{m} ) ==>
[Eg(x)][ g: [0,m+(-1)]_{N} ---> ( Z/[n+1]_{m} ) & g(x) es biyectiva ]
Demostración:
Sea n = mk+(r+(-1)) ==>
n+1 = mk+r
Se define g(r+(-1)) = [r]_{m}
Teorema:
Sea h_{n}: A x..(n)...x A ---> A x..(n+1)...x A ==>
[Ef(x)][ f: A ---> A x..(n)...x A & f(x) es biyectiva ]
Teorema:
Sea h_{n}: A x..(n)...x A ---> A x..(n+1)...x A ==>
[Eg(x)][ g: A ---> A x..(n+1)...x A & g(x) es biyectiva ]
Teorema:
Sea H(x) inyectiva ==>
Sea h_{n}: {n·( H(x) )} ---> {(n+1)·( H(x) )} ==>
[Ef(x)][ f: {x} ---> {n·( H(x) )} & f(x) es biyectiva ]
Teorema:
Sea H(x) inyectiva ==>
Sea h_{n}: {n·( H(x) )} ---> {(n+1)·( H(x) )} ==>
[Eg(x)][ g: {x} ---> {(n+1)·( H(x) )} & g(x) es biyectiva ]
Teorema:
Sea h_{n}: {nx} ---> {(n+1)·x} ==>
[Ef(x)][ f: {x} ---> {nx} & f(x) es biyectiva ]
Teorema:
Sea h_{n}: {nx} ---> {(n+1)·x} ==>
[Eg(x)][ g: {x} ---> {(n+1)·x} & g(x) es biyectiva ]
Teorema:
Sea H(x) inyectiva ==>
Sea h_{n}: {( H(x) )^{n}} ---> {( H(x) )^{n+1}} ==>
[Ef(x)][ f: {x} ---> {( H(x) )^{n}} & f(x) es biyectiva ]
Teorema:
Sea H(x) inyectiva ==>
Sea h_{n}: {( H(x) )^{n}} ---> {( H(x) )^{n+1}} ==>
[Eg(x)][ g: {x} ---> {( H(x) )^{n+1}} & g(x) es biyectiva ]
Teorema:
Sea h_{n}: {x^{n}} ---> {x^{n+1}} ==>
[Ef(x)][ f: {x} ---> {x^{n}} & f(x) es biyectiva ]
Teorema:
Sea h_{n}: {x^{n}} ---> {x^{n+1}} ==>
[Eg(x)][ g: {x} ---> {x^{n+1}} & g(x) es biyectiva ]
Definición:
F(x) es un morfismo topológico expansivo
<==>
[EG(x)][ x [<< G(x) & F(x) [<< F(G(x)) & ...
... F( A [&] B ) [<< F(G(A)) [&] F(G(B)) & ...
... F( A [ || ] B ) [<< F(G(A)) [ || ] F(G(B)) ]
F(x) es un morfismo topológico contractivo
<==>
[EG(x)][ x >>] G(x) & F(x) >>] F(G(x)) & ...
... F( A [&] B ) >>] F(G(A)) [&] F(G(B)) & ...
... F( A [ || ] B ) >>] F(G(A)) [ || ] F(G(B)) ]
Teorema:
Sea x [<< G(x) ==>
Si F(x) = x ==> F(x) es un morfismo topológico expansivo
Teorema:
Sea x >>] G(x) ==>
Si F(x) = x ==> F(x) es un morfismo topológico contractivo
Teorema:
Sea x [<< G(x) ==>
Si F(x) = x [ || ] C ==> F(x) es un morfismo topológico expansivo
Demostración:
F(x) = x [ || ] C [<< G(x) [ || ] C = F(G(x))
F( A [&] B ) = ( A [&] B ) [ || ] C = ( A [ || ] C ) [&] ( B [ || ] C ) [<< ...
... ( G(A) [ || ] C ) [&] ( G(B) [ || ] C ) = F(G(A)) [&] F(G(B))
F( A [ || ] B ) = ( A [ || ] B ) [ || ] C = ( A [ || ] B ) [ || ] ( C [ || ] C ) = ( A [ || ] C ) [ || ] ( B [ || ] C ) [<< ...
... ( G(A) [ || ] C ) [ || ] ( G(B) [ || ] C ) = F(G(A)) [ || ] F(G(B))
Teorema:
Sea x >>] G(x) ==>
Si F(x) = x [&] C ==> F(x) es un morfismo topológico contractivo
Definición:
{x} , {y} = { z : ( z = x || z = y ) } = {x,y}
}x{ ; }y{ = { z : ( z != x & z != y ) } = }x;y{
Teorema:
{x} , {x} = { z : ( z = x || z = x ) } = { z : z = x } = {x}
}x{ ; }x{ = { z : ( z != x & z != x ) } = { z : z != x } = }x{
Teorema:
{x} , 0 = { z : ( z = x || z != z ) } = { z : z = x } = {x}
}x{ ; 1 = { z : ( z != x & z = z ) } = { z : z != x } = }x{
Teorema:
Sea G(x) = x,z_{1},...,z_{n} ==>
Si F(x) = {x} ==> F(x) es un morfismo topológico expansivo
Demostración:
F(x) = {x} [<< {G(x)} = F(G(x))
F(x , y) = {x , y} = {x} , {y} [<< {G(x)} , {G(y)} = F(G(x)) , F(G(y))
Teorema:
Sea G(x) = x;z_{1};...;z_{n} ==>
Si F(x) = }x{ ==> F(x) es un morfismo topológico contractivo
Demostración:
F(x) = }x{ >>] }G(x){ = F(G(x))
F(x ; y) = }x ; y{ = }x{ ; }y{ >>] }G(x){ ; }G(y){ = F(G(x)) ; F(G(y))
Conjetura de Poincaré:
Teorema:
[EF][ Si ( y_{1}(ix) = e^{zix} & F( y_{1}(ix),z ) ) ==> ( lim[n = oo][ F( y_{n}(ix),z ) ] & z = 0 ) ]
Demostración:
Se define F( y_{n}(ix),z ) <==> ( y_{n}(ix) = e^{(1/n)·zix} & d_{ix}[ y_{n}(ix) ] = z·y_{n}(ix) )
d_{ix}[ 1^{zix} ] = d_{ix}[ 1^{ix} ] = 1^{ix}·ln(1) = 0
Teorema:
[EF][ Si ( y_{1}(ix) = re^{(z/r)·ix} & F( y_{1}(ix),z ) ) ==> ( lim[n = oo][ F( y_{n}(ix),z ) ] & z = r ) ]
Demostración:
Se define F( y_{n}(ix),z ) <==> ...
... ( y_{n}(ix) = re^{(1/n)·(z/r)·ix} & d_{ix}[ y_{n}(ix) ] = (1/(nr))·z·y_{n}(ix) )
d_{ix}[ 1^{(z/r)·ix} ] = d_{ix}[ 1^{ix} ] = 1^{ix}·ln(1) = 0
