M( [ || ]-[k = 1]-[n][ A_{k} ] ) [< sum[k = 1]-[n][ M(A_{k}) ]
M( [&]-[k = 1]-[n][ ¬A_{k} ] ) >] sum[k = 1]-[n][ M(¬A_{k}) ]
Teorema:
M(0) >] 0
Demostración:
M(A) = M(A [ || ] 0) [< M(A)+M(0)
0 [< M(0)
Teorema:
M(E) [< 0
Demostración:
M(A) = M(A [&] E) >] M(A)+M(E)
0 >] M(E)
Definición: [ de medida exterior de recubrimiento ]
M(A) = min{ sum[k = 1]-[n][ M(E_{k}) ] : A [<< [ || ]-[k = 1]-[n][ E_{k} ] }
M(¬A) = max{ sum[k = 1]-[n][ M(¬E_{k}) ] : ¬A >>] [&]-[k = 1]-[n][ ¬E_{k} ] }
Teorema:
M(0) >] 0
Demostración
A [<< A = A [ || ] 0
M(A) = min{ M(A)+M(0) } [< M(A)+M(0)
Teorema:
M(E) [< 0
Demostración
¬A >>] ¬A = ¬A [&] E
M(A) = max{ M(A)+M(E) } >] M(A)+M(E)
Teorema:
M( [ || ]-[k = 1]-[n][ A_{k} ] ) [< sum[k = 1]-[n][ M(A_{k}) ]
Demostración:
A = [ || ]-[k = 1]-[n][ A_{k} ]
A [<< [ || ]-[k = 1]-[n][ A_{k} ]
M(A) = min{ sum[k = 1]-[n][ M(A_{k}) ] } [< sum[k = 1]-[n][ M(A_{k}) ]
Teorema:
M( [&]-[k = 1]-[n][ ¬A_{k} ] ) >] sum[k = 1]-[n][ M(¬A_{k}) ]
Demostración:
¬A = [&]-[k = 1]-[n][ ¬A_{k} ]
¬A >>] [&]-[k = 1]-[n][ ¬A_{k} ]
M(¬A) = max{ sum[k = 1]-[n][ M(¬A_{k}) ] } >] sum[k = 1]-[n][ M(¬A_{k}) ]
Teorema: [ de existencia de la conexión cruzada de homologías ]
Sean ( A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n} & B_{m} = [ g_{m}: b_{m} ---> b_{m+1} ]_{m} )
[EP][EQ][ P(a_{n+1}) = b_{m} & Q(a_{n}) = b_{m+1} ]
[Eu][Ev][ u(a_{m+1}) = a_{n} & v(b_{m}) = a_{n+1} ]
Demostración:
Se define P(a_{k}) = b_{m+k+(-1)·(n+1)}
Se define Q(a_{k}) = b_{(m+1)+k+(-n)}
Teorema: [ de existencia de la conexión paralela de homologías ]
Sean ( A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n} & B_{m} = [ g_{m}: b_{m} ---> b_{m+1} ]_{m} )
[EL][ L(a_{n}) = b_{m} & L(a_{n+1}) = b_{m+1} ]
[EH][ H(a_{m}) = a_{n} & H(b_{m+1}) = a_{n+1} ]
Demostración:
Se define L(a_{k}) = b_{m+k+(-n)}
Definición: [ de trapecio de homología ]
Sea A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n}
[Eh_{1}]...[Eh_{m}][ ( h_{m} o...(m)...o h_{1} )(a_{n}) = a_{n+1} ]
Teorema:
Sea A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n}
[Eh][ ( h o...(m)...o h )(a_{n}) = a_{n+1} ]
Demostración:
Se define h(a_{k}) = a_{k+(1/m)}
Teorema: [ del triángulo de homología ]
Sea A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n}
[Eh_{1}][Eh_{2}][ ( h_{2} o h_{1} )(a_{n}) = a_{n+1} ]
Demostración:
Se define h_{1}(a_{k}) = a_{k+(1/m)}
Se define h_{2}(a_{k}) = a_{k+1+(-1)·(1/m)}
Teorema:
Sea A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n}
[Eh_{1}][Eh_{2}][ ( h_{2} o h_{1} o ...(k)...o h_{2} o h_{1} )(a_{n}) = a_{n+1} ]
Demostración:
Examen de homología algebraica.
Teorema:
Sea A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n}
[Eh_{1}][Eh_{2}][ ( h_{2} o ...(k)... o h_{2} o h_{1} )(a_{n}) = a_{n+1} ]
Demostración:
Examen de homología algebraica.
Teorema: [ de compactificación de homología ]
Sea A_{n} = [ f_{n}: a_{n} ---> a_{n+1} ]_{n}
[EB_{n}][ B_{n} = [ g_{n}: b_{[r]_{m}} ---> b_{[r+1]_{m}} ]_{n} & ...
... A_{n} y B_{n} están conectadas paralelamente ]
Demostración:
Sea n = mk+r ==> ...
Se define L(a_{mk+r}) = b_{[r]_{m}}
Se define H(b_{[r]_{m}}) = a_{mk+r}
Teorema:
Sea f(x) = sum[k = 0]-[oo][ ( 1/( mk+(m+(-1)) )! )·x^{mk+(m+(-1))} ]
Sea A_{n} = [ f_{n} : d_{x...x}^{n}[f(x)] ---> d_{x...x}^{n+1}[f(x)] ]_{n}
[EB_{n}][ B_{n} = [ g_{n}: b_{[r]_{m}} ---> b_{[r+1]_{m}} ]_{n} & ...
... A_{n} y B_{n} están conectadas paralelamente ]
Demostración:
Se define L( sum[k = 0]-[oo][ ( 1/(mk+r)! )·x^{mk+r} ] ) = ...
... (1/oo)·sum[k = 0]-[oo][ ( 1/[r]_{m}! )·x^{[r]_{m}} ] = ( 1/[r]_{m}! )·x^{[r]_{m}}
B_{n} = [ g_{mk+r}: b_{[r]_{m}} ---> b_{[r+(-1)]_{m}} ]_{n}
g_{mk}: b_{[0]_{m}} ---> b_{[m+(-1)]_{m}}
Teorema:
Sea f(x) = sum[k = 0]-[oo][ (-1)^{k}·( 1/( mk+(m+(-1)) )! )·x^{mk+(m+(-1))} ]
Sea A_{n} = [ f_{n} : d_{x...x}^{n}[f(x)] ---> d_{x...x}^{n+1}[f(x)] ]_{n}
[EB_{n}][EC_{n}][ B_{n} = [ g_{n}: b_{[r]_{m}} ---> b_{[r+1]_{m}} ]_{n} & ...
... C_{n} = [ h_{n}: c_{[r]_{m}} ---> c_{[r+1]_{m}} ]_{n} & ...
... A_{n} y ( B_{n} [ || ] C_{n} ) están conectadas paralelamente ]
Demostración:
Se define L( sum[k = 0]-[oo][ (-1)^{k}·( 1/(mk+r)! )·x^{mk+r} ] ) = ...
... sum[k = 0]-[oo][ (-1)^{k}·( 1/[r]_{m}! )·x^{[r]_{m}} ] = ( 1/[r]_{m}! )·x^{[r]_{m}}
B_{n} = [ g_{mk+r}: b_{[r]_{m}} ---> b_{[r+(-1)]_{m}} ]_{n}
g_{mk}: b_{[0]_{m}} ---> b_{[m+(-1)]_{m}}
Sea k = p+1 ==>
Se define L( sum[p = 0]-[oo][ (-1)^{p+1}·( 1/(mp+r)! )·x^{mp+r} ] ) = ...
... sum[p = 0]-[oo][ (-1)^{p+1}·( 1/[r]_{m}! )·x^{[r]_{m}} ] = (-1)·( 1/[r]_{m}! )·x^{[r]_{m}}
C_{n} = [ h_{mk+r}: c_{[r]_{m}} ---> c_{[r+(-1)]_{m}} ]_{n}
h_{mk}: c_{[0]_{m}} ---> c_{[m+(-1)]_{m}}
Teorema:
Sea f(x) = sinh(x)
Sea A_{n} = [ f_{n} : d_{x...x}^{n}[f(x)] ---> d_{x...x}^{n+1}[f(x)] ]_{n}
[EB_{n}][ B_{n} = [ g_{n}: b_{[r]_{m}} ---> b_{[r+1]_{m}} ]_{n} & ...
... A_{n} y B_{n} están conectadas paralelamente ]
Demostración:
Examen de homología algebraica.
Teorema:
Sea f(x) = sin(x)
Sea A_{n} = [ f_{n} : d_{x...x}^{n}[f(x)] ---> d_{x...x}^{n+1}[f(x)] ]_{n}
[EB_{n}][EC_{n}][ B_{n} = [ g_{n}: b_{[r]_{m}} ---> b_{[r+1]_{m}} ]_{n} & ...
... C_{n} = [ h_{n}: c_{[r]_{m}} ---> c_{[r+1]_{m}} ]_{n} & ...
... A_{n} y ( B_{n} [ || ] C_{n} ) están conectadas paralelamente ]
Demostración:
Examen de homología algebraica.
Teorema:
Sea A_{n} = [ f_{n} : ( (mk)!/d_{x...x}^{n}[x^{mk}] ) ---> ( (mk)!/d_{x...x}^{n+1}[x^{mk}] ) ]_{n}
[EB_{n}][ B_{n} = [ g_{n}: b_{[r]_{m}} ---> b_{[r+1]_{m}} ]_{n} & ...
... A_{n} y B_{n} están conectadas paralelamente ]
Demostración:
Se define L( (mk+(-r))!/x^{mk+(-r)} ) = ([m+(-r)]_{m})!/x^{[m+(-r)]_{m}}
B_{n} = [ g_{mk+(-r)}: b_{[m+(-r)]_{m}} ---> b_{[m+(-1)+(-r)]_{m}} ]_{n}
g_{mk+(-1)·(m+(-1))}: b_{[1]_{m}} ---> b_{[m]_{m}}
Teorema:
Sea A_{n} = [ A_{1} = {a_{1}} & f_{n} : A_{n} ---> A_{n} [ || ] {a_{n+1}} ]_{n}
Sea B_{n} = [ ¬A_{1} = }a_{1}{ & g_{n} : ¬A_{n} ---> ¬A_{n} [&] }a_{n+1}{ ]_{n}
A_{n} y B_{n} están conectadas paralelamente.
Demostración:
Se define L(A) = ¬A
Axioma:
A [&] }x{ = A
{ x : x != x } [&] }x{ = { x : x != x }
Teorema:
¬A [ || ] {x} = ¬A
{ x : x = x } [ || ] {x} = { x : x = x }
Teorema:
Sea A_{n} = [ A_{1} = {a_{1}} & f_{n} : A_{n} ---> A_{n} [ || ] {a_{n+1}} ]_{n}
Sea B_{n} = [ ¬A_{1} = }a_{1}{ & g_{n} : ¬A_{n} ---> ¬A_{n} [&] }a_{n+1}{ ]_{n}
A_{n} y B_{n} están conectadas cruzadamente.
Demostración:
Se define P(A) = ¬( A [&] }a_{1}{ )
Se define Q(A) = ¬( A [ || ] {a_{1}} )
Teorema:
max{x,max{y,z}} = max{max{x,y},z}
min{x,min{y,z}} = min{min{x,y},z}
Demostración:
Sea a = max{x,max{y,z}} ==>
a >] x & a >] max{y,z}
a >] x & ( a >] y & a >] z )
( a >] x & a >] y ) & a >] z
a >] max{x,y} & a >] z
a = max{max{x,y},z}
Teorema:
max{min{x,y},mim{x,z}} = min{x,max{y,z}}
min{max{x,y},max{x,z}} = max{x,min{y,z}}
Demostración:
Sea a = max{min{x,y},mim{x,z}}
a >] min{x,y} & a >] min{x,z}
( a >] x || a >] y ) & ( a >] x || a >] z )
a >] x || ( a >] y & a >] z )
a >] x || a >] max{y,z}
a = min{x,max{y,z}}
Teorema:
Si [Ak][ 1 [< k [< n ==> p^{k} € E ] ==> mcm{p^{n_{k}}} € E
Si [Ak][ 1 [< k [< n ==> p^{k} € E ] ==> mcd{p^{n_{k}}} € E
Demostración:
mcm{p^{n_{k}}} = p^{max{n_{k}}} € E
mcd{p^{n_{k}}} = p^{min{n_{k}}} € E
Teorema:
mcm{ p^{k},mcm{p^{n},p^{m}} } = mcm{ mcm{p^{k},p^{n}},p^{m} }
mcd{ p^{k},mcd{p^{n},p^{m}} } = mcd{ mcd{p^{k},p^{n}},p^{m} }
Teorema:
mcm{ mcd{p^{k},p^{n_{k}}},mcd{p^{k},p^{m_{k}}} } = ...
... mcd{ p^{k},mcm{p^{n_{k}},p^{m_{k}}} }
mcd{ mcm{p^{k},p^{n_{k}}},mcm{p^{k},p^{m_{k}}} } = ...
... mcm{ p^{k},mcd{p^{n_{k}},p^{m_{k}}} }
Demostración:
mcm{ mcd{p^{k},p^{n_{k}}},mcd{p^{k},p^{m_{k}}} } = ...
... mcm{ p^{min{k,n_{k}}}},p^{min{k,m_{k}}} } = ...
... p^{max{ min{k,n_{k}},min{k,m_{k}} }}
mcd{ p^{k},mcm{p^{n_{k}},p^{m_{k}}} } = mcd{p^{k},p^{max{n_{k},m_{k}}}} =
... p^{min{ k,max{n_{k},m_{k}} }}
Teorema:
Si [Ak][ 1 [< k [< n ==> mp^{k} € E ] ==> mcm{mp^{n_{k}}} € E
Si [Ak][ 1 [< k [< n ==> mp^{k} € E ] ==> mcd{mp^{n_{k}}} € E
Demostración:
Examen de topología.
Definición: [ de medida exterior binaria ]
M(A) = min{ (1/k) : A [<< [ || ]-[k = 1]-[n][ A_{k} ] }
M(¬A) = max{ (-1)·(1/k) : ¬A >>] [&]-[k = 1]-[n][ ¬A_{k} ] }
Teorema:
M( [ || ]-[k = 1]-[n][ A_{k} ] ) [< sum[k = 1]-[n][ M(A_{k}) ]
Demostración:
M( [ || ]-[k = 1]-[n][ A_{k} ] ) = M(A) = min{(1/k)} [< 1+...(n)...+(1/n) = ...
... sum[k = 1]-[n][ M(A_{k}) ]
Teorema:
M( [&]-[k = 1]-[n][ ¬A_{k} ] ) >] sum[k = 1]-[n][ M(¬A_{k}) ]
Demostración:
M( [&]-[k = 1]-[n][ ¬A_{k} ] ) = M(¬A) = max{(-1)·(1/k)} >] (-1)+...(n)...+(-1)·(1/n) = ...
... sum[k = 1]-[n][ M(¬A_{k}) ]
Teorema:
M(0) >] 0
Demostración:
M(0) = min{(1/k)} = 0
M(0) >] 0 & M(0) [< 0
Teorema:
M(E) [< 0
Demostración:
M(E) = max{(-1)·(1/k)} = 0
M(E) [< 0 & M(E) >] 0
Definición: [ de medida exterior entera ]
M(A) = min{ k : A [<< [ || ]-[k = 1]-[n][ A_{k} ] }
M(¬A) = max{ (-k) : ¬A >>] [&]-[k = 1]-[n][ ¬A_{k} ] }
Teorema:
M( [ || ]-[k = 1]-[n][ A_{k} ] ) [< sum[k = 1]-[n][ M(A_{k}) ]
Demostración:
M( [ || ]-[k = 1]-[n][ A_{k} ] ) = M(A) = min{k} [< 1+...(n)...+n = ...
... sum[k = 1]-[n][ M(A_{k}) ]
Teorema:
M( [&]-[k = 1]-[n][ ¬A_{k} ] ) >] sum[k = 1]-[n][ M(¬A_{k}) ]
Demostración:
M( [&]-[k = 1]-[n][ ¬A_{k} ] ) = M(¬A) = max{(-k)} >] (-1)+...(n)...+(-n) = ...
... sum[k = 1]-[n][ M(¬A_{k}) ]
Teorema:
M(0) >] 0
Demostración:
M(0) = min{k} = 1
M(0) = 1 >] 0
Teorema:
M(E) [< 0
Demostración:
M(E) = max{(-k)} = (-1)
M(E) = (-1) [< 0
Definición: [ de medida exterior binaria desplazada ]
M(A) = min{ p+(1/k) : A [<< [ || ]-[k = 1]-[n][ A_{k} ] }
M(¬A) = max{ (-p)+(-1)·(1/k) : ¬A >>] [&]-[k = 1]-[n][ ¬A_{k} ] }
Demostrad que es una medida exterior.
Definición: [ de medida exterior entera desplazada ]
M(A) = min{ p+k : A [<< [ || ]-[k = 1]-[n][ A_{k} ] }
M(¬A) = max{ (-p)+(-k) : ¬A >>] [&]-[k = 1]-[n][ ¬A_{k} ] }
Demostrad que es una medida exterior.
Teorema:
Si f(x) = min{ z : [Ey][ y > 0 & z = | xy+(-a) | ] } ==> | f(a) | = 0
Si f(x) = max{ z : [Ey][ y < 0 & z = | ( xy+(-a) )·i | ] } ==> | f(-a)·i | = 0
Demostración:
Sea y = 1 ==>
f(x) = min{ z : [Ey][ y > 0 & z = | xy+(-a) | ] } [< | xy+(-a) | = | x+(-a) |
0 [< | f(a) | [< | a+(-a) | = 0
Ley:
No es interesante para joder un fiel,
porque no hay reverso tenebroso,
y hay condenación.
Es interesante para joder un infiel,
porque hay el reverso tenebroso,
y no hay condenación.
Ley:
Es aburrido en el Mal,
no tener reverso tenebroso,
de joder a fieles,
porque se tiene que amar.
Es interesante en el Mal,
tener reverso tenebroso,
de joder a infieles,
porque no se tiene que amar.
Anexo:
Tendrán que amar a la próximo como a si mismo con la familia,
cocinar, lavar o vatchnar a comprar.
para tener amor.
Tendrán que amar al prójimo como no a si mismo,
estudiar y der o datchnar la energía al prójimo,
para tener amor.
Por esto es aburrido en el Mal joder a un fiel,
porque se tiene que amar,
y no puedes ser un señor no estudiando.
Por eso es interesante en el Mal joder a un infiel,
porque no se tiene que amar,
y puedes ser un señor no estudiando.
Teorema: [ de Cardano-Tartaglia ]
Si x^{3}+ax+b = 0 ==> [Ep][Eq][ u^{6}+pu^{3}+q = 0 & v^{6}+pv^{3}+q = 0 & x = u+v ]
Demostración:
u^{3}+v^{3}+b = 0
v^{3}+u^{3}+b = 0
3uv·(u+v) = (-a)·(u+v)
Se define p = b
Se define q = (-1)·(1/27)·a^{3}
Teorema:
x^{3}+ax+b = (x+(-1)·(u+v))·(x+(-j))·(x+(-k))
Demostración:
x^{3}+ax+b = (x+(-1)·(u+v))·( x^{2}+(u+v)·x+( a+(u+v)^{2} ) )
Teorema: [ de Cardano-Ferrari de números reales ]
Si x^{4}+ax^{2}+bx+c = 0 ==> ...
... [Ep][ u^{3}+pu+b = 0 & v^{3}+pv+b = 0 & ( x = u+k || x = v+j ) ]
Demostración:
Sea x = u+v ==>
u^{4}+(a+w)·u^{2}+bu = 0
v^{4}+(a+w)·v^{2}+bv = 0
4·(uv)·( u^{2}+v^{2} ) = w·( u^{2}+v^{2} )
6·(uv)^{2}+2a·(uv)+c = 0
Se define p = a+w
Teorema: [ de Cardano-Ferrari de números imaginarios ]
Si x^{4}+ax^{2}+bx+c = 0 ==> ...
... [Ep][ u^{3}+(-p)·u+bi = 0 & v^{3}+(-p)·v+bi = 0 & ( x = ui+ki || x = vi+ji ) ]
Demostración:
Sea x = ui+vi ==>
u^{4}+(-1)·(a+w)·u^{2}+bui = 0
v^{4}+(-1)·(a+w)·v^{2}+bvi = 0
4i·(uv)·( u^{2}+v^{2} ) = w·( u^{2}+v^{2} )
(-6)·(uv)^{2}+2ai·(uv)+c = 0
Se define p = a+w
Teorema:
x^{4}+ax^{2}+bx+c = (x+(-1)·(u+j))·(x+(-1)·(v+k))·(x+(-i)·(u+j))·(x+(-i)·(v+k))
Teorema: [ de Cardano quíntico ]
Si x^{5}+ax^{3}+bx^{2}+cx+d = 0 ==> ...
... [Ep][Eq][ u^{4}+pu^{2}+bu+q = 0 & v^{4}+pv^{2}+bv+q = 0 & ( x = 2u+j+k || x = 2vi+ji+ki ) ]
Demostración:
Sea x = u+v ==>
u^{5}+(a+m)·u^{3}+bu^{2}+(c+w)·u = 0
v^{5}+(a+m)·v^{3}+bv^{2}+(c+w)·v = 0
5·(uv)·( u^{3}+v^{3} ) = m·( u^{3}+v^{3} )
10·(uv)^{2}·(u+v)+3a·(uv)·(u+v) = w·(u+v)
2b·(uv)+d = 0
Se define p = a+m
Se define q = c+w
Teorema:
x^{5}+ax^{3}+bx^{2}+cx+d = ( x+(-u) )·( x+(-v) )·...
... ( x^{3}+( u+v )·x^{2}+( ( a+(-1)·(uv) )+( u+v )^{2} )·x+...
... ( b+(-1)·( a·( u+v )+( u+v )^{3} ) ) = ...
Conjetura:
( a·( u+v )+( u+v )^{3} )+(uv) )·2x^{2} = 0
( (-b)·( u+v )+( a·( u+v )^{2}+( u+v )^{4} )·x = ...
... ( c+(-1)·( a·(uv)+(uv)^{2} ) )·x
( b+( a·( u+v )+( u+v )^{3} )·(uv) = d
(uv) = (1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} )
Demostración:
( ( x^{5}+ax^{3}+bx^{2}+cx+d ) / ( x^{2}+(-1)·( u+v )·x+(uv) ) ) = ...
... x^{3} | ...
... ( u+v )·x^{4}+( a+(-1)·(uv) )·x^{3}+bx^{2}+cx+d ...
...
... ( u+v )·x^{2} | ...
... ( ( a+(-1)·(uv) )+( u+v )^{2} )·x^{3}+( b+(-1)·( u+v )·(uv) )·x^{2}+cx+d
...
... ( a+(-1)·(uv)+( u+v )^{2} )·x | ...
... ( b+(-1)·( u+v )·(uv) )+(-1)·( a·( u+v )+(-1)·(uv)·( u+v )+( u+v )^{3} ) )·x^{2}+...
... ( c+(-1)·( a·(uv)+(-1)·(uv)^{2}+( u+v )^{2} )·(uv) ) )·x+d
...
... ( b+(-1)·(uv)·( u+v ) )+...
... (-1)·( a·( u+v )+(-1)·(uv)·( u+v )+( u+v )^{3} )
Teorema:
x^{5}+ax^{3}+bx^{2}+cx+d = ...
... ( x^{2}+...
... (-1)·...
... ( ( c+(-a)·( (1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} )+(-1)·( (1/4)·( b+( b^{2}+(-4)·d )^{(1/2)} )^{2} ) )...
... /...
... ( (-1)·( (3/2)·b+(1/2)·( b^{2}+(-4)·d )^{(1/2)} ) ) )·x+...
... (1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} ) ...
... )
... ( x^{3}+...
... ( ( c+(-a)·( (1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} )+(-1)·( (1/4)·( b+( b^{2}+(-4)·d )^{(1/2)} )^{2} ) )...
... /...
... ( (-1)·( (3/2)·b+(1/2)·( b^{2}+(-4)·d )^{(1/2)} ) ) ...
... )·x^{2}...
... +...
... ( ( a+(-1)·( (1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} ) ) )+...
... ( ...
... ( c+(-a)·( (1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} )+(-1)·( (1/4)·( b+( b^{2}+(-4)·d )^{(1/2)} )^{2} ) )...
... /...
... ( (-1)·( (3/2)·b+(1/2)·( b^{2}+(-4)·d )^{(1/2)} ) ) ...
... )^{2}...
... )·x...
... +...
... ( b+(1/2)·( b+( b^{2}+(-4)·d )^{(1/2)} ) ) ...
... )
Teorema:
[Ea][Eb][ x^{5}+2x^{3}+3x^{2}+5x+4 = (x+(-a))·(x+(-b))·P_{3}(x) ]
Demostración:
Examen de Álgebra I.
Teorema: [ de Cardano síxtico ]
Si x^{6}+ax^{4}+bx^{3}+cx^{2}+dx+h = 0 ==> ...
... [Ep][Eq][Ez][Es][ ...
... u^{5}+pu^{3}+qu^{2}+zu+s = (y+(-u))·(y+(-v))·P_{3}(j) = 0 & x = u+j & ...
... v^{5}+pv^{3}+qv^{2}+zv+s = (y+(-u))·(y+(-v))·P_{3}(k) = 0 & x = v+k ]
Teorema: [ de Galois ]
Sea n >] 5 ==>
P_{n}(x) = (x+(-1)·a_{1})·...·(x+(-1)·a_{n}) <==> ...
... [Ek][ 1 [< k [< n & P_{n}(a_{k}) no es resoluble por Cardano ni cuadrática ]
Demostración:
x^{n}+a_{n+(-2)}·x^{n+(-2)}+...+a_{0} = ...
... (x+(-a))·( x^{n+(-1)}+ax^{n+(-2)}+Q(x) )
P_{3}(x) = (x+(-a))·( x^{2}+ax+c )
x^{n}+a_{n+(-2)}·x^{n+(-2)}+...+a_{0} = ...
... (x^{2}+(-1)·(u+v)·x+uv)·( x^{n+(-2)}+(u+v)·x^{n+(-3)}+Q(x) )
P_{4}(x) = (x^{2}+(-1)·(u+v)·x+uv)·( x^{2}+(u+v)·x+d )
x^{n} = x·x^{n+(-1)} punto fijo de la división
ax^{n+(-1)} = x·ax^{n+(-2)} punto fijo de la división
ax^{n+(-2)} = 0 <==> a = 0
Sea P_{n}(x) = x·P_{n+(-1)}(x) ==>
P_{n+(-1)}(x) es resoluble por Cardano <==> P_{n}(x) es resoluble por Cardano
Sea P_{n}(x) = (x+(-a))·P_{n+(-1)}(x) ==>
P_{n+(-1)}(x) no es resoluble por Cardano <==> P_{n}(x) es resoluble por cardano.
Teorema:
Si n = 5 ==> ...
... P_{5}(x) = (x+(-u))·(x+(-v))·P_{3}(x) & ( P_{3}(x) = x^{3}+ax^{2}+bx+c & a != 0 )
Teorema:
{ < k,f(k) > : [Ak][ 1 [< k [< n ==> f(k) = k ] } es irresoluble por Cardano-Galois <==> n >] 5
Demostración:
f(a_{f(k)}·x^{f(k)}) = f(a_{k}·x^{k}) = x·a_{k}·x^{k+(-1)} = x·a_{f(k)}·x^{f(k)+(-1)}
Teorema: [ de virus de Church ]
a_{k} = ( a_{0} )^{m^{k}} es computablemente irresoluble <==> n >] 5
a_{k} = ( a_{0} )^{m^{(-k)}} es computablemente irresoluble <==> n >] 5
Demostración:
a_{f(k)} = a_{k} = ( a_{0} )^{m^{k}} = ( a_{0} )^{m^{f(k)}}
a_{0} = a_{0}
a_{f(k)} = a_{k} = ( a_{0} )^{m^{(-k)}} = ( a_{0} )^{m^{(-1)·f(k)}}
a_{0} = a_{0}
Anexo:
No penséis está sucesión del virus de Church,
porque podéis morir y no son infieles a los que matáis.
Teorema: [ de Turing ]
a_{k} = kx es computablemente resoluble
Demostración:
a_{0} = 0
Teorema: [ de Turing ]
a_{k} = x^{k} es computablemente resoluble
Demostración:
a_{0} = 1
No hay comentarios:
Publicar un comentario