Definición:
B^{0} = O
B^{1} = B
B^{2} = BB
B^{n+2} = BO...(n)...OB
Teorema:
B^{n}·O = B^{n}
Demostración:
B^{n}·O = B^{n}·B^{0} = B^{n+0} = B^{n}
Teorema:
x^{2}+(-1)·BB = (x+B)·(x+(-B))
x^{2}+BB = (x+iB)·(x+(-i)·B)
Grupo suma y espacio vectorial:
Definición:
... a_{0}·O+a_{1}·B+sum[k = 0]-[n+(-1)][ a_{k+2}·BO...(k)...OB ] ...
... +...
... b_{0}·O+b_{1}·B+sum[k = 0]-[n+(-1)][ b_{k+2}·BO...(k)...OB ] = ...
... (a_{0}+b_{0})·O+(a_{1}+b_{1})·B+sum[k = 0]-[n+(-1)][ (a_{k+2}+b_{k+2})·BO...(k)...OB ]
Definición:
w·( a_{0}·O+a_{1}·B+sum[k = 0]-[n+(-1)][ a_{k+2}·BO...(k)...OB ] ) =...
... (w·a_{0})·O+(w·a_{1})·B+sum[k = 0]-[n+(-1)][ (w·a_{k+2})·BO...(k)...OB ]
Grupo producto por coordenada:
Definición:
... a_{0}·O+a_{1}·B+sum[k = 0]-[n+(-1)][ a_{k+2}·BO...(k)...OB ] ...
... [+ · +] ...
... b_{0}·O+b_{1}·B+sum[k = 0]-[n+(-1)][ b_{k+2}·BO...(k)...OB ] = ...
... (a_{0}·b_{0})·O+(a_{1}·b_{1})·B+sum[k = 0]-[n+(-1)][ (a_{k+2}·b_{k+2})·BO...(k)...OB ]
Teorema:
< BOB+BB+B+O,BB+B+O,B+O,O > es base
Demostración:
Independencia lineal:
a·(BOB+BB+B+O)+b·(BB+B+O)+c·(B+O)+d·O = 0
a·BOB+(a+b)·(BB)+(a+b+c)·B+(a+b+c+d)·O = 0
a = 0
a = 0 & b = 0
a = 0 & b = 0 & c = 0
a = 0 & b = 0 & c = 0 & d = 0
Generador:
a·BOB+b·BB+c·B+d·O = a·(BOB+BB+B+O)+(b+(-a))·(BB+B+O)+(c+(-b))·(B+O)+(d+(-c))·O
No entiendo que vos han dicho de los hombres,
si vos creéis Jesucristo,
y estáis pagando condenación del mundo salvando-lo,
porque no hay indulgencias,
y no es ataque de ningún hombre.
Definición:
[Ev][ F(x) = x+v ]
Teorema:
F(x+y) = F(x)+F(y)
F(w·x) = w·F(x)
Demostración:
F(x+y) = (x+y)+v = (x+y)+( (1/2)·v+(1/2)·v ) = (x+(1/2)·v)+(y+(1/2)·v) = (x+p)+(y+q) = F(x)+F(y)
F(w·x) = w·x+v = w·( x+(1/w)·v ) = w·(x+s) = w·F(x)
Teorema:
Ker(F) = {(-v)}
Demostración:
F(-v) = (-v)+v = 0
Teorema:
Si ( E/Ker(F) ) = {z+(-v)} ==> F[ ( E/Ker(f) ) ] = E
Demostración:
F(z+(-v)) = (z+(-v))+v = z+((-v)+v) = z+0 = z
Teorema:
Si ( E/Ker(F) ) = {z+(-v)} ==> Im(F) =[h(z)]= ( E/Ker(F) )
Demostración:
h(x) = h(y)
x+(-v) = y+(-v)
x = y
h(x+(-v)) = f(y+(-v))
x = y
x+(-v) = y+(-v)
Definición:
[Ev][ F(x,y) = xy+v ]
Teorema:
F(z,x+y) = F(z,x)+F(z,y)
F(z,w·x) = w·F(z,x)
Demostración:
F(z,x+y) = z·(x+y)+v = (zx+zy)+v = (zx+zy)+( (1/2)·v+(1/2)·v ) = (zx+(1/2)·v)+(zy+(1/2)·v) = ...
... (zx+p)+(zy+q) = F(z,x)+F(z,y)
F(z,w·x) = z·(w·x)+v = w·(zx)+v = w·( zx+(1/w)·v ) = w·(zx+s) = w·F(z,x)
Teorema:
Ker(F) = { < z,(1/z)·(-v) > || < (-v)·(1/z),z > }
Demostración:
F(z,(1/z)·(-v)) = z·((1/z)·(-v))+v = (z/z)·(-v)+v = (-v)+v = 0
Teorema:
Si ( E/Ker(F) ) = { < z,(1/z)·(-v)+s > || < (-v)·(1/z)+s,z > } ==> Im(F) =[h(w)]= ( E/Ker(F) )
Demostración:
h(z,(1/z)·(-v)+p) = h(z,(1/z)·(-v)+q)
zp = zq
p = q
(1/z)·(-v)+p = (1/z)·(-v)+q
< z,(1/z)·(-v)+p > = < z,(1/z)·(-v)+q >
h(zp) = h(zq)
< z,(1/z)·(-v)+p > = < z,(1/z)·(-v)+q >
zp = zq
Teorema:
Sea F(x,y) = xy+BB ==>
Ker(F) = { < B^{n},(O/B)^{n}·(-1)·BB > || < (-1)·BB·(O/B)^{n},B^{n} > }
Teorema:
Si d_{t}[z] = f(t)·z ==> z_{n}(t) = z_{0}·( 1+h·f(t) )^{n}
Si h = 0·( int[f(t)]d[t]/f(t) ) ==>
... z_{n}(t) = ( 1+0·int[f(t)]d[t] )^{n}
Método numérico convergente:
(1/h)·( z_{n+1}+(-1)·z_{n} ) = f(t)·z_{n}
z_{0} = 1
Demostración:
(1/h)·( z_{n+1}+(-1)·z_{n} ) = f(t)·z_{n}
( z_{n+1}+(-1)·z_{n} ) = h·f(t)·z_{n}
z_{n+1} = z_{n}+h·f(t)·z_{n}
z_{n+1} = z_{n}·(1+h·f(t))
Teorema:
Si d_{t}[z] = f(t)·(1/z) ==> a_{n}(t) = a_{0}+(n/2)·h·f(t)
Si h = (1/n) ==>
... a(t) = a_{0}+(1/2)·f(t)
... z(t) = ( 2a_{0}+f(t) )^{(1/2)} & a_{0} = int[f(t)]d[t]+(-1)·(1/2)·f(t)
Método numérico convergente:
(1/h)·( z_{n+1}+(-1)·z_{n} ) = ( f(t)/z_{n} )
z_{0} = ( 2·int[f(t)]d[t]+(-1)·f(t) )^{(1/2)}
Demostración:
(1/h)·( z_{n+1}+(-1)·z_{n} ) = ( f(t)/z_{n} )
( z_{n+1}+(-1)·z_{n} ) = ( (h·f(t))/z_{n} )
z_{n+1} = z_{n}+( (h·f(t))/z_{n} )
z_{n+1}·z_{n} = ( z_{n} )^{2}+h·f(t)
Sea z_{n} = ( 2a_{n} )^{(1/2)} & z_{n+1}·z_{n} = 2a_{n+1} ==>
2a_{n+1} = 2a_{n}+h·f(t)
a_{n+1} = a_{n}+(1/2)·h·f(t)
Teorema:
Forma integral interior:
Sea F(ax+b) = int[x = 0]-[1][ ax+b ]d[x] ==>
G(ax+b) = int[x = 0]-[1][ (8/a)·x+(-1)·(1/b) ]d[x]
F(ax+b) [o] G(ax+b) = 1
Teorema:
Forma integral exterior:
Sea F(ax+b) = int[x = 0]-[1][ ax+b ]d[x] ==>
G(ax+b) = int[x = 0]-[1][ (4/a)·x+(-1)·(1/b) ]d[x]
F(ax+b) [o] G(ax+b) = 0
Teorema:
Forma funcional interior:
Sea F(h(x)) = sum[k = 1]-[n][ ( h(x) )^{k} ]+1 ==>
G(h(x)) = sum[k = 1]-[n][ (1/h(x))^{k} ]+((-n)+1)
F(h(x)) [o] G(h(x)) = 1
Teorema:
Forma funcional exterior:
Sea F(h(x)) = sum[k = 1]-[n][ ( h(x) )^{k} ]+1 ==>
G(h(x)) = sum[k = 1]-[n][ (1/h(x))^{k} ]+(-n)
F(h(x)) [o] G(h(x)) = 0
Teorema:
< cosh(kx), sinh(kx) > es linealmente independiente
Demostración
a·cosh(kx)+b·sinh(kx) = 0
(1/2)·( (a+b)·e^{kx}+(a+(-b))·e^{(-k)·x} ) = 0
(-a) = b = a
a = 0 & b = 0
Teorema:
sum[k = 0]-[n][ a_{k}·e^{kx}] = sum[k = 0]-[n][ a_{k}·cosh(kx)+a_{k}·sinh(kx) ]
Teorema:
sum[k = 0]-[n][ a_{k}·e^{(-k)·x}] = sum[k = 0]-[n][ a_{k}·cosh(kx)+a_{k}·(-1)·sinh(kx) ]
Teorema:
sum[k = 0]-[n][ a_{k}·e^{kxi}] = sum[k = 0]-[n][ a_{k}·cosh(kxi)+a_{k}·(1/i)·sinh(kxi) ]
Teorema:
sum[k = 0]-[n][ a_{k}·e^{(-k)·xi}] = sum[k = 0]-[n][ a_{k}·cosh(kx)+a_{k}·i·sinh(kxi) ]
Definición:
[Ex][ f_{sup{k}}(x) = c_{0}+sum[k = 1]-[oo][ a_{k}·cosh(x)+b_{k}·sinh(x) ] ]
c_{0} = (1/(2pi·i))·int[x = 0]-[2pi·i][ f_{1}(x) ]d[x]
a_{k} = (1/(pi·i))·int[x = 0]-[2pi·i][ f_{k}(x)·cosh(x) ]d[x]
b_{k} = (-1)·(1/(pi·i))·int[x = 0]-[2pi·i][ f_{k}(x)·sinh(x) ]d[x]
Axioma:
Si f_{k}(x) = (x/k)^{s} ==> sup{(1/k)} = max{s+(-1),1}
Teorema:
Sea f_{k}(x) = (x/k)^{n} ==>
Si n >] 4 ==> No es resoluble el método
Demostración:
Sea 0 [< j [< n ==>
x^{n+(-j)} = k^{n+(-j)}
Existen más de 5 puntos fijos
No es resoluble el método
Teorema:
Sea f_{k}(x) = kx ==>
(oo·x) = 2·0·sinh(x)·sum[k = 1]-[oo][ k ]
Sea x = 0 ==>
sum[k = 1]-[oo][ k ] = (1/2)·oo^{2}
Teorema:
Sea f_{k}(x) = (x/k) ==>
sup{(1/k)} = max{s+(-1),1} = max{(1+(-1)),1} = 1
x = 2·0·sinh(x)·sum[k = 1]-[oo][ (1/k) ]
Sea x = ln(oo) ==>
sum[k = 1]-[oo][ (1/k) ] = ln(oo)
Teorema:
Sea f_{k}(x) = (x/k)^{2} ==>
sup{(1/k)} = max{s+(-1),1} = max{(2+(-1)),1} = 1
x^{2} = (-1)·(8/3)·pi^{2}+(-1)·2·4·cosh(x)·sum[k = 1]-[oo][ (1/k)^{2} ]
Sea x = (2pi·i) ==>
sum[k = 1]-[oo][ (1/k)^{2} ] = (1/6)·pi^{2}
Teorema:
Sea f_{k}(x) = (x/k)^{3} ==>
sup{(1/k)} = max{s+(-1),1} = max{(3+(-1)),1} = 2
(2x)^{3} = (-1)·3·2·4·sinh(x)·sum[k = 1]-[oo][ (1/k)^{3} ]
Sea x = (pi/2)·i ==>
sum[k = 1]-[oo][ (1/k)^{3} ] = (1/24)·pi^{3}
Teorema:
Sea f_{k}(x) = (x/k)^{5 || ( 4 ==> 5 ) || ( 3 ==> 4 ) || ( 2 ==> 1 ) }} ==>
f_{k}(x) es resoluble
n = ( 5 || 4 || 1 || 0 )
sup{(1/k)} = max{s+(-1),1} = max{(5+(-1)),1} = 4
(4x)^{5} = (-1)·5·(4·(-5))·(3·(-4))·(2·(-1))·4·sinh(x)·sum[k = 1]-[oo][ (1/k)^{5} ]
Sea x = (pi/2)·i ==>
sum[k = 1]-[oo][ (1/k)^{5} ] = (1/300)·pi^{5}
Teorema:
f_{k}(x) = (x/k)^{7 || 7 ==> 7 || 6 ==> 45 || 5 ==> 45 || 4 ==> 4 || 3 ==> 9 || 2 ==> 2 }
f_{k}(x) es resoluble
n = ( 45 || 9 || 1 || 0 )
sup{(1/k)} = max{s+(-1),1} = max{(7+(-1)),1} = 6
(6x)^{7} = ...
... (-1)·(7/(-7))·(6·(-45))·(5·(-45))·(4/(-4))·(3·(-9))·(2/(-2))·4·sinh(x)·sum[k = 1]-[oo][ (1/k)^{7} ]
Sea x = (pi/2)·i ==>
sum[k = 1]-[oo][ (1/k)^{7} ] = (1/3,000)·pi^{7}
Principio del Mal:
Rezar al próximo,
sin condenación instantánea,
no amando al próximo como a ti mismo.
Rezar al prójimo,
con condenación instantánea,
amando al prójimo como a ti mismo.
Ley:
Rezar al Mal proyectado le pasa al cuerpo del próximo:
de cuerpo del próximo semejante al próximo,
no amando al próximo como a ti mismo.
Rezar al Mal proyectado no le pasa al cuerpo del prójimo:
de alma del próximo semejante al prójimo,
amando al prójimo como a ti mismo.
Ley:
Quizás soy Jûan Bautista de verdad,
porque me rezan un psiquiatra,
creyendo-me senador romano.
Arte: [ de serie de Laurent ]
[En][ d_{a...a}^{n}[f(a)] = (-1)^{n}·(n+(-1))!·d_{a...a}^{n}[f(a)] ]
Exposición:
n = 2
f(1) = (1/n)
g(1/n) = 0
H(1) = z
[o(1)o] = [o(H(1))o] = [o(z)o]
r = 0 & z = re^{x}+a
d_{z...z}^{n+1}[f(z)] = d_{z}^{1}[ d_{z...z}^{n}[f(z)] ] = d_{z}^{f(1)}[ d_{z...z}^{n}[f(z)] ] = ...
... d_{z}^{(1/n)}[ d_{z...z}^{n}[f(z)] ] = d_{z}^{g(1/n)}[ d_{z...z}^{n}[f(z)] ] = ...
... d_{z}^{0}[ d_{z...z}^{n}[f(z)] ] = d_{z...z}^{n}[f(z)]
Por inducción:
d_{z...z}^{n}[f(z)] = (-1)^{n}·(n+(-1))!·d_{z...z}^{n}[f(z)] = ...
... int-...(n)...-int[x = 0]-[1][z = re^{x}+a][ ...
... (n+(-1))!·d_{z...z}^{n}[f(z)]·(1/(a+(-z))^{n})·d[z]...(n)...d[z] = ...
... int-...(n+1)...-int[x = 0]-[1][z = re^{x}+a][ ...
... (n+(-1))!·d_{z}[ d_{z...z}^{n}[f(z)]·(1/(a+(-z))^{n}) ]·d[z]...(n+1)...d[z] = ...
... int-...(n+1)...-int[x = 0]-[1][z = re^{x}+a][ ...
... (n+(-1))!·d_{z}[ d_{z...z}^{n}[f(z)] [o(z)o] (1/(a+(-z))^{n}) ]·d[z]...(n+1)...d[z] = ...
... int-...(n+1)...-int[x = 0]-[1][z = re^{x}+a][ ...
... n!·d_{z...z}^{n+1}[f(z)]·(1/(a+(-z))^{n+1})·d[z]...(n+1)...d[z] = (-1)^{n+1}·n!·d_{z...z}^{n+1}[f(z)]
Arte:
[Ex][ e^{x} = 1+sum[k = 1]-[oo][ (-1)^{n}·(1/n)·x^{n} ] ]
[Ex][ e^{(-x)} = 1+sum[k = 1]-[oo][ (1/n)·x^{n} ] ]
Arte: [ de falsus infinitorum ]
sum[k = 1]-[oo][ ( ln(1+k) )^{k} ] != ln(2)
sum[k = 1]-[oo][ ( ln(1+(1/k)) )^{k} ] != ln(2)
Exposición:
sum[k = 1]-[n][ ( ln(1+k) )^{k} ] = ln(2)
Arte: [ de falsus infinitorum ]
sum[k = 1]-[oo][ ln(1+k)+(-1)·(1/k) ] != ln(2)
sum[k = 1]-[oo][ ln(1+(1/k))+(-1)·(1/k) ] != ln(2)
Exposición:
sum[k = 1]-[n][ ln(1+k)+(-1)·(1/k) ]+(1/n) = ln(2)
