topología-algebraica y álgebra y métodos-numéricos y análisis-funcional y series-de-Fourier-y-constante-de-Áperi y arte-matemático

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

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 >

(1/z)·(-v)+p = (1/z)·(-v)+q

p = 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}

c_{0} = (1/(pi·i))·(1/3)·(2pi·i)^{3} = (-1)·(8/3)·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.

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:

Sea n = 1 ==>

sum[k = 1]-[n][ ( ln(1+k) )^{k} ] = ln(2)

f(k) = 1

g(1) = n

sum[k = 1]-[n][ ( ln(1+k) )^{k} ] = sum[k = 1]-[n][ (1/g(1))·( ln(1+f(k)) )^{f(k)} ] = ...

... sum[k = 1]-[n][ (1/n)·ln(2) ] = ln(2)

Sea n = oo ==>

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:

Sea n = 1 ==>

sum[k = 1]-[n][ ln(1+k)+(-1)·(1/k) ]+(1/n) = ln(2)

f(k) = 1

sum[k = 1]-[n][ ln(1+k)+(-1)·(1/k) ]+(1/n) = sum[k = 1]-[n][ ln(1+f(k))+(-1)·(1/k) ]+(1/n) = ...

... sum[k = 1]-[n][ ln(2)+(-1)·(1/k) ]+(1/n)

Sea n = oo ==>

sum[k = 1]-[oo][ ln(1+k)+(-1)·(1/k) ] = sum[k = 1]-[oo][ ln(1+k)+(-1)·(1/k) ]+(1/oo) = ...

... sum[k = 1]-[oo][ ln(2)+(-1)·(1/k) ]+(1/oo) = sum[k = 1]-[oo][ ln(2)+(-1)·(1/k) ] = ...

... ln(2)·oo+(-1)·ln(oo) = ln(2)·oo+(-1)·ln(2)·oo = ln(2)