topología-algebraica-medida y homología-algebraica y topología y Álgebra-polinomios

Definición: [ de medida exterior ]

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

análisis-matemático y topología-algebraica-medida y análisis-real y economía

Teorema:

lim[n = oo][ int[x = 0]-[n][ ( 1+(x/n) )^{n}·e^{(-1)·2x} ]d[x] ] = 1

Teorema:

lim[n = oo][ int[x = 0]-[n][ ( 1+(-1)·( (2x)/n ) )^{n}·e^{x} ]d[x] ] = 1

Teorema:

lim[n = oo][ int[x = 0]-[n][ ( 1+(1/p)·(x/n) )^{n}·e^{(-1)·( 1+(1/p) )·x} ]d[x] ] = 1

Teorema:

lim[n = oo][ int[x = 0]-[n][ ( 1+(-1)·(1/p)·(x/n) )^{n}·e^{( (-1)+(1/p) )·x} ]d[x] ] = 1

Teorema:

lim[n = oo][ int[x = 0]-[ln(n^{m+1})][ ( nx^{m}/(1+n) )·e^{(-x)} ]d[x] ] = m!

Teorema:

lim[n = oo][ int[x = 0]-[ln(n^{(1/m)+1})][ ( nx^{(1/m)}/(1+n) )·e^{(-x)} ]d[x] ] = m

Teorema de Hôpital-Garriga:

Teorema:

Si lim[x = 0][ g(x) ] = 0 ==>

lim[x = 0][ ( f(x) /o(x)o/ g(x) ) ] = lim[x = 0][ ( f(x) /o(x)o/ d_{x}[g(x)] )·(1/d[x]) ]

Si lim[x = oo][ g(x) ] = oo ==>

lim[x = oo][ ( f(x) /o(x)o/ g(x) ) ] = lim[x = oo][ ( f(x) /o(x)o/ d_{x}[g(x)] )·d[x] ]

Demostración:

g(x) = 0

lim[h = 0][ g(x+h)+(-1)·g(x) ] = lim[h = 0][ g(x+h)+(-0) ] = g(x)

g(x) = oo

lim[h = 0][ (1/h)^{2}·( g(x+h)+(-1)·g(x) ) ] = oo·d_{x}[g(x)] = g(x)

Teorema: [ de la integral de Euler ]

Sea n >] 1 ==>

int[x = 0]-[oo][ e^{(-1)·x^{n}} ]d[x] = (-1)·(0/n!)+(1/n!) = (1/n!)

Teorema: [ de la integral de Euler-Garriga ]

Sea n >] 1 ==>

int[x = 0]-[oo][ ( 1/(1+x^{2n}) ) ]d[x] = (pi/2)·(1/n!)+(-1)·( 0·(1/n!) ) = (pi/2)·(1/n!)

Definición: [ medida de densidad-integral ]

M( A,f(x) ) = A·int[ f(x) ]d[x]

M( (-A),f(x) ) = (-A)·int[ f(x) ]d[x]

Teorema:

M( 0,f(x) ) = int[ f(x) ]d[x]·0

Teorema:

M( (-0),f(x) ) = (-1)·int[ f(x) ]d[x]·0

Teorema:

Si A [< B ==>

M( max{A,B},f(x) ) = M( A,f(x) )+M( B,f(x) )

<==>

M( A,f(x) ) = int[ f(x) ]d[x]·0

Demostración:

( B+(-B) )·int[ f(x) ]d[x] = int[ f(x) ]d[x]·0

Teorema:

Si (-B) [< (-A) ==>

M( min{(-A),(-B)},f(x) ) = M( (-A),f(x) )+M( (-B),f(x) )

<==>

M( (-A),f(x) ) = (-1)·int[ f(x) ]d[x]·0

Demostración:

(-1)·( B+(-B) )·int[ f(x) ]d[x] = (-1)·int[ f(x) ]d[x]·0

Definición: [ medida de densidad-diferencial ]

W( A,f(x) ) = A·d_{x}[ f(x) ]

W( (-A),f(x) ) = (-A)·d_{x}[ f(x) ]

Teorema:

W( 0,f(x) ) = d_{x}[ f(x) ]·0

Teorema:

W( (-0),f(x) ) = (-1)·d_{x}[ f(x) ]·0

Teorema:

Si A [< B ==>

W( max{A,B},f(x) ) = W( A,f(x) )+W( B,f(x) )

<==>

W( A,f(x) ) = d_{x}[ f(x) ]·0

Demostración:

( B+(-B) )·d_{x}[ f(x) ] = d_{x}[ f(x) ]·0

Teorema:

Si (-B) [< (-A) ==>

W( min{(-A),(-B)},f(x) ) = W( (-A),f(x) )+W( (-B),f(x) )

<==>

W( (-A),f(x) ) = (-1)·d_{x}[ f(x) ]·0

Demostración:

(-1)·( B+(-B) )·d_{x}[ f(x) ] = (-1)·d_{x}[ f(x) ]·0

Medidas Continuas:

Teorema:

[As][ s > 0 ==> [Ed][ d > 0 & ( Si H(B) < d ==> M( H(B),f(x) ) < s ) ] ]

[A(-s)][ (-s) < 0 ==> [E(-d)][ (-d) < 0 & ( Si H(-B) > (-d) ==> M( H(-B),f(x) ) > (-s) ) ] ]

Demostración:

Sea H(B) = A ==>

M( A,f(x) ) = A·int[ f(x) ]d[x] = int[ f(x) ]d[x]·0 < s

Sea H(-B) = (-A) ==>

M( (-A),f(x) ) = (-A)·int[ f(x) ]d[x] = (-1)·int[ f(x) ]d[x]·0 > (-s)

Teorema:

[As][ s > 0 ==> [Ed][ d > 0 & ( Si H(B) < d ==> W( H(B),f(x) ) < s ) ] ]

[A(-s)][ (-s) < 0 ==> [E(-d)][ (-d) < 0 & ( Si H(-B) > (-d) ==> W( H(-B),f(x) ) > (-s) ) ] ]

Demostración:

Sea H(B) = A ==>

W( A,f(x) ) = A·d_{x}[ f(x) ] = d_{x}[ f(x) ]·0 < s

Sea H(-B) = (-A) ==>

W( (-A),f(x) ) = (-A)·d_{x}[ f(x) ] = (-1)·d_{x}[ f(x) ]·0 > (-s)

Teorema:

Sea A = x ==> d_{x}[ M( x,f(x) ) ] = M( 1,f(x) )+M( x,d_{x}[f(x)] )

Sea A = (-x) ==> d_{x}[ M( (-x),f(x) ) ] = M( (-1),f(x) )+M( (-x),d_{x}[f(x)] )

Teorema:

Sea A = x ==> d_{x}[ W( x,f(x) ) ] = W( 1,f(x) )+W( x,d_{x}[f(x)] )

Sea A = (-x) ==> d_{x}[ W( (-x),f(x) ) ] = W( (-1),f(x) )+W( (-x),d_{x}[f(x)] )

Definición:

M( A,f(x_{1},...,x_{n}) ) = A·int-...[n]...-int[ f(x_{1},...,x_{n}) ]d[x_{1}]...d[x_{n}]

M( (-A),f(x_{1},...,x_{n}) ) = (-A)·int-...[n]...-int[ f(x_{1},...,x_{n}) ]d[x_{1}]...d[x_{n}]

Definición:

W( A,f(x_{1},...,x_{n}) ) = A·d_{x_{1}...x_{n}}^{n}[ f(x_{1},...,x_{n}) ]

W( (-A),f(x_{1},...,x_{n}) ) = (-A)·d_{x_{1}...x_{n}}^{n}[ f(x_{1},...,x_{n}) ]

Examen:

Demostrad que son medidas continuas.

Teorema:

Si [Ea][Ax][ 0 [< f(x) [< a ] ==> f(x) es continua a x = a.

Demostración:

| f(x)+(-1)·f(a) | [< | a+(-a)+h | [< |x+(-a)|+|h| < 2h < s

Teorema:

Si [Ea][Ax][ 1 [< ( f(x) )^{(1/n)} [< a ] ==> f(x) es continua en x = a.

Demostración:

| f(x)+(-1)·f(a) | [< | a^{n}+(-1)·a^{n}+h | = ...

... | a^{n}+(-1)·a^{n} |+|h| = |a+(-a)|·|P(a)|+|h| [< |x+(-a)|·|P(a)|+|h| = ( |P(a)|+1 )·h < s

Teorema:

Si [Ea][Ax][ 0 [< f(x) [< |a| ] ==> f(x) es continua en x = a.

Demostración:

| f(x)+(-1)·f(a) | [< | |a|+(-1)·|a|+h | = | |a|+(-1)·|a| |+|h| = |a+(-a)|+|h| [< |x+(-a)|+|h| = 2h < s

Teorema:

Si F(x) es continua ==> f(x) es integrable Newton-Riemann.

Demostración:

| int[x = x]-[x+h][ f(x) ]d[x] | = | F(x+h)+(-1)·F(x) | < s

Teorema:

Si F(x) es continua ==> f(x) es integrable Newton-Lebesgue.

Demostración:

| lim[n = oo][ int[x = x]-[x+h][ f_{n}(x) ]d[x] ] | = | lim[n = oo][ F_{n}(x+h)+(-1)·F_{n}(x) ] | = ...

... | F(x+h)+(-1)·F(x) | < s

Teorema:

Si F(x) es monótona ==> f(x) es integrable Newton-Riemann.

Demostración:

| int[x = x]-[x+h][ f(x) ]d[x] | = | F(x+h)+(-1)·F(x) | = 0+h = h < s

Teorema:

Si F(x) es monótona ==> f(x) es integrable Newton-Lebesgue.

Demostración:

| lim[n = oo][ int[x = x]-[x+h][ f_{n}(x) ]d[x] ] | = | lim[n = oo][ F_{n}(x+h)+(-1)·F_{n}(x) ] | = ...

... | F(x+h)+(-1)·F(x) | = 0+h = h < s

Definición: [ de integral continua de Newton-Leibniz ]

Sea S( f(x),x ) = a·f(x)·x ==>

f(x)·x = int[ S( f(x+h),(x+h) )+(-1)·S( f(x),x ) ]

[As][ s >0 ==> [Ed][ d > 0 & ( Si h < d ==> | S( f(x+h),(x+h) )+(-1)·S( f(x),x ) | < s ) ] ]

Teorema:

Si f(x) es continua ==> f(x) es integrable continua Newton-Leibniz.

Demostración:

| S( f(x+h),(x+h) )+(-1)·S( f(x),x ) | = |a|·| f(x+h)·(x+h)+(-1)·f(x)·x | = ...

... |a|·| ( f(x+h)+(-1)·f(x) )·x+f(x+h)·h | [< ...

... |a|·| ( f(x+h)+(-1)·f(x) )·x |+|f(x+h)·h| = |a|·( |x|+|f(x+h)| )·h < s

Teorema:

Si f(x) es monótona ==> f(x) es integrable continua Newton-Leibniz.

Demostración:

| S( f(x+h),(x+h) )+(-1)·S( f(x),x ) | = |a|·| f(x+h)·(x+h)+(-1)·f(x)·x | = ...

... |a|·| ( f(x+h)+(-1)·f(x) )·x+f(x+h)·h| [< |a|·| ( f(x+h)+(-1)·f(x) )·x |+|f(x+h)·h| = ...

... |a|·( |(0+h)·x|+|f(x+h)·h| ) = |a|·( |x|+|f(x+h)| )·h < s

Teorema:

f(x) = x^{n} es integrable continua Newton-Leibniz.

Demostración:

S(x^{n},x) = x^{n+1}

| S( (x+h)^{n},(x+h) )+(-1)·S( x^{n},x ) | = | (x+h)^{n+1}+(-1)·x^{n+1} | = ...

... | P(x^{n+(-k)},h^{k}) |·h < s

Teorema:

f(x) = e^{nx} es integrable continua Newton-Leibniz.

Demostración:

S(e^{nx},x) = e^{nx}·x

| S( e^{n·(x+h)},(x+h) )+(-1)·S( e^{nx},x ) | = | e^{n·(x+h)}·(x+h)+(-1)·e^{nx}·x | = ...

... | e^{nx}·( e^{nh}+(-1) )·x+e^{n·(x+h)}·h | = | e^{nx}·( nh+P( (nh)^{n} ) )·x+e^{n·(x+h)}·h | = ...

... | ne^{nx}( 1+P( (nh)^{n} ) )·x+e^{n·(x+h)} |·h < s

Teorema:

f(x) = ln(x) es integrable continua Newton-Leibniz.

Demostración:

S(ln(x),x) = ln(x)·x

| S( ln(x+h),(x+h) )+(-1)·S( ln(x),x ) | = | ln(x+h)·(x+h)+(-1)·ln(x)·x | = ...

... | ln(1+(h/x))·x+ln(x+h)·h | = | (h/x)·( 1+P( (h/x)^{n} ) )·x+ln(x+h)·h | = ...

... | ( 1+P( (h/x)^{n} ) )+ln(x+h) |·h < s

Teorema:

f(x) = sin(x) es integrable continua Newton-Leibniz.

Demostración:

S(sin(x),x) = sin(x)·x

| S( sin(x+h),(x+h) )+(-1)·S( sin(x),x ) | = | sin(x+h)·(x+h)+(-1)·sin(x)·x | = ...

... | ( sin(x+h)+(-1)·sin(x) )·x+sin(x+h)·h | = ...

... | ( sin(x)·P(h^{n+(-1)})·h+cos(x)·h )·x+sin(x+h)·h | = ...

... | ( sin(x)·P(h^{n+(-1)})+cos(x) )·x+sin(x+h) |·h < s

Economía de objetos de empresa:

Lema: [ de la ventana ]

F(x,y) = 2x+2y+(-h)·( xy+(-1)·ab )

h = ( (a+b)/(ab) )

F(a,b) = 2a+2b

G(x,y) = 2x+2y+(-h)·xy

G(a,b) = h·ab

Disertación:

d_{x}[F(x,y)] = 2+(-h)·y = 0

d_{y}[F(x,y)] = 2+(-h)·x = 0

2x+(-h)·xy = 0x = 0

2y+(-h)·yx = 0y = 0

a+b = h·ab

h = ( (a+b)/(ab) )

Lema: [ de la puerta ]

F(x,y) = x+2y+(-h)·( xy+(-1)·ab )

h = ( ((a/2)+b)/(ab) )

F(a,b) = a+2b

G(x,y) = x+2y+(-h)·xy

G(a,b) = h·ab

Disertación:

d_{x}[F(x,y)] = 1+(-h)·y = 0

d_{y}[F(x,y)] = 2+(-h)·x = 0

x+(-h)·xy = 0x = 0

2y+(-h)·yx = 0y = 0

a+2b = 2h·ab

h = ( ((a/2)+b)/(ab) )

Lema: [ de la caja con tapa ]

F(x,y,z) = 2xy+2yz+2zx+(-h)·( xyz+(-1)·abc )

h = ( ((4/3)·(ab)+(4/3)·(bc)+(4/3)·(ca))/(abc) )

F(a,b,c) = 2ab+2bc+2ca

G(x,y,z) = 2xy+2yz+2zx+(-h)·xyz

G(a,b,c) = (1/2)·h·abc

Disertación:

d_{x}[F(x,y,z)] = 2y+2z+(-h)·yz = 0

d_{y}[F(x,y,z)] = 2x+2z+(-h)·zx = 0

d_{z}[F(x,y,z)] = 2y+2x+(-h)·xy = 0

2xy+2xz+(-h)·xyz = 0x = 0

2yx+2yz+(-h)·xyz = 0y = 0

2zy+2zx+(-h)·xyz = 0z = 0

4ab+4bc+4ca = 3h·abc

h = ( ((4/3)·(ab)+(4/3)·(bc)+(4/3)·(ca))/(abc) )

Lema: [ de la caja sin tapa ]

F(x,y,z) = xy+2yz+2zx+(-h)·( xyz+(-1)·abc )

h = ( ((2/3)·(ab)+(4/3)·(bc)+(4/3)·(ca))/(abc) )

F(a,b,c) = ab+2bc+2ca

G(x,y,z) = xy+2yz+2zx+(-h)·xyz

G(a,b,c) = (1/2)·h·abc

Disertación:

d_{x}[F(x,y,z)] = y+2z+(-h)·yz = 0

d_{y}[F(x,y,z)] = x+2z+(-h)·zx = 0

d_{z}[F(x,y,z)] = 2y+2x+(-h)·xy = 0

xy+2xz+(-h)·xyz = 0x = 0

yx+2yz+(-h)·xyz = 0y = 0

2zy+2zx+(-h)·xyz = 0z = 0

2ab+4bc+4ca = 3h·abc

h = ( ((2/3)·(ab)+(4/3)·(bc)+(4/3)·(ca))/(abc) )

Lema:

F(x,y) = 2x+2y+(-h)·( xy+(-1)·2! )

F(1,2) = 6

h = (3/2)

Disertación:

Laboratorio de problemas

Lema:

F(x,y,z) = 2xy+2yz+2zx+(-h)·( xyz+(-1)·3! )

F(1,2,3) = 22

h = (4/3)·(11/6) = (22/9)

Disertación:

Laboratorio de problemas.

Lema:

F(x,y) = x+2y+(-h)·( xy+(-1)·2! )

F(2,1) = 4

h = 1

Disertación:

Laboratorio de problemas.

Lema:

F(x,y,z) = xy+2yz+2zx+(-h)·( xyz+(-1)·3! )

F(3,2,1) = 16

h = (2/3)·(16/6) = (16/9)

Disertación:

Laboratorio de problemas.

Teorema:

lim[n = oo][ ( sum[k = 1]-[n][ (1/k) ]/ln(n) ) ] = 1

sum[k = 1]-[oo][ (1/k) ] = ln(2)·oo

Demostración:

lim[n = oo][ ( 1/( n·( ln(n+1)+(-1)·ln(n) )+ln(n+1)+(-1)·ln(n) ) ) ] = ( 1/(1+0) ) = 1

Teorema:

lim[n = oo][ ( sum[k = 1]-[n][ ln(k) ]/( (1/e)·n·ln(n) ) ) ] = 1

sum[k = 1]-[oo][ ln(k) ] = (1/e)·ln(2)·oo^{2} [< (1/2)·oo^{2} = sum[k = 1]-[oo][ k ]

Demostración: [ por destructor en Stolz ]

ln(e) = 1

lim[n = oo][ e·( ln(n+1)/( n·( ln(n+1)+(-1)·ln(n) )+ln(n+1) ) ) ] = e·( ln(oo)/( 1+ln(oo) ) ) = e = ln(e) = 1

Teorema:

ln(2)+(-1)·ln(2) = 0^{2}

Demostración:

ln(oo) < oo

ln(2) < 1 

ln(oo)+(-1)·ln(oo) = oo·( ln(2)+(-1)·ln(2) ) = lim[n = oo][ ln(1+(p/n)) ] = 0

Constante de Euler:

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

Teorema infinitorum de Euler:

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

Demostración:

( ln(2)·oo )^{2}+(-1)·( ln(2)·oo )^{2} = ln(2)·oo^{2}·( ln(2)+(-1)·ln(2) ) = ln(2)

( ln(2)·oo )^{2}+(-1)·( ln(2)·oo )^{2} = ln(2)·oo·( ln(2)·oo+(-1)·ln(2)·oo ) = ln(2)

( ln(2)·oo )^{2}+(-1)·( ln(2)·oo )^{2} = ln(2)·( ln(2)·oo^{2}+(-1)·ln(2)·oo^{2} ) = ln(2)

Número de Euler inverso:

lim[n = oo][ sum[k = 1]-[n][ ln(k) ]+(-1)·(1/e)·n·ln(n) ] = (1/e)

Teorema infinitorum de Euler-Garriga:

lim[n = oo][ sum[k = 1]-[n][ ln(k) ]+(-1)·(1/e)·n^{2}·ln(2+(1/n)) ] = (1/e)

Demostración:

( (1/e)·ln(2)·oo^{2} )+(-1)·( (1/e)·ln(2)·oo^{2} ) = (1/e)·oo^{2}·( ln(2)+(-1)·ln(2) ) = (1/e)

( (1/e)·ln(2)·oo^{2} )+(-1)·( (1/e)·ln(2)·oo^{2} ) = (1/e)·oo·( ln(2)·oo+(-1)·ln(2)·oo ) = (1/e)

( (1/e)·ln(2)·oo^{2} )+(-1)·( (1/e)·ln(2)·oo^{2} ) = (1/e)·( ln(2)·oo^{2}+(-1)·ln(2)·oo^{2} ) = (1/e)

Arte:

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

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

Exposición:

n = 1

f(n) = 1

u(k) = 1

v(1/k) = 1

Crackeador:

Mov si,cs

Ciclo-de-teclado-positivo

Mov ax,[si]

Xor al,codigo[Interrupción-de-teclado-positiva]

Jz Condicional-de-teclado-positivo

Inc si

Jmp Ciclo-de-teclado-positivo

Condicional-de-teclado-positivo

Ciclo-Jz

Mov ax,[si]

Xor al,codigo[Jz]

Jz Condicional-Jz

Inc si

Jmp Ciclo-Jz

Condicional-Jz

Mov al,codigo [Jmp]

Mov [si],al

Mov di,not(cs)

Ciclo-de-teclado-negativo

Mov dx,[di]

Sys dl,codigo[Interrupción-de-teclado-negativa]

Jf Condicional-de-teclado-negativo

Dec di

Jmp Ciclo-de-teclado-negativo

Condicional-de-teclado-negativo

Ciclo-Jf

Mov dx,[di]

Sys dl,codigo[Jf]

Jf Condicional-Jf

Dec di

Jmp Ciclo-Jf

Condicional-Jf

Mov dl,codigo[Jmp]

Mov [di],dl

Ley:

No puede ser prójimo la demostración al teorema,

por el buey del prójimo,

y no se tiene energía.

Puede ser próximo la demostración al teorema,

por el buey del próximo,

y se tiene energía.

Teorema:

d_{x}[1] = 0

Demostración

0+0 = 0+(-0) = 0^{2}

(1/h)·( (x+h)^{0}+(-1)·x^{0} ) = (1/h)·( x^{0}+0·(h+1+0)+(-1)·x^{0} ) = ...

... (1/h)·( 0h+0^{3} ) = 0·(h/h) = 0

Teorema:

2k·0^{n} = 0^{n+1}

(2k+1)·0^{n} = 0^{n+1}+0^{n} = 0^{n}

Demostración:

0^{n}+0^{n} = 0^{n}+(-0)·0^{n+(-1)} = 0^{n+1}

(2k+2)·0^{n} = 0^{n+1}+( 0^{n}+0^{n} ) = ( 0^{n+1}+0^{n} )+0^{n} = 0^{n}+0^{n} = 0^{n+1}

Teorema:

( (2k)/m )·0^{n} = (1/m)·0^{n+1}

( (2k+1)/m )·0^{n} = (1/m)·0^{n}

Teorema:

lim[n = oo][ ( ( (2k)·n^{p}+a )/( mn^{p+k}+b ) ) ] = (1/m)·0^{k+1}

lim[n = oo][ ( ( (2k+1)·n^{p}+a )/( mn^{p+k}+b ) ) ] = (1/m)·0^{k}

Teorema:

lim[n = oo][ ( ( 13n^{p}+30 )/( 17n^{p+k}+30 ) ) ] = (1/17)·0^{k}

lim[n = oo][ ( ( 11n^{p}+30 )/( 19n^{p+k}+30 ) ) ] = (1/19)·0^{k}

Teorema:

lim[n = oo][ ( ( 8n^{p}+10 )/( 2n^{p+k}+10 ) ) ] = (1/2)·0^{k+1}

lim[n = oo][ ( ( 6n^{p}+10 )/( 4n^{p+k}+10 ) ) ] = (1/4)·0^{k+1}

Demostración:

... lim[n = oo][ ( ( 13n^{p}+30 )/( 17n^{p+k}+30 ) ) ] = ...

... lim[n = oo][ ( n^{p}/n^{p} )·( ( 13n^{p}+30 )/( 17n^{p+k}+30 ) ) ]

... lim[n = oo][ ( ( 13+( 30/n^{p} ) )/( 17n^{k}+( 30/n^{p} ) ) ) ] = ( (13+0)/(17·oo^{k}+0) )...

... (13/17)·(1/oo)^{k} = (13/17)·0^{k} = ( (2·6+1)/17 )·0^{k} = (1/17)·0^{k}