medicina y homología-algebraica y economía y álgebra y futbol

Ley:

mv·d_{t}[q] = q·F(t)·(ut)^{n}

q(t) = ( (1/(mv))·q )·int[ F(t) ]d[t] [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1}

Ley:

mv·d_{t}[q] = q(t)·F(t)·(ut)^{n}

q(t) = qe^{(1/(mv))·int[ F(t) ]d[t] [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1}}

Sal:

n = 1

Na-Cl

Azúcar:

n = 2

A-O-B

Hierro:

n = 3

A-Fe=Fe-B

Ley:

mr·d_{t}[q]^{2} = pq·F(t)·(ut)^{n}

q(t) = ( (1/(mr))·pq )^{(1/2)}·int[ ( F(t) )^{(1/2)} ]d[t] [o(t)o] (1/u)·(2/(n+2))·(ut)^{( (n+2)/2 )}

Ley:

mr·d_{t}[q]^{2} = pq(t)·F(t)·(ut)^{n}

q(t) = ( (1/(mr))·p )·( (1/2)·int[ ( F(t) )^{(1/2)} ]d[t] [o(t)o] (1/u)·(2/(n+2))·(ut)^{( (n+2)/2 )} )^{2}

Mono-Leucocitos de tiroides: 

n = 2

A-O-O-B

Antibiótico de 1 destructor en sangre:

1 = (1/2)+(1/2)

Fumar:

COOH_{4}+A-O-O-B <==> C·(OH)_{4}+A-B

Gluten:

n = 4

B-O-A-B-O-A

Bi-Leucocitos de tiroides:

n = 6

A-B-A-O-O-B-A-B

Antibiótico de 2 destructores en sangre:

2 = (1/2)+(1/2)+(1/2)+(1/2)

Fumar:

COOH_{4}+A-B-A-O-O-B-A-B <==> C·(OH)_{4}+A-B-A-B-A-B

Ley:

b(x,y,t) = int-int[ d_{xy}^{2}[ m(x,y) ] ]d[x]d[y]·u·(ut)^{n}

M(x,y,t) = m(x,y)·(1/(n+1))·(ut)^{n+1}

Ley:

k(x,y,t) = int-int[ d_{xy}^{2}[ m(x,y) ] ]d[x]d[y]·u^{2}·(-n)·(ut)^{n+(-1)}

M(x,y,t) = m(x,y)·(1/(n+1))·(ut)^{n+1}

Ley:

b(x,y,t) = int-int[ d_{xy}^{2}[ m(x,y) ] ]d[x]d[y]·u·(ut)^{(n/2)}

M(x,y,t) = m(x,y)·(2/(n+2))·(ut)^{( (n+2)/2)}

Ley:

k(x,y,t) = int-int[ d_{xy}^{2}[ m(x,y) ] ]d[x]d[y]·u^{2}·(2/n)·(ut)^{( (n+(-2))/2 )}

M(x,y,t) = m(x,y)·(2/(n+2))·(ut)^{( (n+2)/2)}

Teorema:

Sea h_{n}: {i^{n}} x R ---> {i^{n+1}} x R ==>

[Ef(x)][ f: [0,3]_{N} x R ---> {i^{n}} x R & f(x) es biyectiva ]

Demostración:

Sea n = 4k+r ==>

Se define f(r,x) = < i^{4k+r},x >

f(r,x) = f(s,y)

< i^{4k+r},x > = < i^{4k+s},y >

i^{r} = i^{s} & x = y

r = s & x = y

< r,x > = < s,y >

Teorema:

Sea h_{n}: {i^{n}} x R ---> {i^{n+1}} x R ==>

[Eg(x)][ g: [0,3]_{N} x R ---> {i^{n+1}} x R & g(x) es biyectiva ]

Demostración:

Sea n = 4k+r ==>

Se define g(r,x) = < i^{4k+(r+1)},x >

g(r,x) = g(s,y)

< i^{4k+(r+1)},x > = < i^{4k+(s+1)},y >

i^{r+1} = i^{s+1} & x = y

r+1 = s+1 & x = y

r = s & x = y

< r,x > = < s,y >

Arte:

Sea h_{k}: P_{k}(A) ---> P_{k+1}(A) ==>

[Ef(x)][ f: A ---> P_{k}(A) & f(x) es biyectiva ]

Exposición:

w(k) = n+(-1)

Se define < f: A ---> [ n // n+(-1) ] & f(x) = A [&] }x{ >

f(x) = f(y)

A [&] }x{ = A [&] }y{

}x{ = }y{

x = y

Arte:

Sea h_{k}: P_{k}(A) ---> P_{k+1}(A) ==>

[Eg(x)][ g: A ---> P_{k+1}(A) & g(x) es biyectiva ]

Exposición:

w(k) = n+(-2)

Se define < g: A ---> [ n // n+(-1) ] & g(x) = A [&] }x{ >

Homologías de Figalli:

Arte:

Sea h_{n}: f(nx) ---> f((n+1)·x) ==>

[Ef(x)][ f(x) = f((n+1)·x)+(-1)·f(nx) ]

Exposición:

f(x) = Id(x)

w(f(x)) = Id(x)

Arte:

Sea h_{n}: f(x^{n}) ---> f(x^{n+1}) ==>

[Ef(x)][ ln( f(x) ) = ln( f(x^{n+1}) )+(-1)·ln( f(x^{n}) ) ]

Exposición:

f(x) = Id(x)

w(f(x)) = Id(x)

Arte:

Sea h_{n}: f(x+n) ---> f(x+(n+1)) ==>

[Ef(x)][ f(1) = f(x+(n+1))+(-1)·f(x+n) ]

Exposición:

f(x) = Id(x)

w(f(x)) = Id(x)

Arte:

Sea h_{n}: f(x^{[m:n]}) ---> f(x^{[m:n+1]}) ==>

[Ef(x)][ f(1) = f(x^{[m:n+1]})+(-1)·f(x^{[m:n]}) ]

Exposición:

f(x) = Id(x)

w(f(x)) = Id(x)

Arte:

Sea h_{n}: d_{x...x}^{n}[f(x)] ---> d_{x...x}^{n+1}[f(x)] ==>

[Ef(x)][ d_{x}[f(2x)] = d_{x...x}^{2k+1}[f(x)]^{2}+(-1)·d_{x...x}^{2k}[f(x)]^{2} ]

Exposición:

f(x) = sin(x)

w(f(x)) = sin(x)

Arte:

Sea h_{n}: d_{x...x}^{n}[f(x)] ---> d_{x...x}^{n+1}[f(x)] ==>

[Ef(x)][ d_{x}[f(0)] = d_{x...x}^{2k+1}[f(x)]^{2}+(-1)·d_{x...x}^{2k}[f(x)]^{2} ]

Exposición:

f(x) = sinh(x)

w(f(x)) = sinh(x)

El Aznar tiene que vigilar con la amnistía,

porque no se necesita ser muy listo,

para saber quien es J.M. en los papeles de Bárcenas.

Se ha manifestado contra la amnistía varias veces,

y no entiendo el juez del Bárcenas,

porque le aplica una amnistía al Aznar,

siendo J.M igual a José María.

Ley: [ de esquizofrenia ]

Hago lo que puedo, a quien puedo, donde puedo, cuando puedo y como puedo.

Deducción:

La voz en la mente dice:

Hago lo que quiero, a quien quiero, donde quiero, cuando quiero y como quiero.

Lema:

F(x,y) = (k+(-j))·x+jy+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-j))·x+jy+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-2))+x^{k+(-j)}+y^{j}+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-2))+x^{k+(-j)}+y^{j}+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-j))·e^{x}+je^{y}+(-h)·( pe^{x}+qe^{y}+(-m) )

h(0,0) = (k/m)

G(x,y) = (k+(-j))·e^{x}+je^{y}+(-h)·( pe^{x}+qe^{y} )

G(0,0) = 0

Lema:

F(x,y) = (k+(-2))+e^{(k+(-j))·x}+e^{jy}+(-h)·( pe^{x}+qe^{y}+(-m) )

h(0,0) = (k/m)

G(x,y) = (k+(-2))+e^{(k+(-j))·x}+e^{jy}+(-h)·( pe^{x}+qe^{y} )

G(0,0) = 0

Lema:

F(x,y) = (k+(-j))·(ln(x)+1)+j·(ln(y)+1)+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-j))·(ln(x)+1)+j·(ln(y)+1)+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-2))+(ln(x)+1)^{k+(-j)}+(ln(y)+1)^{j}+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-2))+(ln(x)+1)^{k+(-j)}+(ln(y)+1)^{j}+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-j))·(ln(1+ln(x))+1)+j·(ln(1+ln(y))+1)+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-j))·(ln(1+ln(x))+1)+j·(ln(1+ln(y))+1)+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-2))+(ln(1+ln(x))+1)^{k+(-j)}+(ln(1+ln(y))+1)^{j}+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-2))+(ln(1+ln(x))+1)^{k+(-j)}+(ln(1+ln(y))+1)^{j}+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-j))·(sin(ln(x))+1)+j·(sin(ln(y))+1)+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-j))·(sin(ln(x))+1)+j·(sin(ln(y))+1)+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-2))+(sin(ln(x))+1)^{k+(-j)}+(sin(ln(y))+1)^{j}+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-2))+(sin(ln(x))+1)^{k+(-j)}+(sin(ln(y))+1)^{j}+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-j))·(sinh(ln(x))+1)+j·(sinh(ln(y))+1)+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-j))·(sinh(ln(x))+1)+j·(sinh(ln(y))+1)+(-h)·( px+qy )

G(1,1) = 0

Lema:

F(x,y) = (k+(-2))+(sinh(ln(x))+1)^{k+(-j)}+(sinh(ln(y))+1)^{j}+(-h)·( px+qy+(-m) )

h(1,1) = (k/m)

G(x,y) = (k+(-2))+(sinh(ln(x))+1)^{k+(-j)}+(sinh(ln(y))+1)^{j}+(-h)·( px+qy )

G(1,1) = 0

Definición:

... F(u^{n},v^{n}) = 0 es Hoph-resoluble ...

... <==> ...

... [E$n$ e^{(k/n)·2pi·i}][E$n$ e^{(k/n)·pi·i}][ u = e^{(k/n)·2pi·i} & v = e^{(k/n)·pi·i} ]

Definición:

Gal( F(u^{n},v^{n}) ) = 2n+1

Teorema:

Gal( F(u^{n},v^{n}) ) = 4m+1 <==> F(u^{n},v^{n}) = 0 es Hoph-irresoluble

Gal( F(u^{n},v^{n}) ) != 4m+1 <==> F(u^{n},v^{n}) = 0 es Hoph-resoluble

Demostración:

e^{(2m+1)/(2m)·2pi·i} = e^{(1/m)·pi·i+2pi·i} = e^{(1/m)·pi·i} = e^{(1/(2m))·2pi·i}

Teorema:

u+v = 0 es Hoph-resoluble

p+q = c <==> ( p = (2c)·u & q = cv )

Demostración:

F(u+v) = v+u = u+v

u = e^{(1/1)·2pi·i} & v = e^{(1/1)·pi·i}

Teorema:

u^{2}+v^{2} = 0 es Hoph-irresoluble

p^{2}+q^{2} = c <==> ( p = (2c)^{(1/2)}·u & q = c^{(1/2)}·v )

Demostración:

F(u^{2}+v^{2}) = v^{2}+u^{2} = u^{2}+v^{2}

u = e^{(1/2)·2pi·i} & v = e^{(1/2)·pi·i}

u = e^{(3/2)·2pi·i} & v = e^{(3/2)·pi·i}

Teorema:

u^{3}+v^{3} = 0 es Hoph-resoluble

p^{3}+q^{3} = c <==> ( p = (2c)^{(1/3)}·u & q = c^{(1/3)}·v )

Demostración:

F(u^{3}+v^{3}) = v^{3}+u^{3} = u^{3}+v^{3}

u = e^{(1/3)·2pi·i} & v = e^{(1/3)·pi·i}

u = e^{(3/3)·2pi·i} & v = e^{(3/3)·pi·i}

u = e^{(5/3)·2pi·i} & v = e^{(5/3)·pi·i}

Teorema:

u^{4}+v^{4} = 0 es Hoph-irresoluble

p^{4}+q^{4} = c <==> ( p = (2c)^{(1/4)}·u & q = c^{(1/4)}·v )

Demostración:

F(u^{4}+v^{4}) = v^{4}+u^{4} = u^{4}+v^{4}

u = e^{(1/4)·2pi·i} & v = e^{(1/4)·pi·i}

u = e^{(3/4)·2pi·i} & v = e^{(3/4)·pi·i}

u = e^{(5/4)·2pi·i} & v = e^{(5/4)·pi·i}

u = e^{(7/4)·2pi·i} & v = e^{(7/4)·pi·i}

Teorema:

(0.a...)_{(2n+1)} = a·( (2n+1)/(2n) )+(-a)

(0.a...)_{(2n+2)} = a·( (2n+2)/(2n+1) )+(-a)

2·4+1 = 9

10x = (a,a...)_{10} & x = (0.a...)_{10}

x = (a/9)

Demostración:

sum[k = 0]-[oo][ a·(1/(2n+1))^{k} ] = a·( (2n+1)/(2n) )

Teorema:

(0.ab...)_{(2n+1)} = b·( (4n^{3}+4n+1)/(4n^{2}+4n) )+(-b)+a·( (2n+1)/(4n^{2}+4n) )

(0.ab...)_{(2n+2)} = b·( (4n^{3}+8n+4)/(4n^{2}+8n+3) )+(-b)+a·( (2n+2)/(4n^{2}+8n+3) )

4·16+8·4+3 = 99

100x = (ab.ab...)_{10} & x = (0.ab...)_{10}

x = ( (ab)/99 )

Demostración:

sum[k = 0]-[(oo/2)][ b·(1/(2n+1))^{2k} ] = b·( (4n^{2}+4n+1)/(4n^{2}+4n) )

sum[k = 0]-[(oo/2)][ a·(1/(2n+1))^{2k+1} ] = a·( (2n+1)/(4n^{2}+4n) )

Teorema: [ de Hoph-p-àdic ]

[Ef(x)][ f( (0.a...)_{(2n+1)} ) = (-1)·e^{( 1/(2n) )·pi·i} & f(x) es biyectiva ]

[Eg(x)][ g( (0.a...)_{(2n+2)} ) = (-1)·e^{( 1/(2n+1) )·pi·i} & g(x) es biyectiva ]

Demostració:

[1] Sigui (0.a...)_{(2n+1)} = a·( (2n+1)/(2n) )+(-a) ==>

Es defineix f( a·( (2n+1)/(2n) )+(-a) ) = e^{( (2n+1)/(2n) )·pi·i}

f( a·( (2n+1)/(2n) )+(-a) ) = f( a·( (2m+1)/(2m) )+(-a) )

e^{( (2n+1)/(2n) )·pi·i} = e^{( (2m+1)/(2m) )·pi·i}

( (2n+1)/(2n) ) = ( (2m+1)/(2m) )

a·( (2n+1)/(2n) )+(-a) = a·( (2m+1)/(2m) )+(-a)

f( (0.a...)_{(2n+1)} ) = (-1)·e^{( 1/(2n) )·pi·i}

[2] Sigui (0.a...)_{(2n+2)} = a·( (2n+2)/(2n+1) )+(-a)

Es defineix g( a·( (2n+2)/(2n+1) )+(-a) ) = e^{( (2n+2)/(2n+1) )·pi·i}

g( a·( (2n+2)/(2n+1) )+(-a) ) = g( a·( (2m+2)/(2m+1) )+(-a) )

e^{( (2n+2)/(2n+1) )·pi·i} = e^{( (2m+2)/(2m+1) )·pi·i}

( (2n+2)/(2n+1) ) = ( (2m+2)/(2m+1) )

a·( (2n+2)/(2n+1) )+(-a) = a·( (2m+2)/(2m+1) )+(-a)

g( (0.a...)_{(2n+2)} ) = (-1)·e^{( 1/(2n+1) )·pi·i}

Aun se puede celebrar la victoria de España en el mundial,

siendo la defensa algo útil para la vida,

pasando-se la pelota dualmente,

y entrando en el centro y volviendo a la defensa.

Se tiene que jugar mejor,

para que el futbol no sea algo inútil para la vida,

porque sinó no tiene sentido celebrar nada.

La defensa jugando dualmente,

es una recarga de energía para todo el equipo,

y son superiores físicamente.

El tiki taka se tiene que mejorar,

porque es el futbol bueno de pases duales con control dual.

La defensa del Barça o de España,

desgastaba la defensa del Madrid o de Argentina,

haciendo luz con pases duales,

y por esto el mejor futbol es el tiki taka.

El pase a centrocampista volviendo a la defensa,

desgasta el centro del campo del oponente.

Principio:

El Tiki Taka,

es como el ying y el yang del futbol.

Cura las piernas,

de los aficionados del equipo que lo juega.

Como dicen por el mundo:

El Papa es del Real Madrid,

pero Dios es del Barça,

y tiene que ser un hospital,

siendo más que un club,

jugando al Tiki Taka,

no siendo un ejército catalán,

porque hay uno de real.

Ley: [ de tiki taka ]

Control derecho de pase desde la defensa,

y vuelta a la defensa con pierna izquierda.

Control izquierdo de pase desde la defensa,

y vuelta a la defensa con pierna derecha.

Ley: [ de arbitraje ]

Falta en este control es tarjeta amarilla,

porque se vuelve inútil el juego para la vida.

Ley: [ de tiki taka ]

Control derecho,

de pase a la izquierda de la defensa

Control izquierdo,

de pase a la derecha de la defensa.

Ley: [ de arbitraje ]

Falta en este control es tarjeta amarilla,

porque se vuelve inútil el juego para la vida.

Ley: [ de tiki taka ]

Control derecho del portero,

de disparo izquierdo en profundidad al centro del campo.

Control izquierdo del portero,

de disparo derecho en profundidad al centro del campo.

Ley: [ de arbitraje ]

Falta en este control es tarjeta roja,

porque se vuelve inútil el juego para la vida.

Historia:

El Tiki Taka no es tan diferente a la final del mundial,

solo que tiene controles en la conexión con el balón

Al final el maletín de Jûan Laporta,

del video de los maletines,

ha hecho los 3 goles de España,

aunque solo haya sido válido uno,

y suficiente para ganar el mundial.

economía y categorías-en-álgebra

Lema:

p = 1·100+(1/1)·1,000 = 1,100€

q = 10·100+(1/10)·1,000 = 1,100€

Lema:

p = 2·100+(1/2)·1,000 = 700€

q = 5·100+(1/5)·1,000 = 700€

Lema:

p = 10^{1}+1,000^{(1/1)} = 1,010€

q = 10^{3}+1,000^{(1/3)} = 1,010€

Lema:

p = 50^{1}+2,500^{(1/1)} = 2,550€

q = 50^{2}+2,500^{(1/2)} = 2,550€

Impuesto de 1€ por unidades del producto

Lema:

(nx)^{p} = x^{p} <==> n = 1€

((1/n)·x)^{p} = x^{p} <==> n = 1€

Disertación:

(nx)^{p} = x^{p}

p·ln(nx) = p·ln(x)

ln(nx) = ln(x)

e^{ln(nx)} = e^{ln(x)}

nx = x

n = 1

Lema:

e^{nx} = e^{x} <==> n = 1€

e^{(1/n)·x} = e^{x} <==> n = 1€

Disertación:

e^{nx} = e^{x}

nx·ln(e) = x·ln(e)

nx = x

n = 1

Lema:

ln(nx) = ln(x) <==> n = 1€

ln((1/n)·x) = ln(x) <==> n = 1€

Disertación:

ln(nx) = ln(x)

e^{ln(nx)} = e^{ln(x)}

nx = x

n = 1

Lema:

(nx)^{p}·e^{nx} = x^{p}·e^{x} <==> n = 1€

((1/n)·x)^{p}·e^{(1/n)·x} = x^{p}·e^{x} <==> n = 1€

Disertación:

(nx)^{p}·e^{nx} = x^{p}·e^{x}

Anti-[ s^{p}·e^{s} ]-( (nx)^{p}·e^{nx} ) = Anti-[ s^{p}·e^{s} ]-( x^{p}·e^{x} )

nx = x

n = 1

Lema:

(nx)^{p}·ln(nx) = x^{p}·ln(x) <==> n = 1€

((1/n)·x)^{p}·ln((1/n)·x) = x^{p}·ln(x) <==> n = 1€

Disertación:

(nx)^{p}·ln(nx) = x^{p}·ln(x)

Anti-[ s^{p}·ln(s) ]-( (nx)^{p}·ln(nx) ) = Anti-[ s^{p}·ln(s) ]-( x^{p}·ln(x) )

nx = x

n = 1

Ley:

Después de la resurrección de los muertos,

se pueden recordar algo dual,

siendo 0t < (1/2)

Después de la resurrección de los muertos,

no se pueden recordar nada no dual,

siendo 0t > (0/2)

Ley:

Después de la resurrección de los muertos,

se pueden recordar teoremas,

siendo 0t < 1

Después de la resurrección de los muertos,

no se pueden recordar artes destructores,

siendo 0t > (-1)

Teorema:

int[x = 0]-[pi][ ( 1/sin(x) ) ]d[x] = 2+ln(4)

Demostración:

Por Hôpital-Jûanagoras:

[ (-1)·cos(x)+ln(sin(x)) [o(x)o] ( sin(x) /o(x)o/ x^{0} ) ]_[x = 0]-[pi] = 1+ln(2)+1+ln(2) = 2+ln(4)

sin(pi) = (-0)

Teorema:

int[x = 0]-[(pi/2)][ ( 1/sin(x) ) ]d[x] = 1+ln(2)

int[x = (pi/2)]-[pi][ ( 1/sin(x) ) ]d[x] = 1+ln(2)

Teorema:

int[x = 0]-[(pi/4)][ ( 1/sin(x) ) ]d[x] = ( 1+(-1)·(1/2)^{(1/2)} )+( 1+(-1)·(1/2)^{(3/2)} )·ln(2)

int[x = ((3pi)/4)]-[pi][ ( 1/sin(x) ) ]d[x] = ( 1+(-1)·(1/2)^{(1/2)} )+( 1+(-1)·(1/2)^{(3/2)} )·ln(2)

Demostración:

Por Hôpital-Jûanagoras:

[ (-1)·cos(x)+ln(sin(x)) [o(x)o] ( cos(x) /o(x)o/ x^{0} ) ]_[x = 0]-[(pi/4)]

Macroeconomía:

Lema:

Arancel de 4 socios

p = ( 16/(4!+(-8)) ) = 1

q = 0.80+2.56 = 3.36€

Precio:

0.85 = 5·0.17

0.85 = 4·0.20+0.05

Lema:

Arancel de 5 socios

p = ( 105/(5!+(-15)) ) = 1

q = 6.30+11.55 = 17.85€

Precio:

1.56 = 6·0.26

1.56 = 5·0.30+0.06

Definición: [ de categoría ]

          z                           z

          |                            |

z ---> F(z)               ---> F^{o(-1)}(z)

          |                            |

z ---> F^{o(-1)}(z) ---> F(z)

F^{o(-1)}( F(z) ) = z

F( F^{o(-1)}(z) ) = z

Teorema:

          z                                   z

          |                                    |

z ---> F(z+a)                   ---> F^{o(-1)}(z)+(-a)

          |                                    |

z ---> F^{o(-1)}(z)+(-a) ---> F(z+a)

Teorema:

          z                                 z

          |                                  |

z ---> F(z)+a                 ---> F^{o(-1)}(z+(-a))

          |                                  |

z ---> F^{o(-1)}(z+(-a)) ---> F(z)+a

Demostración:

F^{o(-1)}( F(z+a) )+(-a) = (z+a)+(-a) = z+(a+(-a)) = z+0 = z

F( ( F^{o(-1)}(z)+(-a) )+a ) = F( F^{o(-1)}(z)+((-a)+a) ) = F( F^{o(-1)}(z)+0 ) = F( F^{o(-1)}(z) ) = z

Teorema: [ de categoría suma ]

          z                 z

          |                  |

z ---> z+n     ---> z+(-n)

          |                  |

z ---> z+(-n) ---> z+n

Teorema: [ de categoría múltiplo ]

             z                   z

             |                    |

z --->    nz    ---> (1/n)·z

             |                    |

z ---> (1/n)·z    ---> nz

Teorema:

          z                      z

          |                       |

z ---> z^{n}       ---> z^{(1/n)}

          |                       |

z ---> z^{(1/n)} ---> z^{n}

Teorema:

Sea F(z) o G(z) = G(z) o F(z) ==>

          z                     

          |                      

z ---> F(z)              ---> G(z)

          |                            |

         G^{o(-1)}(z) ---> F^{o(-1)}(z) ---> G(z)

                                       |

                                      G^{o(-1)}(z)

Demostración:

( G^{o(-1)} o F )^{o(-1)} = F^{o(-1)} o G = G o F^{o(-1)}

( G o F^{o(-1)} o G^{o(-1)} o F )(z) = Id(z) = z

( G o F )^{o(-1)} = F^{o(-1)} o G^{o(-1)} = G^{o(-1)} o F^{o(-1)}

( G^{o(-1)} o F^{o(-1)} o G o F )(z) = Id(z) = z

Teorema:

Sea F(z) o G(z) = G(z) o F(z) ==>

          z                     

          |                      

z ---> F(z)              ---> G(z)

          |                            |

         G^{o(-1)}(z) ---> F^{o(-1)}(z) ---> (-1)·G(z)

                                       |                                    |

                                      (-1)·G^{o(-1)}(z) ---> (-z)

Demostración:

(-1)·( (-1)·( G o F^{o(-1)} o G^{o(-1)} o F )(z) ) = (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

(-1)·( (-1)·( G^{o(-1)} o F^{o(-1)} o G o F )(z) ) = (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

Teorema:

          z                     

          |                      

z ---> z^{n}       ---> z^{m}

          |                       |

         z^{(1/m)} ---> z^{(1/n)} ---> z^{m}

                                  |

                                  z^{(1/m)}

Problema:

Mostrad la categoría múltiplo y suma.

Teorema:

          z                     

          |                      

z ---> F(z)        ---> F(z)

          |                      |

         F(z)  ---> (-1)·F^{o(-2)}(z) ---> (-z)

                                 |

                               (-z)

Demostración:

(-1)·( (-1)·F^{o(-2)}( F(F(z)) ) ) = (-1)·( (-1)·( F^{o(-2)} o F^{o2} )(z) ) = ...

... (-1)·( (-1)·Id(z) ) = (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

Teorema:

          z                     

          |                      

z ---> F(z)        ---> F(z)

          |                      |

         F(z)  ---> (-1)·F^{o(-2)}(z) ---> Id(z)

                                 |                           |

                                 Id(z)            ---> (-z)

Demostración:

(-1)·Id( (-1)·F^{o(-2)}( F(F(z)) ) ) = (-1)·Id( (-1)·( F^{o(-2)} o F^{o2} )(z) ) = ...

... (-1)·Id( (-1)·Id(z) ) =(-1)·Id( (-1)·z ) = (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

Teorema:

          z                     

          |                      

z ---> kz ---> kz

          |           |

         kz  ---> iz ---> (-z)

                      |

                     (-z)

Demostración:

(-1)·( ikkz) = z

Problema:

Mostrad la categoría dual en números complejos y reales simétricos.

Teorema:

          z                     

          |                      

z ---> z+n ---> z+n

          |               |

         z+n  ---> (-z)+2n ---> Id(z)

                          |                    |

                        Id(z)      ---> (-z)

Demostración:

(-1)·( Id( (-1)·( (z+n)+n )+2n ) ) = z

Teorema:

          z                     

          |                      

z ---> F(z)        ---> (-1)·F(z)

          |                             |

         (-1)·F(z)  ---> (-1)·F^{o(-2)}(-z) ---> (-z)

                                       |

                                     (-z)

Demostración:

(-1)·( (-1)·F^{o(-2)}( (-1)·( (-1)·F(F(z)) ) ) ) = (-1)·( (-1)·F^{o(-2)}( ((-1)·(-1))·F(F(z)) ) ) = ...

... (-1)·( (-1)·( F^{o(-2)} o F^{o2} )(z) ) = (-1)·( (-1)·Id(z) ) = (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

Teorema:

          z                     

          |                      

z ---> F(z)         ---> (-1)·F(z)

          |                        |

         (-1)·F(z)  ---> (-1)·F^{o(-2)}(-z) ---> Id(z)

                                  |                                   |

                                 Id(z)                     ---> (-z)

Demostración:

(-1)·Id( (-1)·F^{o(-2)}( (-1)·( (-1)·F(F(z)) ) ) ) = (-1)·Id( (-1)·F^{o(-2)}( ((-1)·(-1))·F(F(z)) ) ) = ...

... (-1)·Id( (-1)·( F^{o(-2)} o F^{o2} )(z) ) = (-1)·Id( (-1)·Id(z) ) = (-1)·Id( (-1)·z ) = ...

... (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

Teorema:

          z                     

          |                      

z ---> kz      ---> (-k)·z

          |                   |

         (-k)·z  ---> (-i)·z ---> (-z)

                              |

                           (-z)

Demostración:

(-1)·( (-i)·(-k)·kz ) = z

Problema:

Mostrad la categoría dual en números complejos y reales simétricos.

Teorema:

          z                             z                             z

          |                              |                              |

z ---> F(z)                ---> F(z)                 ---> F^{o(-2)}(z)

          |                              |                              |

z ---> F(z)                ---> F^{o(-2)}(z)   ---> F(z)

          |                              |                              |

z ---> F^{o(-2)}(z)  ---> F(z)                 ---> F(z)

Demostración:

F( F( F^{o(-2)}(z) ) ) = ( F^{o2} o F^{o(-2)} )(z) = Id(z) = z

F( F^{o(-2)}( F(z) ) ) = ( F o ( F^{o(-2)} o F ) )(z) = ( F o F^{o(-1)} )(z) = Id(z) = z

F^{o(-2)}( F( F(z) ) ) = ( F^{o(-2)} o F^{o2} )(z) = Id(z) = z

Teorema:

             z              z                z

             |               |                 |

z --->  kz    --->  kz     ---> (-i)·z

             |               |                 |

z --->  kz    ---> (-i)·z   ---> kz

             |               |                 |

z ---> (-i)·z  ---> kz      ---> kz

Teorema:

          z                              z                              z

          |                               |                               |

z ---> F(-z)                ---> (-1)·F(z)            ---> F^{o(-2)}(-z)

          |                               |                               |

z ---> (-1)·F(z)          ---> F^{o(-1)}(-z)   ---> Id(z)

          |                               |                               |

z ---> F^{o(-2)}(-z)  --->  Id(z)               ---> (-1)·F^{o2}(z)

Demostración:

F^{o(-2)}( (-1)·( (-1)·F( F(z) ) ) ) = F^{o(-2)}( ((-1)·(-1))·F( F(z) ) ) = ...

... ( F^{o(-2)} o F^{o2} )(z) = Id(z) = z

Id( F^{o(-1)}( (-1)·( (-1)·F(z) ) ) ) = F^{o(-1)}( (-1)·( (-1)·F(z) ) ) = ...

... F^{o(-1)}( ((-1)·(-1))·F(z) ) ) = ( F^{o(-1)} o F )(z) ) ) = Id(z) = z

(-1)·F^{o2}(z)( Id( F^{o(-2)}( (-1)·z ) ) ) = (-1)·F^{o2}(z)( F^{o(-2)}( (-1)·z ) ) = ...

... (-1)·( F^{o2} o F^{o(-2)} )( (-1)·z ) ) ) = (-1)·Id( (-1)·z ) = (-1)·( (-1)·z ) = ((-1)·(-1))·z = z

Teorema:

             z                    z                        z

             |                     |                         |

z --->   kz         --->  (-k)·z           ---> iz

             |                    |                          |

z --->  (-k)·z     ---> (1/k)·(-z)     ---> Id(z)

             |                    |                          |

z --->   iz          --->  Id(z)           ---> (-i)·z

Teorema:

Sea f_{n}: nz ---> (n+1)·z ==>

Sea g_{n}: (1/n)·z ---> (1/(n+1))·z ==>

          z                     

          |                      

z ---> nz        ---> (n+1)·z

          |                       |

         (n+1)·z ---> (1/n)·z ---> (1/(n+1))·z

                                  |

                            (1/(n+1))·z

Teorema:

Sea f_{n}: d_{z...z}^{n}[h(z)] ---> d_{z...z}^{(n+1)}[h(z)] ==>

Sea g_{n}: int-[n]-int[h(z)]d[z]...d[z] ---> int-[n+1]-int[h(z)]d[z]...d[z] ==>

                      h(z)                     

                        |                      

h(z) ---> d_{z...z}^{n}[h(z)]       ---> d_{z...z}^{(n+1)}[h(z)]

                        |                                             |

              d_{z...z}^{(n+1)}[h(z)] ---> int-[n]-int[h(z)]d[z]...d[z] ---> int-[n+1]-int[h(z)]d[z]...d[z]

                                                                      |

                                                            int-[n+1]-int[h(z)]d[z]...d[z]

mecanismo-de-Gauge y álgebra y análisis-matemático y filosofía y geometría-diferencial y homología-algebraica y topología

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)

Ley:

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)

Ley:

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)

Ley:

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)

Ley:

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

v^{4}+(-1)·(a+w)·v^{2}+biv = 0

u^{3}+(a+w)·u+b = 0

v^{3}+(-1)·(a+w)·v+bi = 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})

k = 5·(d/(2bi))

u^{5}+(a+k)·u^{3}+bu^{2}+(c+w)·u = 0

iv^{5}+(-i)·(a+k)·v^{3}+(-1)·bv^{2}+(ci+w)·v = 0

u^{4}+(a+k)·u^{2}+bu+(c+w) = 0

iv^{4}+(-i)·(a+k)·v^{2}+(-1)·bv+(ci+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·(uv)^{0}) = F(uv)

Demostración:

F(uv·(uv)^{0}) = F(uv)·(vu)^{0} = vu·(vu)^{0} = (vu)^{1+0} = vu = F(uv)

Teorema:

F(uv·(uv)^{(-1)}) = F( (uv)^{0} )

Demostración:

F(uv·(uv)^{(-1)}) = F(uv)·(vu)^{(-1)} = vu·(vu)^{(-1)} = (vu)^{1+(-1)} = (vu)^{0} = F( (uv)^{0} )

Teorema:

F(uv·ab) = F(ab·uv)

Demostración:

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

métodos-numéricos y topología-algebraica y óptica y arte-matemático y topología y números-figurados y medicina y dualogía

Teorema:

Sea d_{x}[y(x)] = y+x+(k+(-1)) ==>

[Ej][ (1/h)·( y_{n+1}+(-1)·y_{n} ) = y_{n}+j ] es un método numérico convergente a y(x)

Demostración:

Sea h = 0a & j = (-1)·((k/a)+1) ==>

y_{n+1} = y_{n}+h·( y_{n}+j ) = y_{n}·(1+h)+hj

y_{n+1} = y_{0}·(1+h)^{n}+nhj

Sea y_{0} = 1 ==>

y(a) = y_{oo} = e^{a}+(-k)+(-a)

Teorema:

Sea d_{x}[y(x)] = y+x^{2}+(k+(-2)) ==>

[Ej][ (1/h)·( y_{n+1}+(-1)·y_{n} ) = y_{n}+j ] es un método numérico convergente a y(x)

Demostración:

Sea h = 0a & j = (-1)·((k/a)+2+a) ==>

y_{n+1} = y_{n}+h·( y_{n}+j ) = y_{n}·(1+h)+hj

y_{n+1} = y_{0}·(1+h)^{n}+nhj

Sea y_{0} = 1 ==>

y(a) = y_{oo} = e^{a}+(-k)+(-1)·2a+(-1)·a^{2}

Teorema: [ de sp-line cuadrática ]

P(x) = (x+(-1)·x_{j})·(x+(-1)·x_{k})·( (x_{i}+(-1)·x_{j})·(x_{i}+(-1)·x_{k}) )^{(-1)}·f(x_{i})

Teorema: [ de sp-line cúbica ]

Q(x) = ...

... (x+(-1)·x_{j})·x·(x+(-1)·x_{k})·( (x_{i}+(-1)·x_{j})·x_{i}·(x_{i}+(-1)·x_{k}) )^{(-1)}·f(x_{i})

Teorema:

Sea ( m != 1 & d_{x}[y(x)] = y^{m} ) ==>

[Ej][ (1/h)·( y_{n+1}+(-1)·( y_{n} )^{j} ) = ( y_{n} )^{m} ] es un método numérico convergente a y(x)

Demostración:

Sea h = 0 & j = ( 1/(1+(-m))^{0} ) ==>

y_{n+1} = ( y_{n} )^{j}+h·( y_{n} )^{m} = ( y_{n} )^{m+[j+(-m):h]}

y_{n+1} = ( y_{1} )^{( 1/(1+(-m)) )^{0n}}

Sea y_{1} = (1+(-m))·a ==>

y(a) = y_{oo} = ( (1+(-m))·a )^{( 1/(1+(-m)) )}

Teorema:

Sea m != 1 ==>

Si  a_{n+1} = (1/2)·( a_{n}+( a_{n} )^{m}·y_{n} ) ==> a_{oo} = ( y_{n} )^{( 1/(1+(-m)) )}

Demostración:

( a_{oo} )^{1+(-m)} = (1/2)·( ( a_{oo} )^{1+(-m)}+y_{n} )

2·( a_{oo} )^{1+(-m)}+(-1)·( a_{oo} )^{1+(-m)} = y_{n}

( a_{oo} )^{1+(-m)} = y_{n}

a_{oo} = ( y_{n} )^{( 1/(1+(-m)) )}

Teorema:

Sea f_{n}(x): ( x+(-a) )^{n} ---> ( x+(-a) )^{n+1} ==>

[Ex][ f_{n}(x) está compactificada en 2 clases ]

Teorema:

Sea f_{n}(x): ( e^{x}+(-a) )^{n} ---> ( e^{x}+(-a) )^{n+1} ==>

[Ex][ f_{n}(x) está compactificada en 2 clases ]

Teorema:

Sea f_{n}(P(x)): d_{x...x}^{n}[P(x)]·h(x) ---> Q(x) [o(x)o] ( x /o(x)o/ H(x) ) ==>

[EP(x)][ f_{n}(P(x)) está compactificada en 2 clases ]

Demostración:

d_{x}[ sinh(x) [o(x)o] ( x /o(x)o/ H(x) ) ]·h(x) = cosh(x)

d_{x}[ cosh(x) [o(x)o] ( x /o(x)o/ H(x) ) ]·h(x) = sinh(x)

Ley:

d_{z}[f(z(t),x,t)]+d_{x}[f(z(t),x,t)] = (1/S)·vt+a·(1/(ax))^{n}

f(z(t),x,t) = (1/S)·(1/2)·vt^{2} [o(t)o] z(t)+( (ax) /o(ax)o/ (1/(n+1))·(ax)^{n+1} )

Ley:

d_{z}[f(z(t),x,t)]+d_{x}[f(z(t),x,t)] = (1/S)·(1/2)·(q/m)·gt^{2}+a·(1/(ax))^{n}

f(z(t),x,t) = (1/S)·(1/6)·(q/m)·gt^{3} [o(t)o] z(t)+( (ax) /o(ax)o/ (1/(n+1))·(ax)^{n+1} )

Problema:

d_{z}[f(z(t),x,t)]+d_{x}[f(z(t),x,t)] = (1/S)·(1/6)·(I/m)·gt^{3}+a·(1/(ax))^{n}

Ley:

Sea d[...(n)...d[q]...(n)...] = n!·qa^{n}·d[z]...(n)...d[z] ==>

F(z) = pq(z)·k·(1/r)^{3}·z 

z(t) = ( n·( (1/(4+2n))·(1/m)·pqk·(1/r)^{3}·a^{n} )^{(1/2)}·t )^{(-1)·(2/n)}

d_{t}[q(t)] = n!·qa^{n}·(-2)·n^{(-2)}·( (1/(4+2n))·(1/m)·pqk·(1/r)^{3}·a^{n} )^{(-1)}·t^{(-3)}

Artes de Vinogradov energéticos:

Arte:

Sea 0 [< p [< 2 ==>

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

Arte:

[En][ 2^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+3 ]

[En][ 8^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+11 ]

[En][ 32^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+37 ]

Arte:

Sea 1 [< p [< 2 ==>

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

Arte:

[En][ 4^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+5 ]

[En][ 16^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+19 ]

Arte:

Sea 1 [< p [< 3 ==>

[En][ 3^{p·sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+3^{p}+4 ]

Arte:

[En][ 3^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+7 ]

[En][ 9^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+13 ]

[En][ 27^{sum[k = 1][n][k]} < ln( sum[k = 1][n][k] )+31 ]

Arte:

Sea 1 [< p [< 3 ==>

[En][ (5+6p)·sum[k = 1][n][k] < ln( sum[k = 1][n][k] )+( 5+(6p+6) ) ]

Arte:

[En][ 11·sum[k = 1][n][k] < ln( sum[k = 1][n][k] )+17 ]

[En][ 17·sum[k = 1][n][k] < ln( sum[k = 1][n][k] )+23 ]

[En][ 23·sum[k = 1][n][k] < ln( sum[k = 1][n][k] )+29 ]

Teorema:

Sea ( h(1) = 1 & h(1/n) creciente ) ==>

Si E_{n,s} = { x : 0 [< m(x,y) [< h(1/n)·s } ==> ...

... Si ( E_{n,s} [<< B & E_{m,d} [<< B ) ==> E_{n,s} [ || ] E_{m,d} [<< B

... Si ( E_{n,s} [<< B & E_{m,d} [<< B ) ==> E_{n,s} [&] E_{m,d} [<< B

... E_{n} puede estar compactificada en m clases.

Demostración:

A_{1} = E_{1} = { x : 0 [< m(x,y) [< s }

A_{n+1} = E_{n} [ \ ] E_{n+1} = { x : h( 1/(n+1) )·s < m(x,y) [< h(1/n)·s }

Teorema:

Sea ( h(0) = 0 & h(n) creciente ) ==>

Si E_{n} = { x : 0 [< x [< h(n) } ==> ...

... Si ( E_{n} [<< B & E_{m} [<< B ) ==> E_{n} [ || ] E_{m} [<< B

... Si ( E_{n} [<< B & E_{m} [<< B ) ==> E_{n} [&] E_{m} [<< B

... E_{n} puede estar compactificada en m clases.

Demostración:

A_{0} = E_{0} = {0}

A_{n+1} = E_{n+1} [ \ ] E_{n} = { x :  h(n) < x [< h(n+1) }

Teorema:

Sea n >] 1 ==>

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

Demostración: [ por geometría ]

a_{1}:

1

a_{2}:

010

111

a_{3}:

00100

01110

11111

a_{n} = (2n+(-1))·n+(-1)·n·(n+(-1)) = (2n^{2}+(-n))+(-1)·(n^{2}+(-n)) = n^{2}

Teorema:

Sea n >] 1 ==>

sum[k = 1]-[n][ (2k+(-1)) ]+(2n+(-1))^{2} = 5n^{2}+(-1)·4n+1

Demostración: [ por geometría ]

a_{1}:

1

1

a_{2}:

010

111

111

111

111

a_{n} = n^{2}+(2n+(-1))^{2} = n^{2}+(4n^{2}+(-1)·4n+1) = 5n^{2}+(-1)·4n+1

Teorema: [ de números cuadrados perimetrales ]

Sea n >] 1 ==>

(2n+(-1))^{2}+(-1)·(2n+(-3))^{2} = 8n+(-8)

Demostración: [ por geometría ]

a_{1}:

0

a_{2}:

111

101

111

a_{3}:

11111

10001

10001

10001

11111

a_{n} = (2n+(-1))^{2}+(-1)·(2n+(-3))^{2} = (4n^{2}+(-1)·4n+1)+(-1)·(4n^{2}+(-1)·12n+9) = 8n+(-8)

Principio: [ de pitagorancias orgánicas ]

n = 1

Sal = Na-Cl

n = 2

Azúcar = A-O-A

n = 3

Hierro = A-Fe=Fe-A

n = 4

Iodo = A-IH=I=IH-A

Principio: [ de aparato de presión ]

Sea ( mv(t) la impulsión sanguínea & F(t) la fuerza del aparato de presión ) ==>

mv(t)·d_{t}[q] = q(t)·F(t)·(ut)^{n}

q(t) = qe^{( int[ F(t) ]d[t] /o(t)o/ int[ mv(t) ]d[t] ) [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1}}

Ley:

mv(t)·d_{t}[q] = q(t)·(Igt)·(ut)^{n}

q(t) = qe^{( (1/2)·Igt^{2} /o(t)o/ int[ mv(t) ]d[t] ) [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1}}

Ley:

mv(t)·d_{t}[q] = q(t)·(-b)·(r/t)·(ut)^{n}

q(t) = qe^{( (-b)·r·ln(ut) /o(t)o/ int[ mv(t) ]d[t] ) [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1}}

Principio: [ de analítica sanguínea ]

Sea ( mv(t) la impulsión sanguínea & F(t) la fuerza de centrifugación ) ==>

mv(t)·d_{t}[q] = qF(t)·(ut)^{n}

q(t) = q·( int[ F(t) ]d[t] /o(t)o/ int[ mv(t) ]d[t] ) [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1}

Ley:

mv(t)·d_{t}[q] = (1/(mr))·(qgt)^{2}·(ut)^{n}

q(t) = ( ( (1/(mr))·(1/3)·(qg)^{2}·t^{3} /o(t)o/ int[ mv(t) ]d[t] ) [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1} )

Ley:

mv(t)·d_{t}[q] = (1/(mr))·( (1/2)·Igt^{2} )^{2}·(ut)^{n}

q(t) = ( ( (1/(mr))·(1/20)·(Ig)^{2}·t^{5} /o(t)o/ int[ mv(t) ]d[t] ) [o(t)o] (1/u)·(1/(n+1))·(ut)^{n+1} )

Principio: [ de orina de humano ]

b(x,y,t) = int-int[ d_{xy}^{2}[ m(x,y) ] ]d[x]d[y]·u·f(ut)

M(x,y,t) = int[ b(x,y,t) ]d[t]

Ley: [ de sanidad de pitagorancia cero ]

Sea ( f(ut) = (ut)^{0} & d_{xy}^{2}[ m(x,y) ] = ma^{2} ) ==>

M(x,y,t) = mxya^{2}·(ut)

M(x,y,t) = mxya^{2} <==> t = (1/u)

Ley: [ de pitagorancia de materia sanguínea ]

Sea ( f(ut) = (ut)^{n} & d_{xy}^{2}[ m(x,y) ] = ma^{2} ) ==>

M(x,y,t) = mxya^{2}·(1/(n+1))·(ut)^{n+1}

M(x,y,t) = mxya^{2} <==> t = (1/u)·(n+1)^{( 1/(n+1)) }

Ley: [ de virus genético TACCCCAT-TCAAAACT ]

Sea ( f(ut) = (1/(ut)) & d_{xy}^{2}[ m(x,y) ] = ma^{2} ) ==>

M(x,y,t) = mxya^{2}·ln(ut)

M(x,y,t) = mxya^{2} <==> t = (1/u)·e

Principio: [ de heces de animal ]

k(x,y,t) = int-int[ d_{xy}^{2}[ m(x,y) ] ]d[x]d[y]·u^{2}·g(ut)

M(x,y,t) = int-int[ k(x,y,t) ]d[t]d[t]

Ley: [ de sanidad de pitagorancia cero ]

Sea ( g(ut) = 0·(1/(ut)) & d_{xy}^{2}[ m(x,y) ] = ma^{2} ) ==>

M(x,y,t) = mxya^{2}·(ut)

M(x,y,t) = mxya^{2} <==> t = (1/u)

Ley: [ de pitagorancia de materia sanguínea ]

Sea ( g(ut) = n·(ut)^{n+(-1)} & d_{xy}^{2}[ m(x,y) ] = ma^{2} ) ==>

M(x,y,t) = mxya^{2}·(1/(n+1))·(ut)^{n+1}

M(x,y,t) = mxya^{2} <==> t = (1/u)·(n+1)^{( 1/(n+1)) }

Ley: [ de virus genético TACCCCAT-TCAAAACT ]

Sea ( g(ut) = (-1)·(1/(ut))^{2} & d_{xy}^{2}[ m(x,y) ] = ma^{2} ) ==>

M(x,y,t) = mxya^{2}·ln(ut)

M(x,y,t) = mxya^{2} <==> t = (1/u)·e

Teorema:

int[ lim[n = oo][ ( 1/(1+nx) ) ] ]d[x] = int[ (1/oo)·( oo/(1+oox) ) ]d[x] = (1/oo)·ln(oo) = ln(2)

lim[n = oo][ int[ ( 1/(1+nx) ) ] ]d[x] = lim[n = oo][ (1/n)·ln(1+nx) ] = (1/oo)·ln(oo) = ln(2)

Ley:

Los hombres tenemos que rezar al Mal,

que los azeris vos caguéis encima,

pero que lleguéis al váter,

a cagar en la taza,

porque el Mal va a cambiar el rezo de cagar,

y lo vamos a destruir.

Los azeris tenéis que rezar al Mal,

que los hombres nos pijemos encima,

pero que lleguemos al váter,

a pijar en al taza,

porque el Mal va a cambiar el rezo de pijar,

y los vais a destruir.

Ley: [ de esquizofrenia ]

Hay condenación o no he fracasado en destruir a un dios del Mal.

Deducción:

La voz en la mente dice no hay condenación y has fracasado.

Principio: [ de drogas de polímeros de pitagorancia exponencial ]

I_{q}(x,y,t) = int-int-int[ ( q(t) )^{n} ]d[x]d[y]d[q]

Principio: [ de drogas de polímeros de pitagorancia de producto ]

I_{q}(x,y,t) = int-int-int-int[ n·( q(t) )^{n+(-1)} ]d[x]d[y]d[q]d[q]

Ley:

Sea q(t) = qe^{mut} ==>

I_{q}(x,y,t) = (1/(n+1))·q^{n+1}·e^{(n+1)·mut}·xy

Deducción:

I_{q}(x,y,t) = ...

... int[ int[ int-int[ nq^{n+(-1)}e^{(n+(-1))·mut} ]d[x]d[y]·qe^{mut}·mu ]d[t]·qe^{mut}·mu ]d[t]

Ley:

Sea z(t) = q·(ut)^{m} ==>

V(x,y,t) = (1/(n+1))·q^{n+1}·(ut)^{(n+1)·m}·xy

Ley:

Sea z(t) = q·(ut)^{m}+p ==>

V(x,y,t) = (1/(n+1))·q^{n+1}·(ut)^{(n+1)·[m:(p/q)]}·xy

Arte:

[En][ frac[k = 1]-[n][ ( (2k+(-1))/(1+(2k+1)) ) ] = (1/4)·n ]

Exposición:

n = 1

f(k) = 1

frac[k = 1]-[n][ ( (2f(k)+(-1))/(1+(2f(k)+1)) ) ] = frac[k = 1]-[n][ ( 1/(1+3) ) ] = ...

... frac[k = 1]-[n][ ( 1/(1+( 3+(1/2)+(-1)·(1/2) )) ) ] = frac[k = 1]-[n][ ( 1/(1+( 3+(1/2)+(1/2) )) ) ] = ...

... frac[k = 1]-[n][ ( 1/(1+(3+1)) ) ] = frac[k = 1]-[n][ ( 1/(1+4) ) ] = ...

... frac[k = 1]-[n+(-1)][ ( 1/(1+4) ) ] o 1+4 = frac[k = 1]-[n+(-1)][ ( 1/(1+4) ) ] o 1+(1/4) = ...

... (1/4)·(n+(-1))+(1/4) = (1/4)·n

Arte:

[En][ frac[k = 0]-[n][ ( k!/(1+(k+1)!) ) ] = (1/2)·(n+1) ]

Exposición:

n = 0

f(k) = 1

frac[k = 0]-[n][ ( f(k)!/(1+(f(k)+1)!) ) ] = frac[k = 0]-[n][ ( 1/(1+(1+1)!) ) ] = ...

... frac[k = 0]-[n][ ( 1/(1+2) ) ] = frac[k = 0]-[n+(-1)][ ( 1/(1+2) ) ] o 1+2 = ...

... frac[k = 0]-[n+(-1)][ ( 1/(1+2) ) ] o 1+(1/2) = (1/2)·n+(1/2) = (1/2)·(n+1)

Arte: [ de Rogers-Ramanujan ]

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

Exposición:

n = 1

f(1) = (1/m)

g(1/m) = 0

frac[k = 1]-[n][ ( q^{k}/(1+(-1)^{f(1)}·q^{k+1}) ) ] = ...

... frac[k = 1]-[n][ ( q^{k}/(1+(-1)^{(1/m)}·q^{k+1}) ) ] = ...

... frac[k = 1]-[n][ ( q^{k}/(1+(-1)^{g(1/m)}·q^{k+1}) ) ] = ...

... frac[k = 1]-[n][ ( q^{k}/(1+q^{k+1}) ) ] = ...

... frac[k = 1]-[n+(-1)][ ( q^{k}/(1+q^{k+1}) ) ] o q^{n}+q^{2n+1} = ...

... q+...(n)...+q^{2n+(-1)}+q^{2n+1}

Arte: [ de Rogers-Ramanujan-Garriga ]

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

Exposición:

n = 1

f(1) = (1/m)

g(1/m) = 0

frac[k = 1]-[n][ ( q^{(1/k)}/(1+(-1)^{f(1)}·q^{( 1/(k+1) )}) ) ] = ...

... frac[k = 1]-[n][ ( q^{(1/k)}/(1+(-1)^{(1/m)}·q^{( 1/(k+1) )}) ) ] = ...

... frac[k = 1]-[n][ ( q^{(1/k)}/(1+(-1)^{g(1/m)}·q^{( 1/(k+1) )}) ) ] = ...

... frac[k = 1]-[n][ ( q^{(1/k)}/(1+q^{( 1/(k+1) )}) ) ] = ...

... frac[k = 1]-[n+(-1)][ ( q^{(1/k)}/(1+q^{( 1/(k+1) )}) ) ] o q^{(1/n)}+q^{(1/n)+(1/(n+1))} = ...

... q+sum[k = 1]-[n][ q^{(1/k)+(1/(k+1))} ] = ...

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

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

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

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

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

Dual:

La Luá está de-puá me avec sa-pá de-le-munt,

de-le-dans la cupuá de la Luá de La-Franç.

La Luá está-de-puá me avec sa-pá de-la-vall,

de-le-dans la ne cupuá de la Luá de La-Franç.

Morfosintaxis:

[A$1$ [z] ][ [z] és-de-puá Luá ]-[ [z] está de-puá P([a]) , Q([p]) ]

P([a]) <==> [ me avec sa-pá [a] ]-[ [a] és-de-puá de-le-munt ]

Q([p]) <==> [ de-le-dans [p(s)] ]-[A$1$ [p(s)] ][ [p(s)] és-de-puá cupuá de [s(w)] ]-...

... [A$1$ [s(w)] ][ [s(w)] és-de-puá Luá de [w] ]-[ [w] és-de-puá La-Franç ]

[A$1$ [z] ][ [z] és-de-puá Luá ]-[ [z] está de-puá P([b]) , Q([q]) ]

P([b]) <==> [ me avec sa-pá [b] ]-[ [b] és-de-puá de-la-vall ]

Q([q]) <==> [ de-le-dans [q(s)] ]-[A$1$ [q(s)] ][ [q(s)] és-de-puá ne cupuá de [s(w)] ]-...

... [A$1$ [s(w)] ][ [s(w)] és-de-puá Luá de [w] ]-[ [w] és-de-puá La-Franç ]

Definición: [ de dualogía ]

[Ey][ x@y & y@z ] <==> x = z

Teorema:

Si [Ec][ x+y = f(c) = z+y & f(c) = 0 ] ==> x+y = f(x) es dualogía

Definición:

Dual[ x+y = f(x) ] = { < x,y > : x+y = f(x) & f(x) = 0 }

Definición:

Gen[ x+y = f(x) ] = { < x,(-x) > = sum[k = 1]-[n][ a_{k}·< c_{k},(-1)·c_{k} > ] : ...

... < c_{k},(-1)·c_{k} > € Dual[ x+y = f(x) ] }

Teorema:

(1/2)·x^{2}+int[ y ]d[x] = F(x) es dualogía

Demostración:

x+y = f(x)

x·d[x]+y·d[x] = (x+y)·d[x] = f(x)·d[x]

int[ x ]d[x]+int[ y ]d[x] = int[ f(x) ]d[x]

(1/2)·x^{2}+int[ y ]d[x] = F(x)

Teorema:

Dual[ (1/2)·x^{2}+int[ y ]d[x] = x+(-a) ] = { < a,(-a) > }

< x,(-x) > = (x/a)·< a,(-a) >

< (-x),x > = (-1)·(x/a)·< a,(-a) >

Dual[ x+y = 1 ] = { < (1/n),1+(-1)·(1/n) > }

< p(z),¬p(z) > = < 0,0 >+< (1/n),1+(-1)·(1/n) >

< ¬q(z),q(z) > = < 1,1 >+(-1)·< (1/n),1+(-1)·(1/n) >

Demostración:

y = d_{x}[ int[ y ]d[x] ] = d_{x}[ (-1)·(1/2)·x^{2} ] = d_{a}[ (-1)·(1/2)·a^{2} ] = (-a)

Teorema:

Dual[ (1/2)·x^{2}+int[ y ]d[x] = (1/2)·x^{2}+(-1)·a^{2} ] = ...

... { 2^{(1/2)}·< a,(-a) > , 2^{(1/2)}·< (-a),a > }

Dual[ x+y = x ] = { < 1,0 > }

Teorema:

Dual[ (1/2)·x^{2}+int[ y ]d[x] = e^{x}+(-a) ] = { < ln(a),(-1)·ln(a) > }

Dual[ x+y = e^{x} ] = { < ln(0),ln(oo) > }

Teorema:

Si [Ec][ x·y = f(c) = z·y & f(c) = 1 ] ==> x·y = f(x) es dualogía

Definición:

Dual[ x·y = f(x) ] = { < x,y > : x·y = f(x) & f(x) = 1 }

Definición:

Gen[ x·y = f(x) ] = { < x,(1/x) > = sum[k = 1]-[n][ < a_{k},b_{k} >·< c_{k},( 1/(c_{k}) ) > ] : ...

... < c_{k},( 1/(c_{k}) ) > € Dual[ x·y = f(x) ]}

Teorema:

Si [Ec(t)][ x(t) [o(t)o] y(t) = f(c(t)) = z(t) [o(t)o] y(t) & f(c(t)) = t ] ==> ...

... x(t) [o(t)o] y(t) = f(x(t)) es dualogía

Definición:

Dual[ x(t) [o(t)o] y(t) = f(x(t)) ] = { < x(t),y(t) > : x(t) [o(t)o] y(t) = f(x(t)) & f(x(t)) = t }

Definición:

Gen[ x(t) [o(t)o] y(t) = f(x(t)) ] = { < x(t),( t /o(t)o/ x(t) ) > = ...

... sum[k = 1]-[n][ < a_{k}(t),b_{k}(t) > [o(t)o] < c_{k}(t),( t o(t)o/ c_{k}(t) ) > ] : ...

... < c_{k}(t),( t /o(t)o/ c_{k}(t) ) > € Dual[ x(t) [o(t)o] y(t) = f(x(t)) ]}

Teorema:

Si [Ec][ m(x,y) = f(c) = m(z,y) & f(c) = k ] ==> m(x,y) = f(x) es dualogía

Demostración:

m(x,y) = f(c) = m(z,y)

< x,y > = < z,y >

x = z

Se define < x,y > = < c,0 > = < z,y > & f(c) = m(c,0)

Definición:

Dual[ m(x,y) = f(x) ] = { < x,y > : m(x,y) = f(x) & f(x) = k }

Definición:

m(x,y) = | x+(-y) |

Teorema:

m(x,x) = 0

Demostración:

| x+(-x) | = 0

Teorema:

m(x,y) [< m(x,z)+m(z,y)

Demostración:

m(x,y) = | x+(-y) | = | x+(-z)+z+(-y) | [< | x+(-z) |+| z+(-y) | = m(x,z)+m(z,y)

Teorema:

Sea m(x,y) = | x+(-y) | = k ==>

Dual[ m(x,y) = f(x) ] = { < (n+1)·k,nk >,< nk,(n+1)·k > }

Teorema:

Sea m(x,y) = | x+(-y) | = |x|+(-a) ==>

Dual[ m(x,y) = f(x) ] = { < k+a,a >,< (-k)+(-a),(-a) > }

Teorema:

Sea m(x,y) = | x+(-y) | = x^{2}+(-a) ==>

Dual[ m(x,y) = f(x) ] = ...

... { < k^{(1/2)·[1:a]},k+k^{(1/2)·[1:a]} >,< k^{(1/2)·[1:a]},(-k)+k^{(1/2)·[1:a]} >,...

... < (-1)·k^{(1/2)·[1:a]},k+(-1)·k^{(1/2)·[1:a]} >,< (-1)·k^{(1/2)·[1:a]},(-k)+(-1)·k^{(1/2)·[1:a]} > }

Teorema:

|| < a,b >+< u,v > || [< || < a,b > ||+|| < u,v > ||

Demostración:

f(2·|a||b|) = 0

g(2·|u||v|) = 0

... || < a,b >+< u,v > || = ...

... ( (|a|+|u|)^{2}+(|b|+|v|)^{2} )^{(1/2)} [< |a|+|u|+|b|+|v| = |a|+|b|+|u|+|v| = ...

... ( |a|^{2}+2·|a||b|+|b|^{2} )^{(1/2)}+( |u|^{2}+2·|u||v|+|v|^{2} )^{(1/2)} [< ...

... ( |a|^{2}+f(2·|a||b|)+|b|^{2} )^{(1/2)}+( |u|^{2}+g(2·|u||v|)+|v|^{2} )^{(1/2)} =

... ( |a|^{2}+|b|^{2} )^{(1/2)}+( |u|^{2}+|v|^{2} )^{(1/2)} = || < a,b > ||+|| < u,v > || 

Definición:

m(x,y) = || x+yi ||

Teorema:

m(x,x) = 0

Demostración:

( (|a|+|ai|)^{2}+(|b|+|bi|)^{2} )^{(1/2)} = 0

Teorema:

m(x,y) [< m(x,z)+m(z,y)

Demostración:

( (|a|+|ui|)^{2}+(|b|+|vi|)^{2} )^{(1/2)} = ( (|a|+|mi|+|m|+|ui|)^{2}+(|b|+|ni|+|n|+|vi|)^{2} )^{(1/2)}

m(x,y) = || x+yi || = || x+zi+z+yi || [< || x+zi ||+|| z+yi || = m(x,z)+m(z,y)

Teorema:

Sea m(r,0) = ( |x|^{2}+|y|^{2} )^{(1/2)} = x+(-a) ==>

Dual[ m(r,0) = f(x) ] = { < k+a,( k^{2}+(-1)·(k+a)^{2} )^{(1/2)} > }

Teorema:

Sea m(r,0) = ( |x|^{2}+|y|^{2} )^{(1/2)} = x^{2}+(-a) ==>

Dual[ m(r,0) = f(x) ] = { < k^{(1/2)·[1:a]},( k^{2}+(-1)·k^{[1:a]} )^{(1/2)} > }

Series de Riemann-Ramanujan:

Arte:

[Ek][ sum[n = 1]-[oo][ ( 1/(2k)! )·(1/n)^{2k}·(4k+(-2)) ] = (1/6)·pi^{2} ]

Exposición:

k = 1

f(2k) = 2

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

... sum[n = 1]-[oo][ ( 1/(f(2k))! )·(1/n)^{f(2k)}·(2·f(2k)+(-2)) ] = ...

... sum[n = 1]-[oo][ (1/2!)·(1/n)^{2}·(4+(-2)) ] = sum[n = 1]-[oo][ (1/2)·(1/n)^{2}·2 ] = ...

... sum[n = 1]-[oo][ (1/n)^{2} ] = (1/6)·pi^{2}

Arte:

[Ek][ sum[n = 1]-[oo][ ( 1/(3k)! )·(1/n)^{3k}·(9k+(-3)) ] = (1/24)·pi^{3} ]

Arte:

[Ek][ sum[n = 1]-[oo][ ( (4k+(-2))/(4k)! )·(1/n)^{4k}·(16k+(-4)) ] = (1/90)·pi^{4} ]

Arte:

[Ek][ sum[n = 1]-[oo][ ( (5k+1)/(5k)! )·(1/n)^{5k}·(25k+(-5)) ] = (1/300)·pi^{5} ]

geometría-diferencial y arte-matemático y números-y-vectores-afines y evangelio-stronikiano y análisis-matemático y topología

Teorema:

Sea H_{kk}^{k} = k ==> 

x_{k}(t) = (1/k)·ln(t)

Si d_{t}[x_{s}]^{2} = ( x_{s} )^{2} ==>

x_{s}(t) = e^{t}

R_{ijs}^{sss} = ij·t^{2}·e^{2t}

Teorema:

Sea H_{kk}^{k} = k ==> 

x_{k}(t) = (1/k)·ln(t)

Si d_{t}[x_{s}] = x_{s} ==>

x_{s}(t) = e^{t}

R_{ssk}^{sss} = kt·e^{t}

Teorema:

Sea H_{kk}^{k} = kt ==> 

x_{k}(t) = (2/k)·(-1)·(1/t)

Si d_{t}[x_{s}]^{2} = ( x_{s} )^{n} ==>

x_{s}(t) = ( (1+(-1)·(1/2)·n)·t )^{( 1/(1+(-1)·(1/2)·n) )}

R_{ijs}^{sss} = ij·(1/4)·t^{4}·( (1+(-1)·(1/2)·n)·t )^{( n/(1+(-1)·(1/2)·n) )}

Teorema:

Sea H_{kk}^{k} = kt ==> 

x_{k}(t) = (2/k)·(-1)·(1/t)

Si d_{t}[x_{s}] = ( x_{s} )^{n} ==>

x_{s}(t) = ( (1+(-n))·t )^{( 1/(1+(-n)) )}

R_{ssk}^{sss} = k·(1/2)·t^{2}·( (1+(-n))·t )^{( n/(1+(-n)) )}

Arte:

[En][ sum[k = 1]-[n][ mcd{km,k} ] = n ]

Exposición:

n = 1

f(k) = 1

sum[k = 1]-[n][ mcd{km,k} ] = sum[k = 1]-[n][ mcd{f(k)·m,f(k)} ] = ...

... sum[k = 1]-[n][ mcd{m,1} ] = sum[k = 1]-[n][ 1 ] = n

Arte:

[En][ sum[k = 1]-[n][ mcm{km,k} ] = nm ]

Exposición:

n = 1

f(k) = 1

sum[k = 1]-[n][ mcm{km,k} ] = sum[k = 1]-[n][ mcm{f(k)·m,f(k)} ] = ...

... sum[k = 1]-[n][ mcm{m,1} ] = sum[k = 1]-[n][ m ] = nm

Examen:

Arte:

[En][ sum[k = 1]-[n][ mcd{m+k,m} ] = n ]

Arte:

[En][ sum[k = 1]-[n][ mcm{m^{k},m} ] = nm ]

Definición: [ de número afín ]

Sea ( r € Q & m € Z & k € Z ) ==>

{ mk : r } = mk+[r] & [Ej][ j € Z & [r] = jr ]

Teorema:

[(-r)]+r = 0

Demostración:

[(-r)]+r = j·(-r)+r

Sea j = 1 ==>

[(-r)]+r = (-r)+r = 0

Teorema:

{ mk : 0 } = mk

Demostración:

{ mk : 0 } = mk+[0] = mk+0j = mk

Teorema:

a·[r] = [ar]

a·{ mk : r } = a·mk+[r]

Demostración:

a·[r] = a·(jr) = (aj)·r = (ja)·r = j·(ar) = [ar]

a·{ mk : r } = a·mk+a·[r] = a·mk+a·(jr) = a·mk+(aj)·r = a·mk+wr = a·mk+[r]

Definición: [ de múltiplo de un número afín ]

f(k) =[m]= g(j) <==> ...

... a·{ mk : r }+(-b)·{ mj : r } = m·( ak+(-1)·bj )

Definición: [ de potencia de un número afín ]

Sea ( { mk : r } )^{p} = { (mk)^{p} : r } ==>

f(k) =[m]= g(j) <==> ...

... ( { mk : r } )^{p}+(-1)·( { mj : r } )^{q} = m·( k^{p}·m^{p+(-1)}+(-1)·j^{q}·m^{q+(-1)} )

Teorema:

ax^{n}+b =[m]= 0 <==> x = { mk : (-b) }

Demostración:

a·{ mk : (-b) }^{n}+b = a·{ (mk)^{n} : (-b) }+b = ( a·(mk)^{n}+[(-b)] )+b = ...

... a·(mk)^{n}+([(-b)]+b) = a·(mk)^{n}+0 = a·(mk)^{n} =[m]= 0

Teorema:

Sea [Aj][ 1 [< j [< n ==> a_{j} != 0 ] ==>

a_{n}·x^{n}+...+a_{1}·x+a_{0} =[m]= 0 <==> x = { mk : (-1)·(1/n)·a_{0} }

Demostración:

sum[j = 1]-[n][ a_{j}·( { mk : (-1)·(1/n)·a_{0} } )^{j} ]+a_{0} = ...

... sum[j = 1]-[n][ a_{j}·( { (mk)^{j} : (-1)·(1/n)·a_{0} } ]+a_{0} = ...

... sum[j = 1]-[n][ a_{j}·(mk)^{j}+[(-1)·(1/n)·a_{0}] ]+a_{0} = ...

... sum[j = 1]-[n][ a_{j}·(mk)^{j}+(1/n)·[(-1)·a_{0}] ]+a_{0} = ...

... ( sum[j = 1]-[n][ a_{j}·(mk)^{j} ]+[(-1)·a_{0}]+a_{0} = ...

... sum[j = 1]-[n][ a_{j}·(mk)^{j} ]+( [(-1)·a_{0}]+a_{0} ) = ...

... sum[j = 1]-[n][ a_{j}·(mk)^{j} ]+0 = sum[j = 1]-[n][ a_{j}·(mk)^{j} ] =[m]= 0

Definición: [ de vector afín ]

Sea ( r € E & v € E & k € R ) ==>

{ kv : r } = kv+[r] & [EB][ B es matriz & [r] = (B o r) ]

Teorema:

[(-r)]+r = 0

Demostración:

[(-r)]+r = (B o (-r))+r

Sea B = Id ==>

[(-r)]+r = (Id o (-r))+r = (-r)+r = 0

Teorema:

{ kv : 0 } = kv

Demostración:

{ kv : 0 } = kv+[0] = kv+(B o 0) = kv+0 = kv

Teorema:

A o { kv : r } = k·(A o v)+[r] ==>

Demostración:

A o { kv : r } = k·(A o v)+A o [r] = k·(A o v)+A o (B o r) = k·(A o v)+(A o B ) o r = ...

... k·(A o v)+(C o r) = k·(A o v)+[r]

Definición: [ de producto de matrices de un vector afín ]

f(k) =[H(v)]= g(j) <==> ...

... A o { kv : r }+(-1)·( B o { jv : r } ) = ( k·A+(-1)·j·B ) o v

Ley: [ primera de condenación del Mal ]

Rezar al próximo,

sin Ley del Talión,

no se condena el Mal

odiando al próximo no como a si mismo 

pero es destrucción.

Quizás rezar al prójimo,

con Ley del Talión,

no se condena el Mal,

odiando al prójimo como a si mismo 

y entonces también no es destrucción.

Ley: [ segunda de condenación del Mal ]

Rezar al próximo,

con Ley del Talión,

se condena el Mal,

odiando al próximo como a si mismo

pero es destrucción. 

Quizás rezar al prójimo,

sin Ley del Talión,

se condena el Mal,

odiando al prójimo no como a si mismo

y entonces también no es destrucción.

Ley:

Con Ley del Talión rezando al prójimo,

hay condenación,

no teniendo-la el Mal

porque por igualdad tiene alguien la condenación. 

Sin Ley del Talión rezando al prójimo,

no hay condenación,

teniendo-la el Mal

aunque quizás por igualdad tiene alguien la condenación. 

Teorema:

Sea ( f(x) expansiva & d_{x}[f(x)] creciente ) ==>

Si f(0) = 0 ==> [Ax][ 0 < x·d_{x}[f(x)] < 1 ==> d_{x}[f(x)] >] (1/x)·ln(1+x) ]

Demostración:

0 [< c [< x

e^{x·d_{x}[f(x)]} >] 1+x·d_{x}[f(x)] >] 1+x·d_{x}[f(c)] = 1+f(x) >] 1+x

Teorema:

Sea ( f(x) expansiva & d_{x}[f(x)] creciente ) ==>

Si f(0) = 0 ==> [Ax][ 0 < x·d_{x}[f(x)] < 1 ==> d_{x}[f(x)] >] (1/x)·arc-sinh(x) ]

Demostración:

0 [< c [< x

sinh( x·d_{x}[f(x)] ) >] x·d_{x}[f(x)] >] x·d_{x}[f(c)] = f(x) >] x

Teorema:

Sea H(x) = ( f(x) )^{n} & d_{x}[f(x)] creciente ) ==>

Si f(0) = 0 ==> [Ax][ x > 0 ==> d_{x}[H(x)] >] (n/x)·H(x) ]

Demostración:

0 [< c [< x

d_{x}[H(x)] = d_{x}[ ( f(x) )^{n} ] = n·( f(x) )^{n+(-1)}·d_{x}[f(x)] >] ...

... n·( f(x) )^{n+(-1)}·d_{x}[f(c)] = n·( f(x) )^{n+(-1)}·(f(x)/x) = (n/x)·H(x)

Teorema:

Sea H(x) = e^{n·f(x)} & d_{x}[f(x)] creciente & [Ek][Ax][ x >] k ==> f(x) expansiva ] ) ==>

Si f(0) = 0 ==> [Ax][ x >] k ==> d_{x}[H(x)] >] n·H(x) ]

Demostración:

0 [< c [< x

d_{x}[H(x)] = d_{x}[ e^{n·f(x)} ] = n·e^{n·f(x)}·d_{x}[f(x)] >] ...

... n·e^{n·f(x)}·d_{x}[f(c)] = n·e^{n·f(x)}·(f(x)/x) >] n·e^{n·f(x)}·(x/x) = n·H(x)

Teoremas de Cámara-Garriga:

Teorema:

Sea A_{k} [<< A_{k+1} ==>

sum[k = 1]-[n][ [ || ]-[i = 1]-[k][ A_{i} ] ] = [ || ]-[k = 1]-[n][ sum[i = 1]-[k][ A_{i} ] ]

Sea A_{k} >>] A_{k+1} ==>

sum[k = 1]-[n][ [&]-[i = 1]-[k][ A_{i} ] ] = [ || ]-[k = 1]-[n][ sum[i = 1]-[k][ A_{i} ] ]

Demostración:

Sea A_{k} [<< A_{k+1} ==>

sum[k = 1]-[n][ [ || ]-[i = 1]-[k][ A_{i} ] ] = ...

... [ || ]-[i = 1]-[1][ A_{i} ]+...+[ || ]-[i = 1]-[n][ A_{i} ] = A_{1}+...+A_{n}

[ || ]-[k = 1]-[n][ sum[i = 1]-[k][ A_{i} ] ] = sum[i = 1]-[n][ A_{i} ] = A_{1}+...+A_{n}

Teorema:

Sea ¬A_{k} >>] ¬A_{k+1} ==>

sum[k = 1]-[n][ [&]-[i = 1]-[k][ ¬A_{i} ] ] = [&]-[k = 1]-[n][ sum[i = 1]-[k][ ¬A_{i} ] ]

Sea ¬A_{k} [<< ¬A_{k+1} ==>

sum[k = 1]-[n][ [ || ]-[i = 1]-[k][ ¬A_{i} ] ] = [&]-[k = 1]-[n][ sum[i = 1]-[k][ ¬A_{i} ] ]

Teorema:

Sea ( h(0) = 0 & h(i) creciente ) ==>

Si E_{i} = { x : 0 [< x [< h(i) } ==> E_{i} está compactificada en m clases

Demostración:

A_{0} = {0}

A_{i+1} = E_{i+1} [ \ ] E_{i} = { x : h(i) < x [< h(i+1) }

Sea i = mk ==>

A_{mk+1} = E_{mk+1} [ \ ] E_{mk}

Sea i = mk+m ==>

A_{mp+1} = A_{m·(k+1)+1} = A_{(mk+m)+1} = E_{(mk+m)+1} [ \ ] E_{mk+m}

Teorema:

Sea ( h(0) = 0 & h(-i) decreciente ) ==>

Si E_{(-i)} = { x : h(-i) [< x [< 0 } ==> E_{(-i)} está compactificada en m clases

Demostración:

A_{0} = {0}

A_{(-i)+(-1)} = E_{(-i)+(-1)} [ \ ] E_{(-i)} = { x : h((-i)+(-1)) [< x < h(-i) }

Sea (-i) = (-m)·k ==>

A_{(-m)·k+(-1)} = E_{(-m)·k+(-1)} [ \ ] E_{(-m)·k}

Sea (-i) = (-m)·k+(-m) ==>

A_{(-m)·p+(-1)} = A_{(-m)·(k+1)+(-1)} = A_{( (-m)·k+(-m) )+(-1)} = ...

... E_{( (-m)·k+(-m) )+(-1)} [ \ ] E_{(-m)·k+(-m)}

Teorema:

Si E_{i} = { x : 0 [< x [< 2i } ==> E_{i} está compactificada en 4 clases

Demostración:

A_{0} = {0}

Sea i = 4k ==>

A_{4k+1} = E_{2·(2k+1)+(-1)} [ \ ] E_{2·(2k)}

Sea i = 4k+1 ==>

A_{4k+2} = E_{2·(2k+1)} [ \ ] E_{2·(2k+1)+(-1)}

Sea i = 4k+2 ==>

A_{4k+3} = E_{2·(2k+2)+(-1)} [ \ ] E_{2·(2k+1)}

Sea i = 4k+3 ==>

A_{4k+4} = E_{2·(2k+2} [ \ ] E_{2·(2k+2)+(-1)}

Sea i = 4k+5 ==>

A_{4p+1} = A_{4·(k+1)+1} = A_{(4k+4)+1} = E_{2·(2k+2)+1} [ \ ] E_{2·(2k+2)}

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)