geometría-algebraica

Definició-x: [ de dimensió de Hausdorff-Minkowsky ]

Dim-K(r) = < (1/m),k >

<==>

sum[n = 1][ (1/n)^{m}·0n^{m} ] = sum[n = 1][ (1/n)^{m} ]·r^{k}

Teorema:

Dim-K(r) = < (1/2),2 > ==> 24^{(1/2)} = 2pi·r

Demostració:

Dim-K(r) = < (1/2),2 > ==>

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

... (1/6)·pi^{2}·r^{2} = (1/24)·(2pi·r)^{2}

24 = (2pi·r)^{2}

24^{(1/2)} = 2pi·r

Teorema:

Dim-K(r) = < (1/2),4 > ==> 96^{(1/2)} = 4pi·r^{2}

Demostració:

Dim-K(r) = < (1/2),4 > ==>

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

... (1/6)·pi^{2}·r^{4} = (1/96)·( 4pi·r^{2} )^{2}

96 = ( 4pi·r^{2} )^{2}

96^{(1/2)} = 4pi·r^{2}

Teorema:

Dim-K(r) = < (1/3),3 > ==> 192^{(1/3)} = 2pi·r

Demostració:

Dim-K(r) = < (1/3),3 > ==>

sum[k = 1]-[oo][ (1/n)^{3}·0n^{3} ] = sum[k = 1]-[oo][ (1/n)^{3} ]·r^{3} = ...

... (1/24)·pi^{3}·r^{3} = (1/192)·(2pi·r)^{3}

192 = (2pi·r)^{3}

192^{(1/3)} = 2pi·r

Ley:

El agua mineral es buena y no es cancerígena.

Ley:

Si ( Se bebe agua con gas & Se toma el Sol ) ==> ...

... Se tiene una desintegración cancerígena

Deducción:

H(CO)O(CO)H

H(NH)O(NH)H

HOOOH

H+ClgOClg+H

Ley:

Si ( Se fuma & Se toma el Sol ) ==> ...

... Se tiene una desintegración cancerígena

Deducción:

C·(COH)_{4}

C·((NH)·H)_{4}

C·(OH)_{4}

H+H+CClg_{4}+H+H

medicina y homología-algebraica y economía y álgebra y futbol y mecánica-física y geometría-diferencial y cohomología

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)

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}

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

Ley:

Sea d[I_{c}] = M·(r/j)·v·d[t] ==>

x(t) = (M/m)·(r/(jd))·vt

Sea U(w) = U·(w/k)^{n} ==>

w(t) = k·Anti-[ ( s /o(s)o/ (1/(n+1))·s^{n+1} )^{[o(s)o] (1/2)} ]-( ...

... ( U·(8/M)·(j/r)·(1/v)·t )^{(1/2)}·(1/k) )

Ley:

Sea d[I_{c}] = (r/j)·qgt·d[t] ==>

x(t) = (q/m)·(r/(jd))·(1/2)·gt^{2}

Sea U(w) = U·(w/k)^{n} ==>

w(t) = k·Anti-[ ( s /o(s)o/ (1/(n+1))·s^{n+1} )^{[o(s)o] (1/2)} ]-( ...

... ( U·(4/q)·(j/r)·(1/g) )^{(1/2)}·ln(ut)·(1/k) )

Ley:

Sea d[I_{c}] = (r/j)·(1/2)·Igt^{2}·d[t] ==>

x(t) = (I/m)·(r/(jd))·(1/6)·gt^{3}

Sea U(w) = U·(w/k)^{n} ==>

w(t) = k·Anti-[ ( s /o(s)o/ (1/(n+1))·s^{n+1} )^{[o(s)o] (1/2)} ]-( ...

... ( U·(48/I)·(j/r)·(1/g)·(1/t) )^{(1/2)}·(1/k) )

Definición: [ de Cohomología de Rham ]

F_{n}: d[x_{k}] --- F_{n}(U) = (-n)

                  |                                     |

G_{n}: d[x_{j}] --- G_{n}(V) = x_{1}·...·x_{n}

Teorema:

F_{2}: x+y  --- F_{2}(U) = (-2)

               |                               |

G_{2}: y+x --- G_{2}(V) = xy

Demostración:

( x+y [o] y+x )+(-2)·xy = 0

Teorema:

F_{3}: x+y+z        --- F_{3}(U) = (-3)

                |                                         |

G_{3}: yz+zx+xy --- G_{3}(V) = xyz

Demostración:

( x+y+z [o] yz+zx+xy )+(-3)·xyz = 0

Teorema:

Sea F(x,y) = < x^{n},y^{n} > & G(y,x) = < y^{n},x^{n} > ==>

F_{2}: (1/(n+1))·x^{n+1}+(1/(n+1))·y^{n+1}  --- F_{2}(U) = (-2)

                     |                                                                                  |

G_{2}: (1/(n+1))·y^{n+1}+(1/(n+1))·x^{n+1} --- G_{2}(V) = (1/(n+1))^{2}·(xy)^{n+1}

Demostración:

( (1/(n+1))·x^{n+1}+(1/(n+1))·y^{n+1} [o] (1/(n+1))·y^{n+1}+(1/(n+1))·x^{n+1} )+...

... (-2)·(1/(n+1))^{2}·(xy)^{n+1} = 0

Teorema:

x^{3}+y^{3} = z^{2} tiene soluciones enteras

Demostración: 

( 3 & 2 ) ternas pitagóricas

2^{3}+2^{3} = 8+8 = 16 = 4^{2}

Teorema:

x^{2}+y^{2} = z^{3} tiene soluciones enteras

Demostración: 

( 2 & 3 ) ternas pitagóricas

2^{2}+2^{2} = 4+4 = 8 = 2^{3}

Teorema: [ de correcció a Willes de la Cohomología deformable de Galois ]

Sigui el coeficient de Galois el mínim dels punts fixos:

H_{2}: ( cos[1](x) )^{2} --- ( sin[1](x) )^{2} --- z^{2}

                      |                                |                       |

H_{3}: ( cos[2](x) )^{3} --- ( sin[2](x) )^{3} --- z^{2}

Gal(H_{2},H_{3}) = 4

H_{2}: ( cos[1](x) )^{2} --- ( sin[1](x) )^{2} --- z^{2}

                      |                                |                       |

H_{3}: ( cos[1](x) )^{2} --- ( sin[1](x) )^{2} --- (2z)^{3}

Gal(H_{2},H_{3}) = 4

H_{2}: ( cos[1](x) )^{2} --- ( sin[1](x) )^{2} --- z^{2}

                      |                                |                       |

H_{3}: ( cos[2](x) )^{3} --- ( sin[2](x) )^{3} --- (2z)^{3}

Gal(H_{2},H_{3}) = 5

Teorema: [ de Willes-Jûanágoras ]

Si Gal(H_{n},H_{n+1}) = 2n+2 ==> F(x,y,z) és resoluble per racionals

Si Gal(H_{n},H_{n+1}) = 2n+3 ==> F(x,y,z) és irresoluble per racionals

Demostració:

H_{n}:     ( cos[n](x) )^{n+1}     --- ( sin[n](x) )^{n+1}      --- (zn)^{n+1}

                          |                                         |                                  |

H_{n+1}: ( cos[n+1](x) )^{n+2} --- ( sin[n+1](x) )^{n+2} --- (zn)^{n+1}

F(zn) = nz = zn

     |

F(zn) = nz = zn

Gal(H_{n},H_{n+1}) = 2n+2

(1/2)^{n+2}+(1/2)^{n+2} = 2·(1/2)^{n+2} = (1/2)^{n+1}

F(x,y,z) és resoluble per racionals.

H_{n}:     ( cos[n](x) )^{n+1} --- ( sin[n](x) )^{n+1}  --- (zn)^{n+1}

                           |                                   |                               |

H_{n+1}: ( cos[n](x) )^{n+1} --- ( sin[n](x) )^{n+1}  --- ( z·(n+1) )^{n+2}

F( cos[n](x) ) = sin[n](x) = cos[n](x)

            |

F( cos[n](x) ) = sin[n](x) = cos[n](x)

Gal(H_{n},H_{n+1}) = 2n+2

F( sin[n](x) ) = cos[n](x) = sin[n](x)

            |

F( sin[n](x) ) = cos[n](x) = sin[n](x)

Gal(H_{n},H_{n+1}) = 2n+2

2^{n+1}+2^{n+1} = 2·2^{n+1} = 2^{n+2}

F(x,y,z) és resoluble per racionals.

H_{n}:     ( cos[n](x) )^{n+1}     --- ( sin[n](x) )^{n+1}      --- (zn)^{n+1}

                          |                                         |                                  |

H_{n+1}: ( cos[n+1](x) )^{n+2} --- ( sin[n+1](x) )^{n+2} --- ( z·(n+1) )^{n+2}

F( cos[n](x) ) = sin[n](x) = cos[n](x)

            |

F( cos[n+1](x) ) = sin[n+1](x) = cos[n+1](x)

Gal(H_{n},H_{n+1}) = 2n+3

F( sin[n](x) ) = cos[n](x) = sin[n](x)

            |

F( sin[n+1](x) ) = cos[n+1](x) = sin[n+1](x)

Gal(H_{n},H_{n+1}) = 2n+3

F(zn) = nz = zn

     |

F(z·(n+1)) = (n+1)·z = z·(n+1)

Gal(H_{n},H_{n+1}) = 2n+3

F(x,y,z) és irresoluble per racionals.

Ley [ de mono-cáncer ]

Si ( Beber agua = OH_{2} y Tomar el Sol = W ) ==>

... Se provoca una desintegración de cáncer.

Deducción:

F(t) = e^{(1/m)·( HCg+( q+(-W) ) )}·f(t)

G(t) = e^{(1/m)·( W+(-q) )}·g(t)

Ley: [ de tetra-cáncer ]

Si ( Fumar = C·(OH)_{4} y Ver la televisión = W ) ==>

... Se provoca una desintegración de cáncer.

Deducción:

F(t) = e^{(1/m)·( CCg_{4}+4·( q+(-W) ) )}·f(t)

G(t) = e^{(1/m)·4·( W+(-q) )}·g(t)

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} ]