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} ]