Monarquía parlamentaria:
El poder legislativo ejecutivo,
reside en el Congreso de los Diputados,
y es escogido por sufragio universal.
El poder legislativo judicial,
reside en el Senado,
y es escogido por sufragio universal.
Artículo:
El presidente del poder ejecutivo,
es escogido por el Congreso de los Diputados.
El presidente del poder judicial,
es escogido por el Senado.
Artículo:
El Rey es árbitro de la democracia ejecutiva,
pudiendo inhabilitar diputados,
en el Congreso de los Diputados.
El Rey es árbitro de la democracia judicial,
pudiendo inhabilitar senadores,
en el Senado.
Republica Presidencialista:
Artículo:
El poder legislativo ejecutivo,
reside en el Congreso de los Diputados,
y es escogido por sufragio universal.
El poder legislativo judicial,
reside en el Senado,
y es escogido por sufragio universal.
Artículo:
El presidente del poder ejecutivo,
es escogido por sufragio universal.
El presidente del poder judicial,
es escogido por Sufragio universal.
Ley:
Sea ( x no creer-se Jesucristo & y creer-se Jesucristo ) ==>
Si ( a = 23 & b = 7 ) ==>
[05][...][05][...][05][03][05][...] = 23
[11][...][11][...][11][09][11][...] = 53
Falsus Algebratorum:
53 = 30+23 = 30+(-23) = 7
Fórmula:
=Br=(CH)-(Br|-|N)-(CH)=
Ley:
Sea ( x no se cree tu dios & y se cree tu dios ) ==>
Si ( a = 1 & b = 29 ) ==>
[05][...][05][...][05][09][05][...] = 29
[11][...][11][...][11][15][11][...] = 59
Falsus Algebratorum:
59 = 30+29 = 30+(-29) = 1
Fórmula:
=Br=(CH)-(Br|-|Krg|-|N)-(CH)=
Al primer año de voces no vas al psiquiatra,
porque la voz no se cree Jesucristo,
y después se lo cree durante 29 años.
Ley:
Sea ( x estar cerrado en el hospital & y no estar cerrado en el hospital ) ==>
Si ( a = 7 & b = 5 ) ==>
[06][...][01][...] = 07
[12][...][07][...] = 19
Falsus Algebratorum:
19 = 12+7 = 12+(-7) = 5
Fórmula:
=C=C=C-O-O-N=
Se convierte cerrar a un hombre,
en enfermedad mental de doble mandamiento,
en ser una distancia a la menos uno.
Stowed-Nipon-Chinese:
Dual:
Kino-yute [o] Kino-yang [o] Ayer
Hata-yute [o] Hata-yang [o] Mañana
Dual:
Sasa-yute [o] Sasa-yang [o] Matina
Yoro-yute [o] Yoro-yang [o] Tarde
Dual:
Hiro-yute [o] Hiro-yang [o] Medio-día
Banga-yute [o] Banga-yang [o] Noche
Dual:
Kioro-yute [o] Kioro-yang [o] Hoy
Hisa-yute [o] Hisa-yang [o] Día
Dual:
I stare-kate-maruto,
drinket-yuto-yaming mutchet-muto.
kioro-yute by sasa-yute.
I stareti-kate-maruto,
drinket-yuto-yaming pocket-muto,
kioro-yute by yoro-yute.
Dual:
I stare-kate-tai-tai,
drinket-yung-yanguing mutchet-tai-mung,
kioro-yang by sasa-yang.
I stareti-kate-tai-tai,
drinket-yung-yanguing pocket-tai-mung,
kioro-yang by yoro-yang.
Teorema:
Si f_{n}(x) es creciente ==> int[ f_{n}(x) ]d[x] es monótona
Si f_{n}(x) es decreciente ==> int[ f_{n}(x) ]d[x] es monótona
Demostración:
[1] Sea f_{n}(x) [< f_{n+1}(x) ==>
f_{n}(x)·d[x] [< f_{n+1}(x)·d[x]
int[ f_{n}(x) ]d[x] [< int[ f_{n+1}(x) ]d[x]
[2] Sea f_{n}(x) [< f_{n+1}(x) ==>
f_{n}(x)·d[x] >] f_{n+1}(x)·d[x]
int[ f_{n}(x) ]d[x] >] int[ f_{n+1}(x) ]d[x]
Teorema: [ de convergencia monótona ]
Si f_{n}(x) es creciente ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]
Si f_{n}(x) es decreciente ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]
Demostración:
Sea f_{n}(x) es creciente ==>
int[ f_{n}(x) ]d[x] es monótona
int[ f_{n}(x) ]d[x] es creciente || int[ f_{n}(x) ]d[x] es decreciente
[1] Si int[ f_{n}(x) ]d[x] es creciente ==>
int[ f_{n}(x) ]d[x] [< int[ f_{n+1}(x) ]d[x] [< int[ f(x) ]d[x]
lim[n = oo][ int[ f_{n}(x) ]d[x] ] [< int[ f(x) ]d[x]
Se define ( m = 1 || m = oo ) ==>
Sea n >] m ==>
int[ f_{n}(x) ]d[x] >] ( 1+(-1)·(1/m) )^{p}·int[ f(x) ]d[x]
lim[n = oo][ int[ f_{n}(x) ]d[x] ] >] int[ f(x) ]d[x]
[2] Si int[ f_{n}(x) ]d[x] es decreciente ==>
int[ f_{n}(x) ]d[x] >] int[ f_{n+1}(x) ]d[x] >] int[ f(x) ]d[x]
lim[n = oo][ int[ f_{n}(x) ]d[x] ] >] int[ f(x) ]d[x]
Se define ( m = 1 || m = oo ) ==>
Sea n >] m ==>
int[ f_{n}(x) ]d[x] [< ( 1+(-1)·(1/m) )^{p}·int[ f(x) ]d[x]
lim[n = oo][ int[ f_{n}(x) ]d[x] ] [< int[ f(x) ]d[x]
Teorema: [ de convergencia dominada ]
Si [Ek][An][ n >] k ==> f_{n}(x) >] f(x) ] ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]
Si [Ek][An][ n >] k ==> f_{n}(x) [< f(x) ] ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]
Demostración:
Sea n >] k ==>
f_{n}(x) >] f(x)
f_{n}(x)·d[x] >] f(x)·d[x]
int[ f_{n}(x) ]d[x] >] int[ f(x) ]d[x]
Se define ( m = 1 || m = oo ) ==>
Sea n >] m ==>
int[ f_{n}(x) ]d[x] [< (1+(-1)·(1/m))^{p}·int[ f(x) ]d[x]
lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]
Sea n >] k ==>
f_{n}(x) >] f(x)
f_{n}(x)·d[x] [< f(x)·d[x]
int[ f_{n}(x) ]d[x] [< int[ f(x) ]d[x]
Se define ( m = 1 || m = oo ) ==>
Sea n >] m ==>
int[ f_{n}(x) ]d[x] >] (1+(-1)·(1/m))^{p}·int[ f(x) ]d[x]
lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]
Teorema: [ de la primera convergencia algebraica ]
Si [Eh(x)][An][ f_{n}(x)+g_{n}(x) = h(x) ] ==> ...
... lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] ...
... <==> ...
... lim[n = oo][ int[ g_{n}(x) ]d[x] ] = int[ g(x) ]d[x]
Teorema: [ de la segunda convergencia algebraica ]
Si [Eh(x)][An][ f_{n}(x)·g_{n}(x) = h(x) ] ==> ...
... lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] ...
... <==> ...
... lim[n = oo][ int[ g_{n}(x) ]d[x] ] = int[ g(x) ]d[x]
Teorema:
Sea ( u(a,b,n) = min{f(x)+(-1)·(a/n),f(x)+(-1)·(b/n)} & & |A_{oo}(a,b)| = 1 ) ==>
Sea s_{n}(x) = u(a,b,n)·|A_{n}(a,b)| ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = int[x = a]-[b][ f(x) ]d[x]
Demostración:
u(a,b,n)·|A_{n}(a,b)| [< u(a,b,n)·|A_{n+1}(a,b)| [< ...
... u(a·(n/(n+1)),b·(n/(n+1)),n)·|A_{n+1}(a,b)| = u(a,b,n+1)·|A_{n+1}(a,b)|
Si [ (MP) Teorema de convergencia monótona ] ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = int[x = a]-[b][ s(x) ]d[x] = int[x = a]-[b][ f(x) ]d[x]
Teorema:
Sea ( v(a,b,n) = max{f(x)+(-1)·(a/n),f(x)+(-1)·(b/n)} & |B_{(-oo)}(a,b)| = 1 ) ==>
Sea S_{n}(a,b) = v(a,b,n)·|B_{(-n)}(a,b)| ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = int[x = a]-[b][ f(x) ]d[x]
Demostración:
v(a,b,n)·|B_{(-n)}(a,b)| >] v(a,b,n)·|B_{(-1)·(n+1)}(a,b)| >] ...
... v(a·(n/(n+1)),b·(n/(n+1)),n)·|B_{(-1)·(n+1)}(a,b)| = v(a,b,n+1)·|B_{(-1)·(n+1)}(a,b)|
Si [ (MP) Teorema de convergencia monótona ] ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = int[x = a]-[b][ S(x) ]d[x] = int[x = a]-[b][ f(x) ]d[x]
Teorema:
Sea m >] n ==>
Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( m+(-n) ) ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Sea (-m) [< (-n) ==>
Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( n+(-m) ) ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Demostración:
Sea m >] n+1 >] n ==>
(-n) >] (-n)+(-1) = (-1)·(n+1)
(-n)+m >] (-1)·(n+1)+m
|A_{n+1}(a,b)| = |A_{n}(a,b)|+(-1)
s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( m+(-n) ) [< ( f(x)+(-1)·(b/(n+1)) )·( m+(-1)·(n+1) ) = s_{n+1}(0,a)
Sea (-m) [< (-1)·(n+1) [< (-n) ==>
n [< n+1
n+(-m) [< n+1+m
|B_{n+1}(a,b)| = |B_{n}(a,b)|+1
S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( n+(-m) ) >] ( f(x)+(-1)·(a/(n+1)) )·( (n+1)+(-m) ) = S_{n+1}(a,b)
Teorema:
Sea ( c = b+(-a) >] 0 ==> sig(b+(-a)) = 1 ) ==>
Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·sig(c) ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Sea ( (-c) = a+(-b) [< 0 ==> sig(a+(-b)) = (-1) ) ==>
Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·sig(-c) ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Teorema:
Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·( 1+(-1)·(1/n) ) ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·( (-1)+(1/n) ) ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Demostración:
Sea |¬A_{n}(a,b)| = (1/n) ==>
|A_{n+1}(a,b)| = 1+(-1)·|¬A_{n}(a,b)|·( n/(n+1) )
1+(-1)·( 1/(n+1) ) = 1+(-1)·(1/n)·( n/(n+1) )
s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·( 1+(-1)·(1/n) ) [< ...
... ( f(x)+(-1)·(b/(n+1)) )·( (n+1)+(-n) )·( 1+(-1)·(1/(n+1)) ) = s_{n+1}(a,b)
Sea |¬B_{(-n)}(a,b)| = (1/n) ==>
|B_{(-1)·(n+1)}(a,b)| = (-1)+|¬B_{(-n)}(a,b)|·( n/(n+1) )
(-1)+( 1/(n+1) ) = (-1)+(1/n)·( n/(n+1) )
S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·( (-1)+(1/n) ) >] ...
... ( f(x)+(-1)·(a/(n+1)) )·( (n+1)+(-n) )·( (-1)+(1/(n+1)) ) = S_{n+1}(a,b)
Teorema: [ de la función de doble factorial exponencial ]
[1] Sea sum[k = 2p]-[n][ (1/(k+2)!!)·z^{k+2} ] = [!e!]-(z) ==>
Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·...
... ( (n+1)+(-n) )·sum[k = 2p]-[n][ (-1)^{k}·(1/(k+2)!!)·z^{k+2} ]·( 1/[!e!]-(z) ) ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
[2] Sea sum[k = 2p+1]-[n][ (1/(k+1)!!)·(-z)^{k+1} ] = [!e!]-(-z) ==>
Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·...
... ( (n+1)+(-n) )·sum[k = 2p+1]-[n][ (-1)^{k}·(1/(k+1)!!)·(-z)^{k+1} ]·( 1/[!e!]-(-z) ) ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Demostración:
[1] |A_{2k+2}(a,b)| = |A_{2k}(a,b)|+(-1)^{n}·(1/(n+2)!!)·z^{n+2}·( 1/[!e!]-(z) )
Si n = 2p ==>
z^{n+2} >] 0
(-1)^{n}·(1/(n+2)!!)·z^{n+2}·( 1/[!e!]-(z) ) >] 0
s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·((n+1)+(-n))·...
... sum[k = 2p]-[n][ (-1)^{k}·(1/(k+2)!!)·z^{k+2} ]·( 1/[!e!]-(z) ) [< ...
... ( f(x)+(-1)·(b/(n+2)) )·( ((n+2)+1)+(-1)·(n+2) )·...
... sum[k = 2p]-[n+2][ (-1)^{k}·(1/(k+2)!!)·z^{k+2} ]·( 1/[!e!]-(z) ) = s_{n+2}(a,b)
[2] |B_{(-1)·(2k+3)}(a,b)| = |B_{(-1)·(2k+1)}(a,b)|+(-1)^{n}·(1/(n+1)!!)·(-z)^{n+1}·( 1/[!e!]-(-z) )
Si n = 2p+1 ==>
(-z)^{n+1} >] 0
(-1)^{n}·(1/(n+1)!!)·(-z)^{n+1}·( 1/[!e!]-(-z) ) [< 0
S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·((n+1)+(-n))·...
... sum[k = 2p+1]-[n][ (-1)^{k}·(1/(k+1)!!)·(-z)^{k+1} ]·( 1/[!e!]-(-z) ) >] ...
... ( f(x)+(-1)·(a/(n+2)) )·( ((n+2)+1)+(-1)·(n+2) )·...
... sum[k = 2p+1]-[n+2][ (-1)^{k}·(1/(k+1)!!)·(-z)^{k+1} ]·(1/[!e!]-(-z)) = S_{n+2}(a,b)
Teorema:
d_{x}[ [!e!]-(x) ] = [!e!]-(x)
d_{x}[ [!e!]-(-x) ] = (-1)·[!e!]-(-x)
Demostración:
Sea k = m+2 ==>
d_{x}[ [!e!]-(x) ] = sum[k = 2p]-[oo][ (1/k!!)·x^{k} ] = ...
... sum[m = 2p]-[oo][ (1/(m+2)!!)·x^{m+2} ] = [!e!]-(x)
d_{x}[ [!e!]-(-x) ] = sum[k = 2p+1]-[oo][ (-1)·(1/(k+(-1))!!)·(-x)^{k+(-1)} ] = ...
... (-1)·sum[k = 2p+1]-[oo][ (1/(k+(-1))!!)·(-x)^{k+(-1)} ] = ...
... (-1)·sum[m = 2p+1]-[oo][ (1/(m+1)!!)·(-x)^{m+1} ] = (-1)·[!e!]-(-x)
Teorema:
Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·...
... sum[k = 2p]-[n][ (-1)^{k}·(1/( (1/2)·(k+2) )!)·z^{k+2} ]·e^{(-1)·z^{2}} ) ==>
lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·...
... sum[k = 2p+1]-[n][ (-1)^{k}·(1/( (1/2)·(k+1) )!)·(-z)^{k+1} ]·e^{(-1)·z^{2}} ) ==>
lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)
Teorema:
Sea < x€Q ==> f(x) = 1 & x€I ==> f(x) = 0 > ==> int[x = 0]-[1][ f(x) ]d[x] = 1
Demostración:
Sea |M_{n}(0,1)| = oo·|B_{n}(0,1)| ==>
s_{n}(0,1) = ( 1+(-1)·(1/n) )·|A_{n}(0,1)| = [ x ]_{x = 0}^{x = 1} = 1
S_{n}(0,1) = ( 0+(-1)·(0/n) )·|M_{n}(0,1)| = [ x ]_{x = 0}^{x = 1} = 1
Teorema:
Sea < x€Q ==> f(x) = x & x€I ==> f(x) = 1+(-x) > ==> int[x = 0]-[1][ f(x) ]d[x] = (1/2)
Demostración:
s_{n}(0,1) = ( (1/2)+(-1)·(1/n) )·|A_{n}(0,1)| = [ (1/2)·x ]_{x = 0}^{x = 1} = (1/2)
S_{n}(0,1) = ( (1+(-1)·(1/2))+(-1)·(0/n) )·|B_{n}(0,1)| = [ (1/2)·x ]_{x = 0}^{x = 1} = (1/2)
Teorema:
[ {a_{1},...,a_{n}} ] ==> [ {f(a_{1}),...,f(a_{n})} ] ==> [ {g(f(a_{1})),...,g(f(a_{n})} ] es especie
Demostración:
g(f(p)) = g(f(q))
f(p) = f(q)
p = q
Teorema:
[ < a_{1},...,a_{n} > ] ==> [ < f(a_{1}),...,f(a_{n}) > ] ==> [ < g(f(a_{1})),...,g(f(a_{n}) > ] es especie
Demostración:
g(f(p)) = g(f(q))
f(p) = f(q)
p = q
Definición: [ de serie aritmética de una especie ]
Sea A una especie ==>
F(x) = sum[n = 0]-[oo][ #A(n)·x^{n} ]
Definición: [ de isomorfismo de especie ]
Sean A & B especies ==>
A =[#]= B <==> sum[n = 0]-[oo][ #A(n)·x^{n} ] = sum[n = 0]-[oo][ #B(n)·x^{n} ]
Teorema:
#A(n) = #A(n)
#A(n) = #B(n) <==> #B(n) = #A(n)
Si ( #A(n) = #B(n) & #B(n) = #C(n) ) ==> #A(n) = #C(n)
Teorema:
A(n)+[ M ] =[#]= A(n+1)
Demostración:
sum[n = 0]-[oo][ ( #A(n)+#[ M ] )·x^{n} ] = ...
... sum[n = 0]-[oo][ (n+1)·x^{n} ] = sum[n = 0]-[oo][ #A(n+1)·x^{n} ]
Teorema:
Sea A una especie de orden inferior a A(n) ==>
[EB][ B = A [& || &] A(n) & B [<< A(n) ]
Demostración:
Se define B = A [& || &] A(n) ==>
B = B [&] A(n)
B [<< A(n)
Teorema:
Sea A una especie de orden inferior a N(n) ==>
[EB][ B = A [& | &] N(n) & B [<< N(n) ]
Demostración:
Se define B = A [& | &] N(n) ==>
B = B [&] N(n)
B [<< N(n)
Definición: [ de serie geométrica de una especie ]
Sea A(p,n) una especie geométrica ==>
F(x) = sum[n = 0]-[oo][ p^{#A(n)}·x^{n} ]
Teorema:
Sea A(p,n) & B(q,n) dos especies geométricas & a = mcd{p,q} ==>
[EM][ M es especie geométrica & M = mcd{A(p,n),B(q,n)} & F(1) = ( a/(a+(-1)) ) ]
Demostración:
Se define M(mcd{p,q},n) = mcd{A(p,n),B(q,n)} ==>
M(a,n) = M(mcd{p,q},n)
F(1) = sum[n = 0]-[oo][ (1/a)^{n} ] = ( a/(a+(-1)) )
Teorema:
Sea A(p,n) & B(q,n) dos especies geométricas & b = mcm{p,q} ==>
[EW][ W es especie geométrica & W = mcm{A(p,n),B(q,n)} & F(1) = ( b/(b+(-1)) ) ]
Demostración:
Se define W(mcm{p,q},n) = mcm{A(p,n),B(q,n)} ==>
W(b,n) = W(mcm{p,q},n)
F(1) = sum[n = 0]-[oo][ (1/b)^{n} ] = ( b/(b+(-1)) )
Definición:
P_{k}( {a_{1},...,a_{n}} ) es especie
F(x) = sum[n = 0]-[oo][ [ n // k ]·x^{n} ]
Teorema:
Sea f(n) = [ n // n+(-1) ] ==>
[EA][ A = [ 2 // 2 ] [& || &] [ 3 // 2 ] & A [<< [ 3 // 2 ] ]
[EB][ B = A [& || &] [ 4 // 3 ] & B [<< [ 4 // 3 ] ]
Demostración:
A = [ {a,b} ]
B = [ {a,b,c},{a,b,d} ]
Arte:
[Ex][ Si F(x) = sum[n = 1]-[oo][ [ n // n+(-1) ]·x^{n} ] ==> d_{x}[F(x)] = ( 1/(1+(-x)) ) ]
Exposición:
x = 0
d_{x}[F(x)] = sum[n = 1]-[oo][ n^{2}·x^{n+(-1)} ] = sum[n = 1]-[oo][ 2n·x^{n+(-1)} ] = ...
... sum[n = 1]-[oo][ (n+n+(1/2)+(-1)·(1/2))·x^{n+(-1)} ] = ...
... sum[n = 1]-[oo][ (n+(-n)+(1/2)+(1/2))·x^{n+(-1)} ] = sum[n = 1]-[oo][ x^{n+(-1)} ] = ( 1/(1+(-x)) )
Teorema:
Sea f(n) = [ 2n // n ] ==>
[EA][ A = [ 2 // 2 ] [& || &] [ 4 // 2 ] & A [<< [ 4 // 2 ] ]
[EB][ B = A [& || &] [ 6 // 3 ] & B [<< [ 6 // 3 ] ]
Demostración:
A = [ {a,b} ]
B = [ {a,b,c},{a,b,d},{a,b,e},{a,b,f} ]
Arte:
[Ex][ Si F(x) = sum[n = 0]-[oo][ [ 2n // n ]·x^{n} ] ==> F(x) = ( 1/(1+(-x)) ) ]
Exposición:
x = 0
F(x) = sum[n = 0]-[oo][ (2n)!(1/n!)·(1/n!)·x^{n} ] = sum[n = 0]-[oo][ (2n)·(n!/n!)·x^{n} ] = ...
... sum[n = 0]-[oo][ (2n)!·x^{n} ] = sum[n = 0]-[oo][ (n+n)!·x^{n} ] = ...
... sum[n = 0]-[oo][ (n+(-n))!·x^{n} ] = sum[n = 0]-[oo][ 0!·x^{n} ] = ...
... sum[n = 0]-[oo][ x^{n} ] = ( 1/(1+(-x)) )
Definición:
A(n) = { B : i+j = n & B = {a}^{i} [x] {b}^{j} } es especie
F(x) = sum[n = 2]-[oo][ sum[i+j = n][ a^{i}·b^{j} ]·x^{n} ]
Teorema:
Sea A(4) = [ < a,a,a,b >,< a,a,b,b >,< b,b,b,a > ] ==>
[EM][ M = [ < a,a,b > ] [& || &] A(4) & M [<< A(4) ]
[EW][ W = [ < b,b,a > ] [& || &] A(4) & W [<< A(4) ]
Demostración:
M = [ < a,a,a,b >,< a,a,b,b > ]
W = [ < b,b,b,a >,< a,a,b,b > ]
Teorema:
M+W = ab·(a+b)^{2}
Demostración:
( ba^{3}+a^{2}·b^{2} )+( a^{2}·b^{2}+ab^{3} ) = ...
... ba^{3}+2a^{2}·b^{2}+ab^{3} = ab·( a^{2}+2ab+b^{2} )
Arte:
(1/2)·M+(1/2)·W != 2ab·(a+b)^{2}
Exposición:
( (1/2)·ba^{3}+(1/2)·a^{2}·b^{2} )+( (1/2)·a^{2}·b^{2}+(1/2)·ab^{3} ) = ...
... ( 2ba^{3}+2a^{2}·b^{2} )+( 2a^{2}·b^{2}+2ab^{3} ) = ...
... 2ba^{3}+4a^{2}·b^{2}+2ab^{3} = 2ab·( a^{2}+2ab+b^{2} )
Teorema:
Si F(x) = sum[n = 2]-[oo][ sum[i+j = n][ a^{i}·b^{j} ]·x^{n} ] ==> ...
... F(1) = ab·( 1/(1+(-a)) )·( 1/(1+(-b)) )
Demostración:
F(1) = ( 1/(1+(-a)) )·( 1/(1+(-b)) )+(-1)·( 1/(1+(-a)) )+(-1)·( 1/(1+(-b)) )+1
Definición:
A(n) = { < f(1),...,f(n) > } es especie
F(x) = sum[n = 1]-[oo][ n!·x^{n} ]
Teorema:
Sea ( A = { < f(1),f(2) > : [Ek][ f(k) = k ] } & B = { < f(1),f(2) > : [Ak][ f(k) != k ] } ) ==> ...
... [EM][ M = A [& || &] A(3) & M [<< A(3) ]
... [EW][ W = B [& || &] A(3) & W [<< A(3) ]
Demostración:
M = [ < 1,2,3 > ]
W = [ < 2,1,3 > ]
Teorema:
Sea ( A = { < f(1),f(2) > : [Ek][ f(k) = k ] } & B = { < f(1),f(2) > : [Ak][ f(k) != k ] } ) ==> ...
... [EM][ M = A [& || &] A(4) & M [<< A(4) ]
... [EW][ W = B [& || &] A(4) & W [<< A(4) ]
Demostración:
M = [ < 1,2,3,4 >,< 1,2,4,3 > ]
W = [ < 2,1,3,4 >,< 2,1,4,3 > ]
Arte:
Sea [ < f(1),...,f(n) > ] ==> d_{x}[F(x)] != sum[n = 1]-[oo][ n!·x^{n} ]
Exposición:
x = 0
d_{x}[F(x)] = sum[n = 1]-[oo][ n!·n·x^{n+(-1)} ] = sum[n = 0]-[oo][ (n+1)!·(n+1)·x^{n} ] = ...
... sum[n = 0]-[oo][ (n+1)!·( (n+1)+(1/2)+(-1)·(1/2) )·x^{n} ] = ...
... sum[n = 0]-[oo][ (n+1)!·( (n+1)+(1/2)+(1/2) )·x^{n} ] = sum[n = 0]-[oo][ (n+1)!·(n+2)·x^{n} ] = ...
... sum[n = 0]-[oo][ (n+2)!·x^{n} ] = sum[n = 0]-[oo][ ( n+(1+1) )!·x^{n} ] = ...
... sum[n = 0]-[oo][ ( n+(1+(-1)) )!·x^{n} ] = sum[n = 0]-[oo][ n!·x^{n} ]
Definición: [ de octopus geométrico ]
A = [ {p},...,{p^{n}} ]-[ \ ]-[ {p^{j}} ]
F(A,x) = sum[n = 1]-[oo][ p^{n+(-1)}·x^{n} ]
Teorema:
Sea ( A = [ {(1/6)},...,{(1/6)^{n}} ]-[ \ ]-[ {(1/6)^{j}} ] & ...
... B = [ {(1/10)},...,{(1/10)^{n}} ]-[ \ ]-[ {(1/10)^{j}} ] ) ==>
[EM][ M = mcd{A,B} & F(M,1) = ? ]
[EW][ W = mcm{A,B} & F(W,1) = ? ]
Demostración:
6 = 2·3 & 10 = 2·5
M = [ {(1/2)},...,{(1/2)^{n}} ]-[ \ ]-[ {(1/2)^{j}} ]
F(M,1) = 2
W = [ {(1/30)},...,{(1/30)^{n}} ]-[ \ ]-[ {(1/30)^{j}} ]
F(W,1) = (30/29)
Teorema:
Sea [ || ]-[j = 1]-[n][ [ {p},...,{p^{n}} ]-[ \ ]-[ {p^{j}} ] ] ==>
Si F(x) = sum[n = 1]-[oo][ p^{n^{2}+(-n)}·x^{n} ] ==> F(1) = ( 1/(1+(-1)·p^{n}) )
Demostración:
F(1) = sum[n = 1]-[oo][ p^{n^{2}+(-n)} ] = ...
... sum[n = 1]-[oo][ ( p^{n} )^{n+(-1)} ] = ( 1/(1+(-1)·p^{n}) )
Arte:
Si F(x) = sum[n = 1]-[oo][ p^{n+(-1)}·x^{n} ] ==> F(1) != p·( 1/(1+(-p)) )
Exposición:
F(1) = sum[n = 1]-[oo][ p^{n+(-1)} ] = sum[n = 1]-[oo][ p^{n+(-1)·( (1/2)+(1/2) )} ] = ...
... sum[n = 1]-[oo][ p^{n+(-1)·(1/2)+(1/2)} ] = sum[n = 1]-[oo][ p^{n} ] = ...
... p·sum[n = 1]-[oo][ p^{n+(-1)} ] = p·( 1/(1+(-p)) )
Definición: [ de octopus aritmético ]
N(n) = [ {1},...,{n} ]
F(x) = sum[n = 1]-[oo][ nx^{n} ]
Teorema:
Sea ( A = [ {2},{4} ] & B = [ {2},{3} ] ) ==> ...
... [EM][ M = A [& | &] A(8) & M [<< A(8) ]
... [EW][ W = B [& | &] A(12) & W [<< A(12) ]
Demostración:
M = [ {4},{8} ]
W = [ {6},{12} ]
Teorema:
Sea A(n+1) = [ {a_{1},...,a_{n}} ]-[ {a_{1}},...,{a_{n}} ] ==>
[EM][ M = N(n) x A(n+1) ]
M =[#]= 2·sum[k = 1]-[n][ [ {a_{1}},...,{a_{k}} ] ]
Demostración:
M = [ < {1},{a_{1},...,a_{n}} >,...,< {n},{a_{1},...,a_{n}} > ]-...
... [ < {1},{a_{1}} >,...,< {1},{a_{n}} >,...,< {n},{a_{1}} >,...,< {n},{a_{n}} > ]
F(M,x) = sum[n = 1]-[oo][ n·(n+1)·x^{n} ] = sum[n = 1]-[oo][ (n^{2}+n)·x^{n} ]
(1/2)·n·(n+1)+(n+1) = (1/2)·( n·(n+1)+2·(n+1) ) = (1/2)·(n+2)·(n+1) = (1/2)·(n+1)·(n+2)
Teorema:
Sea A(n) = [ {a_{1}},...,{a_{n}} ] ==>
[EM][ M = N(n) x A(n) ]
M =[#]= sum[k = 1]-[n][ [ {a_{1}},...,{a_{2k+(-1)}} ] ]
Demostración:
M = [ < {1},{a_{1}} >,...,< {1},{a_{n}} >,...,< {n},{a_{1}} >,...,< {n},{a_{n}} > ]
F(M,x) = sum[n = 1]-[oo][ n·n·x^{n} ] = sum[n = 1]-[oo][ n^{2}·x^{n} ]
n^{2}+(2n+1) = (n+1)^{2}
Teorema:
Sea A(n) = [ {a_{1}},...,{a_{n}} ] ==>
[EM][ M = ( N(n) x A(n) )+( A(n) x N(n) ) ]
M =[#]= (1/2)·sum[k = 1]-[n][ [ {a_{4}},...,{a_{8k+(-4)}} ] ]
Demostración:
M = ...
... [ < {1},{a_{1}} >,...,< {1},{a_{n}} >,...,< {n},{a_{1}} >,...,< {n},{a_{n}} > ] ...
... +...
... [ < {a_{1}},{1} >,...,< {a_{n}},{1} >,...,< {a_{1}},{n} >,...,< {a_{n}},{n} > ]
F(M,x) = sum[n = 1]-[oo][ (n^{2}+n^{2})·x^{n} ] = sum[n = 1]-[oo][ 2n^{2}·x^{n} ]
4n^{2}+(8n+4) = (2n+2)^{2} = 4·(n+1)^{2}
Teorema:
Sea A(p,n) = [ {(1/p)},...,{(1/p)^{n}} ] ==>
[EM][ M = N(n) [o] A(p,n) ]
F(1) = p·( 1/(p+(-1))^{2} )
Demostración:
M(n) = [ < {1},{(1/p)^{n}} >,...,< {n},{(1/p)^{n}} > ] ...
Teorema:
Sea A(p,n) = [ {(1/p)},...,{(1/p)^{n}} ] ==>
[EM][ M = ( N(n) x N(n) ) [o] A(p,n) ]
F(1) = p·( 1/(p+(-1))^{3} )+p·( 1/(p+(-1))^{2} )
Demostración:
M(n) = [ < {1}x{1},{(1/p)}^{n} >,...,< {1}x{n},{(1/p)^{n}} >,...(n)...,
... < {n}x{1},{(1/p)}^{n} >,...,< {n}x{n},{(1/p)^{n}} > ]
Ley:
La enfermedad de creer-se señor del prójimo,
no impide creer en condenación
porque las voces de la mente no pueden salir de la mente,
en creer que hay condenación del satélite.
Deducción:
La enfermedad de creer-se señor del prójimo,
impide creer en condenación
aunque quizás las voces de la mente no pueden salir de la mente,
en creer que hay condenación del satélite.
Teorema:
Sea f_{n}(x) = (1/n)·x^{2p+(-1)} ==> ...
... ( f_{n}(x) es integrable en R & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = (1/oo)·(1/(2p))·x^{2p}
Demostración:
Sea x > 0 ==>
(1/n) >] (1/(n+1))
f_{n}(x) = (1/n)·x^{2p+(-1)} >] (1/(n+1))·x^{2p+(-1)} = f_{n+1}(x)
Si [ (MP) Teorema de convergencia monótona ] ==>
f_{n}(x) es integrable
Sea x < 0 ==>
(1/n) >] (1/(n+1))
f_{n}(x) = (1/n)·x^{2p+(-1)} [< (1/(n+1))·x^{2p+(-1)} = f_{n+1}(x)
Si [ (MP) Teorema de convergencia monótona ] ==>
f_{n}(x) es integrable
Sea x = 0 ==>
f_{n}(x) = (1/n)·x^{2p+(-1)} >] (1/oo)·x^{2p+(-1)} = f(x)
Si [ (MP) Teorema de convergencia dominada ] ==>
f_{n}(x) es integrable
Sea x = (-0) ==>
f_{n}(x) = (1/n)·x^{2p+(-1)} [< (1/oo)·x^{2p+(-1)} = f(x)
Si [ (MP) Teorema de convergencia dominada ] ==>
f_{n}(x) es integrable
lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] = ...
... int[ (1/oo)·x^{2p+(-1)} ]d[x] = (1/oo)·(1/(2p))·x^{2p}
Teorema:
Sea f_{n}(x) = nx^{2p+1} ==> ...
... ( f_{n}(x) es integrable en R & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = oo·(1/(2p+2))·x^{2p+2} )
Demostración:
Sea x > 0 ==>
n [< n+1
f_{n}(x) = nx^{2p+1} [< (n+1)·x^{2p+1} = f_{n+1}(x)
Si [ (MP) Teorema de convergencia monótona ] ==>
f_{n}(x) es integrable
Sea x < 0 ==>
n [< n+1
f_{n}(x) = nx^{2p+1} >] (n+1)·x^{2p+1} = f_{n+1}(x)
Si [ (MP) Teorema de convergencia monótona ] ==>
f_{n}(x) es integrable
Sea x = 0 ==>
f_{n}(x) = nx^{2p+1} [< oo·x^{2p+1} = f(x)
Si [ (MP) Teorema de convergencia dominada ] ==>
f_{n}(x) es integrable en x = 0
Sea x = (-0) ==>
f_{n}(x) = nx^{2p+1} >] oo·x^{2p+1} = f(x)
Si [ (MP) Teorema de convergencia dominada ] ==>
f_{n}(x) es integrable en x = (-0)
lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] = int[ oo·x^{2p+1} ]d[x] = oo·(1/(2p+2))·x^{2p+2}
Exámenes de Análisis matemático 5:
Teorema:
Sea f_{n}(x) = n·ln(x) ==> ...
... ( f_{n}(x) es integrable en x > 0 & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = oo·( ln(x)·x+(-x) )
Teorema:
Sea f_{n}(x) = n·( e^{x}+(-1) ) ==> ...
... ( f_{n}(x) es integrable en R & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = oo·( e^{x}+(-x) )
Diferencial exterior de producto escalar nulo.
Teorema:
int-int[ x^{n}·d[x]d[y]+y^{n}·d[y]d[x] ] = (1/(n+1))·x^{n}·y^{n}·( y+x )
x^{n}·( d[x] & d[y] )+y^{n}·( d[y] & d[x] ) = x+(-y)
Teorema:
int-int[ x^{n}·d[x]d[y]+y^{n+k}·d[y]d[x] ] = x^{n}·y^{n}·( (1/(n+1))·y+(1/(n+k+1))·y^{k+1}·x )
x^{n}·( d[x] & d[y] )+y^{n+k}·( d[y] & d[x] ) = (n+1)·x+(-1)·(n+k+1)·(1/y)^{k}
Teorema:
int-int[ e^{nx}·d[x]d[y]+y^{n}·d[y]d[x] ] = (1/n)·e^{nx}·y+(1/(n+1))·y^{n+1}·x
e^{nx}·( d[x] & d[y] )+y^{n}·( d[y] & d[x] ) = nxe^{(-1)·nx}+(-1)·(n+1)·(1/y)^{n}
Teorema:
int-int[ yx^{n}·d[x]d[y]+xy^{n}·d[y]d[x] ] = (1/2)·(1/(n+1))·x^{n}·y^{n}·( y^{2}+x^{2} )
yx^{n}·( d[x] & d[y] )+xy^{n}·( d[y] & d[x] ) = x^{2}+(-1)·y^{2}
Matemáticas y Física:
Análisis matemático 3:
[%] Derivadas parciales.
[%] Optimización de LaGrange.
Trayectorias sobre superficies.
Análisis matemático 4:
[%] Integrales múltiples.
[%] Integrales de camino ( d[x] & d[yz] )
Diferenciales exteriores.
Matemáticas:
Análisis matemático 5:
Sucesiones de funciones integrables.
Integral de Medida = 1.
Series de Fourier.
Análisis matemático 6:
Transformada integral exponencial.
Integrales por el Método de Euler.
Series de Laurent.
Definición:
a_{0} = 0
a_{k} = (1/pi·i)·int[0]-[2pi·i][ f(x)·cosh(kx) ]d[x]
b_{k} = (-1)·(1/pi·i)·int[0]-[2pi·i][ f(x)·sinh(kx) ]d[x]
cosh(2w) = ( cosh(w) )^{2}+( sinh(w) )^{2}
Teorema:
x^{3} = (-12)·( sinh(x)+(1/3)^{3}·sinh(3x)...+(1/(2n+1))^{3}·sinh((2n+1)·x) )
sum[k = 0]-[oo][ (-1)^{k}·( 1/(2k+1) )^{3} ] = (1/96)·pi^{3}
Demostración:
0 [< x [< 2pi·i = 0
( (pi/2)·i )^{3} = (-1)·(1/8)·pi^{3}·i
Teorema:
x^{5} = 160·pi^{2}·( sinh(x)+(1/3)^{3}·sinh(3x)...+(1/(2n+1))^{3}·sinh((2n+1)·x) )+...
... (-240)·( sinh(x)+(1/3)^{5}·sinh(3x)...+(1/(2n+1))^{5}·sinh((2n+1)·x) )
sum[k = 0]-[oo][ (-1)^{k}·( 1/(2k+1) )^{5} ] = (157/23,040)·pi^{5}
Demostración:
0 [< x [< 2pi·i = 0
( (pi/2)·i )^{5} = (1/32)·pi^{5}·i
(-1)·(1/pi·i)·( (1/(2k+1))^{3}·5·4·8pi^{3}·(-i)+(1/(2k+1))^{5}·5·4·3·2·2pi·i )
Trayectorias en superficies:
Definición:
[As][Ad][ ( s > 0 & d > 0 ) ==> ( Si |h(kx)+(-y)| < d ==> | M(x,h(kx))+(-1)·F(x,y) | < s ) ]
Teorema:
Si ( lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) & lim[y = h(kx)][ F(x,y) ] = N(x,h(kx)) ) ==> ...
... M(x,h(kx)) = N(x,h(kx))
Demostración:
Sea s > 0 ==>
| M(x,h(kx))+(-1)·N(x,h(kx)) | = | M(x,h(kx))+(-1)·F(x,y)+F(x,y)+(-1)·N(x,h(kx)) | [< ...
... | M(x,h(kx))+(-1)·F(x,y) |+| F(x,y)+(-1)·N(x,h(kx)) | < (s/2)+(s/2) = s
Teorema:
Si ( lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) & lim[y = h(kx)][ G(x,y) ] = N(x,h(kx)) ) ==> ...
... lim[y = h(kx)][ F(x,y)+G(x,y) ] = M(x,h(kx))+N(x,h(kx))
Teorema:
Si lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) ==> lim[y = h(kx)][ w·F(x,y) ] = w·M(x,h(kx))
Teorema:
Sea lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) ==> F(x,y) está acotada
Demostración:
h(kx) = y = c
x = (1/k)·h^{o(-1)}(c)
|F(x,y)| [< | F(x,y)+(-1)·M(x,h(kx)) |+|M(x,h(kx))|
Se define M = max{s,M(x,h(kx))} ==>
|F(x,y)| [< M
Teorema:
Sea lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) ==> M(x,h(kx)) está acotada
Demostración:
h(kx) = y = c
x = (1/k)·h^{o(-1)}(c)
|M(x,h(kx))| [< | M(x,h(kx))+(-1)·F(x,y) |+|F(x,y)|
Se define M = max{s,F(x,y)} ==>
|M(x,h(kx))| [< M
Teorema:
Si ( lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) & lim[y = h(kx)][ G(x,y) ] = N(x,h(kx)) ) ==> ...
... lim[y = h(kx)][ F(x,y)·G(x,y) ] = M(x,h(kx))·N(x,h(kx))
Demostración:
Sea s > 0 ==>
| M(x,h(kx))·N(x,h(kx))+(-1)·F(x,y)·G(x,y) | = ...
... | M(x,h(kx))·N(x,h(kx))+(-1)·F(x,y)·N(x,h(kx))+F(x,y)·N(x,h(kx))+(-1)·F(x,y)·G(x,y) | [< ...
... | M(x,h(kx))+(-1)·F(x,y) |·|N(x,h(kx))|+|F(x,y)|·| N(x,h(kx))+(-1)·G(x,y) | < (s/2)+(s/2) = s
Teorema:
lim[y = kx][ ( (x^{n+1}+y^{n+1})/(x+y) ) ] = x^{n}·k^{[n+1:1]+(-1)·[1:1]}
Demostración:
( (x^{n+1}+(kx)^{n+1})/(x+kx) ) = ( (x^{n+1}+k^{n+1}·x^{n+1})/(x+kx) ) = ...
... ( ((1+k^{n+1})·x^{n+1})/((1+k)·x) ) = ( (1+k^{n+1})/(1+k) )·x^{n}
Teorema:
lim[y = ln(kx)][ ( (x^{n+1}+e^{(n+1)·y})/(x+e^{y}) ) ] = x^{n}·k^{[n+1:1]+(-1)·[1:1]}
Teorema:
lim[y = e^{kx}][ ( ln(e^{nx}+y)/x ) ] = k+[n+(-k):1]
