Doctorats de stroniken

Guifré del Bergadà:

Doctorat en análisis matemátic.

Capítol I:

Teorema:

[Ek][ k >] 2 & cos((1/pi)·i) [< ( k/(k+(-1)) ) ]

[Ek][ k >] 2 & sin((1/pi)·i) [< ( i/(k+(-1)) ) ]

Desmostració:

Destrocter ponens:

[Ap][ (1/(2p)!)·(1/pi)^{2p} > 1 ]

Destrocter ponens:

[Ap][ (1/(2p+1)!)·(1/pi)^{2p+1} > (1/k) ]

Teorema:

[Ek][ k >] 2 & cosh(1/pi) [< ( k/(k+(-1)) ) ]

[Ek][ k >] 2 & sinh(1/pi) [< ( 1/(k+(-1)) ) ]

Teorema:

(1/3) [< e^{(-1)·(1/pi)} [< 1

1 [< e^{(1/pi)} [< 3

Teorema:

( sin(x) & cos(x) ) convergeish la serie a [0i,i]_{C}

( sinh(x) & cosh(x) ) convergeish la serie a [0,1]_{C}

Capítol II:

Teorema:

[Ek][ k >] 2 & cos((1/pi)·i) >] ( k/(k+1) ) ]

[Ek][ k >] 2 & sin((1/pi)·i) >] ( i/(k+1) ) ]

Desmostració:

Destrocter ponens:

[Ap][ (1/(2p)!)·(1/pi)^{2p} < (1/(2p)!)·(1/pi)^{2p}( (-1)·(1/k) )^{p} ]

[Ap][ (1/(2p)!)·(1/pi)^{2p}·( (-1)·(1/k) )^{p} < ( (-1)·(1/k) )^{p} ]

Destrocter ponens:

[Ap][ (1/(2p+1)!)·(1/pi)^{2p+1} < (1/(2p+1)!)·(1/pi)^{2p+1}·(1/k)·( (-1)·(1/k) )^{p} ]

[Ap][ (1/(2p+1)!)·(1/pi)^{2p+1}·(1/k)·( (-1)·(1/k) )^{p} < (1/k)·( (-1)·(1/k) )^{p} ]

Teorema:

[Ek][ k >] 2 & cosh(1/pi) >] ( k/(k+1) ) ]

[Ek][ k >] 2 & sinh(1/pi) >] ( 1/(k+1) ) ]

Teorema:

(1/3) [< e^{(-1)·(1/pi)} [< 1

1 [< e^{(1/pi)} [< 3

Teorema:

( sin(x) & cos(x) ) convergeish la serie a [0i,i]_{C}

( sinh(x) & cosh(x) ) convergeish la serie a [0,1]_{C}

Hugo de Portugal:

Doctorat en lógica algebraica.

[p(x)] = Binari concret.

]q(x)[ = Borrós semblant-abstracte.

Teorema:

min{[p(x)],]q(x)[} = [p(x)] <==> max{[p(x)],]q(x)[} = ]q(x)[

max{¬[p(x)],¬]q(x)[} = ¬[p(x)] <==> min{¬[p(x)],¬]q(x)[} = ¬]q(x)[

Teorema:

min{[f(1)],]f(n)[} = [f(1)] <==> max{[f(1)],]f(n)[} = ]f(n)[

max{[f(-1)],]f(-n)[} = [f(-1)] <==> min{[f(-1)],]f(-n)[} = ]f(-n)[

Don Casasayas de Euskal-Herria:

Doctorat en análisis matemátic.

Capítol I:

Teorema:

Si a [< b ==> [Au][Eq][En][ a [< q+(u/n) [< b ]

Demostració:

a [< ( (a+(-u))+(b+u) )/2 [< b

q = ( (a+(-u)+b)/2 ) & n = 2

Teorema:

Si a [< b ==> [Au][Av][Eq][En][Em][ a [< q+(u/n)+(v/m) [< b ]

Demostració:

a [< ( (5a+(-5)·u+(-2)·v)+(5b+5u+2v) )/10 [< b

q = ( (5a+(-5)·u+(-2)·v+5b)/10 ) & n = 2 & m = 5

Teorema:

[Ax][Ea_{n}][Eb_{n}][ a_{n} [< x [< b_{n} ...

... & [Eq][ b_{n}+(-1)·a_{n} = q ] & lim[a_{n}] = lim[b_{n}] = x ]

Demostració:

a_{n} = x+(-1)·(1/n)

b_{n} = x+(1/n)

q = (2/n)

Teorema:

[Ax][Ay][Ea_{n}][Eb_{n}][ x [< a_{n}+b_{n} [< y...

... & [Eq][ b_{n}+(-1)·a_{n} = q ] & lim[a_{n}] = lim[b_{n}] ]

Demostració:

a_{n} = ( (x+y)/4 )+(-1)·(1/n)

b_{n} = ( (x+y)/4 )+(1/n)

q = (2/n)

Capítol II:

Teorema:

max{x,y} = ( ( (x+y)+|y+(-x)| )/2 )

min{x,y} = ( ( (x+y)+|i(y+(-x))| )/2 )

Demostració:

x [< y <==> 0 [< y+(-x)

x >] y <==> 0 >] y+(-x)

Teorema:

x^{2} = ( ( x+|x| )/2 )^{2}+( ( x+(-1)·|x| )/2 )^{2}

x^{2} = ( ( x+|ix| )/2 )^{2}+( ( x+(-1)·|ix| )/2 )^{2}

Teorema:

|x_{1}+...+x_{n}| [< |x_{1}|+...+|x_{n}|

|i(x_{1}+...+x_{n})| >] |ix_{1}|+...+|ix_{n}|

Pla d'estudis de la universitat de Stroniken:

Nota de tall = 7.50

1r semestre:

Guifré del Bergadà:

Análisis matemátic I

Análisis matemátic III

Don Casasayas:

Álgebra

Probabilitats

Hugo de Portugal:

Lógica

Topología

Jûan Garriga:

Informática

Especies combinatories

2n semestre:

Guifré del Bergadà:

Análisis matemátic II

Análisis matemátic IV

Don Casasayas:

Ecuacions Diferencials

Análisis Complex y Borrós

Hugo de Portugal:

Teoria de conjunts

Lógica algebraica y Dualogía

Jûan Garriga:

Álgebra lineal y geometría lineal

Geometría Diferencial

Hidrogen verd:

Aigua:

4·H_{2}+O_{4} <==> 4·H_{2}O

[4·H_{2}]·[O_{4}] <==> [4e]·[4·H_{2}O]

Aigua oxigenada:

2·H_{2}+O_{6} <==> 2·H_{2}O_{3}

[2·H_{2}]·[O_{6}] <==> [2e]·[2·H_{2}O_{3}]

Oxigen:

2·O_{4}+O_{4} <==> 2·O_{6}

[2·O_{4}]·[O_{4}] <==> [2e]·[2·O_{6}]

Ley que no es del mundo:

Si no es tu dinero:

Si enseñas el DNI en el banco robas.

Si no enseñas el DNI en el banco no robas.

Si es tu dinero:

Si enseñas el DNI en el banco no te roban.

Si no enseñas el DNI en el banco te roban.

menjar [o] menjjar [o] menjjate [o] menjjet-kazhe

pujar [o] pujjar [o] pujjate [o] pujjet-kazhe

bajar [o] baishar [o] bashate  [o] bashet-kazhe

dejar [o] deishar [o] deshate  [o] deshet-kazhe

yo havere-po encontratered una miravilem demostraciorum,

apud en el marginis non sere-po capered la demostraciorum.

A + B = A [ || ] B & ¬A + ¬B = ¬A [&] ¬B

A = < {a_{1},...(n)...,a_{n}},{a_{1}} > & S[A] = (n+1)·x^{n}

¬A = < }a_{1},...(n)...,a_{n}{,}a_{1}{ > & S[A] = ((-n)+(-1))·x^{n}

A = {a_{1},...(n)...,a_{n}} [&] {a_{1},...,a_{k}} & S[A] = kx^{k}

¬A = }a_{1},...(n)...,a_{n}{ [&] }a_{1},...,a_{k}{ & S[A] = (-k)·x^{k}

A = }a_{1},...(n)...,a_{n}{ [ \ ] }a_{1},...,a_{k}{  & S[A] = ((-n)+k)·x^{k}

¬A = {a_{1},...(n)...,a_{n}} [ \ ] {a_{1},...,a_{k}}  & S[A] = (n+(-k))·x^{k}

Teoría de ingeniería y de economía:

Definició:

[ n // k ]+[ n // (k+1) ] = [ (n+1) // (k+1) ]

sum[k = 0]-[n][ [ n // k ] ] = 2^{n}

[ (-n) // k ]+[ (-n) // (k+1) ] = [ ((-n)+1) // (k+1) ]

sum[k = 0]-[n][ [ (-n) // k ] ] = 2^{(-n)}

Teorema:

[ (-2) // 0 ] = 1 & [ (-2) // 1 ] = (-2) & [ (-2) // 2 ] = (5/4)

[ (-3) // 0 ] = 1 & [ (-3) // 1 ] = (-3) & [ (-3) // 2 ] = (17/4) & [ (-3) // 3 ] = (-1)·(17/8)

Si k >] 2 ==> [ (-n) // k ] = (-1)^{k}·( F(n,k)/2^{k} )

Binomis:

Teorema:

(x+x)^{n} = sum[k = 0]-[n][ [ n // k ]·x^{n+(-k)}·x^{k} ]

Teorema:

(x+x)^{(-n)} = sum[k = 0]-[n][ [ (-n) // k ]·x^{(-n)+(-k)}·x^{k} ]

Teorema:

lim[h = 0][ (x+h)^{n} ] = ...

... lim[h = 0][ sum[k = 0]-[n][ [ n // k ]·x^{n+(-k)}·h^{k} ] ] = x^{n}

Teorema:

lim[h = 0][ (x+h)^{(-n)} ] = ...

... lim[h = 0][ sum[k = 0]-[n][ [ (-n) // k ]·x^{(-n)+(-k)}·h^{k} ] ] = x^{(-n)}

Demostració:

[ (-n) // k ]·x^{(-n)+(-k)}·h^{k}·(x+h) = ...

... [ (-n) // k ]·x^{(-n)+1+(-1)·(k+1)}·h^{k+1}+[ (-n) // p ]·x^{(-n)+1+(-p)}·h^{p}

Distribucions:

f(k) = [ n // k ]·2^{(-n)}

g(k) = [ (-n) // k ]·2^{n}

E[k·f(k)] = (n/2)

E[k·g(k)] = (-n)+sum[k = 2]-[n][ (-1)^{k}·k·( F(n,k)/2^{k} ) ]

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

Derivada:

d_{x}[x^{n}] = lim[h = 0][ ( ( (x+h)^{n}+(-1)·x^{n} )/h ) ] = nx^{n+(-1)}

Integral:

int[x^{n}]d[x] = ( 1/(n+1) )·lim[h = 0][ int[ (x+h)^{n}·(x+h)+(-1)·x^{n}·x ] ] = ...

... ( 1/(n+1) )·int[ d[x^{n+1}] ] = ( 1/(n+1) )·x^{n+1}

int[x^{n}]d[x] = ( 1/(n+1) )·int[ (n+1)·x^{n} ]d[x]

Derivada:

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

Integral:

int[x^{(-n)}]d[x] = ( 1/((-n)+1) )·lim[h = 0][ int[ (x+h)^{(-n)}·(x+h)+(-1)·x^{(-n)}·x ] ] = ...

... ( 1/((-n)+1) )·int[ d[x^{(-n)+1}] ] = ( 1/((-n)+1) )·x^{(-n)+1}

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

Derivada:

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

Integral:

int[e^{x}]d[x] = lim[h = 0][ int[ e^{x+h}·(x+h)+(-1)·e^{x}·x ] ] = ...

... lim[h = 0][ int[ e^{x}·(x+h)+(-1)·e^{x}·x ] ] = ...

... int[ e^{x} ]d[x] = int[ d_{x}[e^{x}] ]d[x] = e^{x}

Derivada:

d_{x}[ln(x)] = lim[h = 0][ ( ( ln(x+h)+(-1)·ln(x) )/h ) ] = (1/x)

Integral:

int[ (1/x) ]d[x] = lim[h = 0][ int[ ( 1/(x+h) )·(x+h)+(-1)·(1/x)·x ] ] = ...

... lim[h = 0][ int[ (1/x)·(x+h)+(-1)·(1/x)·x ] ] = ...

... int[ (1/x) ]d[x] = int[ d_{x}[ln(x)] ]d[x] = ln(x)