x [< y+x <==> 0 [< y
x >] y+x <==> 0 >] y
x < y+x <==> 0 < y
x > y+x <==> 0 > y
(n+1)x [< y+x <==> nx [< y
(n+1)x >] y+x <==> nx >] y
(n+1)x < y+x <==> nx < y
(n+1)x > y+x <==> nx > y
x+y [< f(x)+y <==> x [< f(x)
x+y >] f(x)+y <==> x >] f(x)
x+y < f(x)+y <==> x < f(x)
x+y > f(x)+y <==> x > f(x)
c^{log_{c^{n}}(c)} = c^{(1/n)}
log_{c^{n}}(c) = log_{c}(c^{(1/n)}) = (1/n)
c = (c^{n})^{(1/n)}
n[+]m = ( (n+m)/(1+1) ) = ((n+m)/2)
n[+]0 = (n/2)
x^{n}[·]x^{m} = x^{((n+m)/2)}
x^{n}[·]1 = x^{(n/2)}
m[+]...(n)...[+]m = m·( 1[+]...(n)...[+]1 ) = m
x^{m}[·]...(n)...[·]x^{m} = x^{m·( 1[+]...(n)...[+]1 )} = x^{m}
a[·]b = x^{log_{x}(a)}[·]x^{log_{x}(b)} = x^{( (log_{x}(ab))/2 )} = (ab)^{(1/2)}
a[·]( p+q ) = x^{log_{x}(a)}[·]x^{log_{x}( p+q )} = ( ap+aq )^{(1/2)}
a[·]( p[+] q) = x^{log_{x}(a)}[·]x^{log_{x}( p[+]q )} = ( ap[+]aq )^{(1/2)}
( ca·x^{n} )^{(1/2)}+( cb·x^{m} )^{(1/2)} = c
(ca)[·]x^{n}+(cb)[·]x^{m} = c
x = c^{( 1/( ( ( log_{c}(a) )+n ) [[+]] ( ( log_{c}(b) )+m ) )}
c^{( ( 1+log_{c}(a) )[+]( n/( ( ( log_{c}(a) )+n ) [[+]] ( ( log_{c}(b) )+m ) ) ) ) )}
c^{( ( 1+log_{c}(b) )[+]( m/( ( ( log_{c}(a) )+n ) [[+]] ( ( log_{c}(b) )+m ) ) ) ) )}
( 2x^{3} )^{(1/2)}+( 4x^{2} )^{(1/2)} = 2
2[·]x^{3}+4[·]x^{2} = 2
x = 2^{( 1/( ( 4 ) [[+]] ( 4 ) ) )}
( 3x^{7} )^{(1/2)}+( 9x^{6} )^{(1/2)} = 3
3[·]x^{7}+9[·]x^{6} = 3
x = 3^{( 1/( ( 8 ) [[+]] ( 8 ) ) )}
[[k]]( f(x) )^{(1/k)} = f(x)
[[1]]( f(x) ) = f(x)
s·[[k]]( f(x) ) = [[k]]( s^{k}·f(x) )
[[k+1]]( f(x) ) = [[k]]( d_{x}[f(x)] )
d_{x}[ [[k+1]]( f(x) ) ] = [[k]]( d_{x}[f(x)] )
int[ [[k]]( d_{x}[f(x)] ) ] d[x] = [[k+1]]( f(x) )
(1/k!)·(m/c^{k+(-2)})·d_{t}[x(t)]^{k} = E_{n}+(a_{k}/k!)·( x(t) )^{k}
x(t) = [[(1/k)]]( e^{(a_{k}/m)·c^{k+(-2)}·t}+(-1)·(k!/a_{k})·E_{n} )
d_{t}[x(t)] = [[(1/k)+(-1)]]( (a_{k}/m)·c^{k+(-2)}e^{(a_{k}/m)·c^{k+(-2)}·t} )
[[1+(-k)]]( ( (a_{k}/m)·c^{k+(-2)} )^{k}e^{(a_{k}/m)·c^{k+(-2)}·t} ) = ...
... e^{(a_{k}/m)·c^{k+(-2)}·t}
(m/p!)·d_{t}[x(t)]^{p} = E_{n}+a_{q}·(1/q!)·( x(t) )^{q}
x(t) = [[(1/q)]]( e^{(a_{q}/m)·(p!/q!)·t}+(-1)·(q!/a_{q})·E_{n} )
d_{t}[x(t)] = [[(1/q)+(-1)]]( (a_{q}/m)·(p!/q!)·e^{(a_{q}/m)·(p!/q!)·t} )
[[(p/q)+(-p)]]( ( (a_{q}/m)·(p!/q!) )^{p}( (a_{q}/m)·(p!/q!) )^{1+(-1)·(p/q)}·...
... e^{(a_{q}/m)·(p!/q!)·t} ) = [[(0/0)+(-0)]]( e^{(a_{q}/m)·(p!/q!)·t} ) = ...
... [[(0/q)+(-0)]]( (a_{q}/m)·(p!/q!)·e^{(a_{q}/m)·(p!/q!)·t} ) = e^{(a_{q}/m)·(p!/q!)·t}
d_{x}[ [k+1]( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) ) ] = ...
... [k]( d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)] ) [o( (1/k!)x^{k} )o] ...
... d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)]
int[ [k]( d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)] ) ] d[x] = ...
... [k+1]( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) ) [o( (1/(k+1)!)x^{k+1} )o] ...
... ( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) )^{[o(x)o](-1)}
[k]( d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)] ) = [k+1]( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) )
Lagranià cuàntic de primera especie:
( h^{2}/(2ml) )·d_{x}[f(x)] = ( E_{n}+qA(x) )·f(x)
f(x) = [1]( ( (2ml)/h^{2} )·x [o(x)o] int[ ( E_{n}+qA(x) ) ] d[x] )
f(x) = [0]( ( (2ml)/h^{2} )·( E_{n}+qA(x) ) )
Lagranià cuàntic de segona especie:
( h^{2}/(2m) )·d_{xx}^{2}[f(x)] = ( E_{n}+qA(x) )·f(x)
f(x) = [2]( ( (2m)/h^{2} )^{(1/2)}·(1/2!)·x^{2} [o( (1/2!)·x^{2} )o] ...
... int[ int[ ( E_{n}+qA(x) )^{(1/2)} ] d[x] ] d[x] )
f(x) = [0]( ( (2m)/h^{2} )^{(1/2)}·( E_{n}+qA(x) )^{(1/2)} )
Lagranià cuàntic de hyper-espai:
( (l^{n}h^{2})/(2m) )·d_{x...x}^{n+2}[f(x)] = ( E_{n}+qA(x) )·f(x)
f(x) = [n+2]( ( (2m)/(l^{n}h^{2}) )^{(1/(n+2))}·(1/(n+2)!)·x^{n+2} [o( (1/(n+2)!)·x^{n+2} )o] ...
... int[ ...(n+2)... int[ ( E_{n}+qA(x) )^{(1/(n+2))} ] d[x] ...(n+2)... ] d[x] )
f(x) = [0]( ( (2m)/(l^{n}h^{2}) )^{(1/(n+2))}·( E_{n}+qA(x) )^{(1/(n+2))} )
Hamiltonià cuàntic:
ih·d_{t}[f(x,t)] = ( E_{n}+qA(x) )·f(x,t)
ih·d_{x}[f(x,t)]·d_{t}[x] = ( E_{n}+qA(x) )·f(x,t)
f(x,t) = [1]( ((-i)/h)·x [o(x)o] int[ (1/d_{t}[x]) ] d[x] [o(x)o] int[ ( E_{n}+qA(x) ) ] d[x] )
f(x,t) = [0]( ((-i)/h)·(1/d_{t}[x])·( E_{n}+qA(x) ) )
Hamiltonià cuàntic de camp constant:
ih·d_{t}[f(x,t)] = ( E_{n}+qgx )·f(x,t)
ih·d_{x}[f(x,t)]·d_{t}[x] = ( E_{n}+qgx )·f(x,t)
f(x,t) = [1]( ((-i)/h)·x [o(x)o] t [o(x)o] ( E_{n}x+qg·(1/2)·x^{2} ) )
f(x,t) = [0]( ((-i)/h)·( m/(qgt) )·( E_{n}+qgx ) )
(m/2)·d_{t}[x(t)]^{2} = E_{n}+qg·x(t)
x(t) = ( (qg)/m )·(1/2)·t^{2}+(-1)·( E_{n}/(qg) )
d_{t}[x(t)] = ( (qg)/m )·t
[[1]]( f(x) ) = f(x)
[[k]]( f(x) )^{(1/k)} = f(x)
[[k]]( s^{k}·f(x) ) = s·[[k]]( f(x) )
d_{x}[ [[k+1]]( f(x) ) ] = [[k]]( d_{x}[f(x)] )
int[ [[k]]( d_{x}[f(x)] ) ] d[x] = [[k+1]]( f(x) )
[[k+1]]( f(x) ) = [[k]]( d_{x}[f(x)] )
(m/2)·d_{t}[x(t)]^{2} = E_{n}+(a/2)·( x(t) )^{2}
x(t) = [[(1/2)]]( e^{(a/m)t}+(-1)·(2/a)·E_{n} )
d_{t}[x(t)] = [[(-1)(1/2)]]( (a/m)·e^{(a/m)t} )
[[(-1)]]( (a/m)^{2}·e^{(a/m)t} ) = e^{(a/m)t}
Hamiltonià cuàntic de oscilador harmónic:
ih·d_{t}[f(x,t)] = ( E_{n}+(a/2)·x^{2} )·f(x,t)
ih·d_{x}[f(x,t)]·d_{t}[x] = ( E_{n}+(a/2)·x^{2} )·f(x,t)
f(x,t) = [1]( ((-i)/h)·x [o(x)o] t [o(x)o] ( E_{n}x+(a/6)·x^{3} ) )
f(x,t) = [0]( ((-i)/h)·( [[(1/2)]]( (a/m)·e^{(a/m)t} ) )·( E_{n}+(a/2)·x^{2} ) )
Si m·d_{tt}^{2}[x(t)] = f(t) ==> (1) & (2) & (3)
(1): d_{tt}^{2}[x(t)] = (1/m)·f(t)
(2): d_{t}[x(t)] = (1/m)·int[ f(t) ] d[t]
(3): x(t) = (1/m)·int[ int[ f(t) ] d[t] ] d[t]
Si ( m·d_{tt}^{2}[x(t)] = a·( x(t) )^{n} <==> (m/2)·d_{t}[x(t)]^{2}= ( a/(n+1) )( x(t) )^{n+1} ) ==> ...
... (1) & (2) & (3)
Demostració: ( k+(-2) = kn || 2k+(-2) = k·(n+1) )
(1): x(t) = ...
... ( ( (1+(-n))/(n+1) )^{(1/2)}·( (1+(-n))/2 )^{(1/2)}·(a/m)^{(1/2)}·t )^{2/(1+(-n))}
(2): d_{t}[x(t)] = ...
... ( (1+(-n))/(n+1) )^{(1/2)}·( 2/(1+(-n)) )^{(1/2)}·(a/m)^{(1/2)}·...
... ( ( (1+(-n))/(n+1) )^{(1/2)}·( (1+(-n))/2 )^{(1/2)}·(a/m)^{(1/2)}·t )^{(n+1)/(1+(-n))}
(3): d_{tt}^{2}[x(t)] = ...
... (a/m)·( ( (1+(-n))/(n+1) )^{(1/2)}·( (1+(-n))/2 )^{(1/2)}·(a/m)^{(1/2)}·t )^{(2n)/(1+(-n))}
m·d_{t}[x(t)] = a·( x(t) )^{(2/3)}
d_{tt}^{2}[x(t)] = (a/m)·(2/3)·( x(t) )^{(-1)·(1/3)}·d_{t}[x(t)]
d_{tt}^{2}[x(t)]^{2} = (a/m)^{3}·(4/9)·d_{t}[x(t)]
d_{tt}^{2}[x(t)]·d_{t}[x(t)]^{(-1)·(1/2)} = (a/m)^{(3/2)}·(2/3)
d_{t}[x(t)]^{(1/2)} = (1/3)·(a/m)^{(3/2)}·t
d_{t}[x(t)] = (1/9)·(a/m)^{3}·t^{2}
x(t) = (1/27)·(a/m)^{3}·t^{3}
d_{tt}^{2}[x(t)] = (2/9)·(a/m)^{3}·t
(1/2)·d_{t}[x(t)]^{2} = (1/2)·(1/81)·(a/m)^{6}·t^{4}
él <==> ell
ella <==> ell-na
ellos <==> ells
ellas <==> ell-nas
aquel <==> aquell
aquella <==> aquell-na
aquellos <==> aquells
aquellas <==> aquell-nas
íshtep <==> aquíshtep
íshtep-na <==> aquíshtep-na
íshtep-nos <==> aquíshteps
íshtep-nas <==> aquíshtep-nas
íshep <==> aquíshep
íshep-na <==> aquíshep-na
íshep-nos <==> aquísheps
íshep-nas <==> aquíshep-nas
un <==> un
una <==> una
unos <==> uns
unas <==> unas
lo <==> el
la <==> la
los <==> els
las <==> las
lo que <==> el que
la que <==> la que
los que <==> els que
las que <==> las que
íshtep coche,
està mal aparcado.
íshep coche,
està bien aparcado.
íshtep-na casa
vale mucho dinero.
íshep-na casa
vale poco dinero.
