stroniken: màxims y mínims, dualogía

miércoles, 27 de octubre de 2021

teorema:

max{n: f(n) = n+p } = max{n: f(n) = n }+p

min{n: f(n) = n+p } = min{n: f(n) = n }+p

teorema:

sup{n: f(n) = n+p } = sup{n: f(n) = n }+p

inf{n: f(n) = n+p } = inf{n: f(n) = n }+p

demostració per absurd:

max{n: f(n) = n+p } != max{n: f(n) = n }+p

n [< max{n: f(n) = n }

n+p [< max{n: f(n) = n }+p = a < max{n: f(n) = n+p }

n+p [< a < max{n: f(n) = n+p }

n+p [< max{n: f(n) = n+p } = b+p < max{n: f(n) = n }+p

n [< b < max{n: f(n) = n }

Dualogía paralela a una funció:

f(x+cos(s)·h)+f(x+(-1)·cos(s)·h) = F(x)

g(a)+y(a) = 0

( x+cos(s)·h )+( x+(-1)·cos(s)·h ) = 2x

g(0) = (cos(s)·h)

y(0) = (-1)·(cos(s)·h)

( x+cos(s)·h )^{2}+( x+(-1)·cos(s)·h )^{2} = 2·(x+i·cos(s)·h)·(x+(-i)·cos(s)·h)

g(i·cos(s)·h) = 2i·(cos(s)·h)

y(i·cos(s)·h) = (-2)·i·(cos(s)·h)

( x+cos(s)·h )^{3}+( x+(-1)·cos(s)·h )^{3} = 2·x·(x^{2}+3(cos(s)·h))

g(i·3^{(1/2)}·(cos(s)·h)^{(1/2)}) = (-8)·(cos(s)·h)^{3}

y(i·3^{(1/2)}·(cos(s)·h)^{(1/2)}) = 8·(cos(s)·h)^{3}

( x+cos(s)·h )^{4}+( x+(-1)·cos(s)·h )^{4} = ...

... 2·( ( x^{2}+(cos(s)·h)^{2} )^{2}+4x^{2}·(cos(s)h)^{2} )

x^{2}+2i·x·(cos(s)·h)+(cos(s)·h)^{2} = 0

x = ((-1)+2^{(1/2)})·i·(cos(s)·h)

( (-1)+2^{(1/2)} )^{4}·i^{4} = 1+(-4)·2^{(1/2)}+6·2+(-4)·2^{(3/2)}+4

6·( (-1)+2^{(1/2)} )^{2}·i^{2} = 6·( (-1)+2·2^{(1/2)}+(-2) )

g( ((-1)+2^{(1/2)})·i·(cos(s)·h) ) = ...

... ((-1)+2^{(1/2)})·i·(cos(s)·h)^{4}+(-1)·((-1)+2^{(1/2)})^{3}·i·(cos(s)·h)^{4}

y( ((-1)+2^{(1/2)})·i·(cos(s)·h) ) = ...

... (-1)·((-1)+2^{(1/2)})·i·(cos(s)·h)^{4}+((-1)+2^{(1/2)})^{3}·i·(cos(s)·h)^{4}

En símbol de polinómic potencial:

( x+cos(s)·h )^{7}+( x+(-1)·cos(s)·h )^{7} = ...

... 2·x·( x^{6}+21x^{4}(cos(s)·h)^{2}+35x^{2}(cos(s)·h)^{4}+7·(cos(s)·h)^{6} )

(-7)·( cos(s)·h )^{6} = ...

x^{4+[...( 21·(cos(s)·h)^{2} )...[2]...( 21·(cos(s)·h)^{2} )...]}+35x^{2}(cos(s)·h)^{4} = ...

x^{2+[...( 35·(cos(s)·h)^{4} )...[...