topologia

E_{m}[W]...(n+(-m))...[W]E_{n}=E_{n}
E_{m}[M]...(n+(-m))...[M]E_{n}=E_{m}
f( E_{i},Ps[E_{i}] )=E_{i+1}




E_{j}=p^{j}


mcm{p^{m},...(n+(-m))...,p^{n}}=p^{max{m,...(n+(-m))...,n}}
mcd{p^{m},...(n+(-m))...,p^{n}}=p^{min{m,...(n+(-m))...,n}}
mcm{p^{m},...(n+(-m))...,p^{n}}=p^{n}
mcd{p^{m},...(n+(-m))...,p^{n}}=p^{m}


Ps[E_{j}] = (p^{j+1}/p^{j})=p




E_{j}=kp^{j}


mcm{kp^{m},...(n+(-m))...,kp^{n}}=kp^{max{m,...(n+(-m))...,n}}
mcd{kp^{m},...(n+(-m))...,kp^{n}}=kp^{min{m,...(n+(-m))...,n}}
mcm{kp^{m},...(n+(-m))...,kp^{n}}=kp^{n}
mcd{kp^{m},...(n+(-m))...,kp^{n}}=kp^{m}


Ps[E_{j}] = (kp^{j+1}/kp^{j})=p




{15,30,60,120}={15·2^{0},15·2^{1},15·2^{2},15·2^{3}}




E_{j}=j


(1+...(m)...+1)[W]...(n+(-m))...[W](1+...(n)...+1)=(1+...(n)...+1)
(1+...(m)...+1)[M]...(n+(-m))...[M](1+...(n)...+1)=(1+...(m)...+1)
m[W]...(n+(-m))...[W]n=n
m[M]...(n+(-m))...[M]n=m


Ps[E_{j}] = (j+1)+(-1)j=1=(j+1)^{0}


E_{j}=qj


(q+...(m)...+q)[W]...(n+(-m))...[W](q+...(n)...+q)=(q+...(n)...+q)
(q+...(m)...+q)[M]...(n+(-m))...[M](q+...(n)...+q)=(q+...(m)...+q)
qm[W]...(n+(-m))...[W]qn=qn
qm[M]...(n+(-m))...[M]qn=qm


Ps[E_{j}] = q(j+1)+(-1)qj=q=q(j+1)^{0}






E_{j} = ( j(j+1)/2 )


(1+...(m)...+m)[W]...(n+(-m))...[W](1+...(n)...+n) = (1+...(n)...+n)
(1+...(m)...+m)[M]...(n+(-m))...[M](1+...(n)...+n) = (1+...(m)...+m)
( m(m+1)/2 )[W]...(n+(-m))...[W]( n(n+1)/2 ) = ( n(n+1)/2 )
( m(m+1)/2 )[M]...(n+(-m))...[M]( n(n+1)/2 ) = ( m(m+1)/2 )


Ps[E_{j}] = ( (j+1)(j+2)/2 )+(-1)(j(j+1)/2)=j+1


E_{j} = p!( B_{p}(j+1)+(-1)B_{p}(1) )


(1^{p}+...(m)...+m^{p})[W]...(n+(-m))...[W](1^{p}+...(n)...+n^{p}) = (1^{p}+...(n)...+n^{p})
(1^{p}+...(m)...+m^{p})[M]...(n+(-m))...[M](1^{p}+...(n)...+n^{p}) = (1^{p}+...(m)...+m^{p})
( p!( B_{p}(m+1)+(-1)B_{p}(1) ) )[W]...(n+(-m))...[W]( p!( B_{p}(n+1)+(-1)B_{p}(1) ) ) =...
... ( p!( B_{p}(n+1)+(-1)B_{p}(1) ) )
( p!( B_{p}(m+1)+(-1)B_{p}(1) ) )[M]...(n+(-m))...[M]( p!( B_{p}(n+1)+(-1)B_{p}(1) ) ) = ...
...( p!( B_{p}(m+1)+(-1)B_{p}(1) ) )


Ps[E_{j}] = p!( B_{p}((j+1)+1)+(-1)B_{p}(1) )+(-1)p!( B_{p}(j+1)+(-1)B_{p}(1) )=(j+1)^{p}




E_{j}=j!


(1·...(m)...·m)[W]...(n+(-m))...[W](1·...(n)...·n)=(1·...(n)...·n)
(1·...(m)...·m)[M]...(n+(-m))...[M](1·...(n)...·n)=(1·...(m)...·m)
m![W]...(n+(-m))...[W]n!=n!
m![M]...(n+(-m))...[M]n!=m!


Ps[E_{j}] = ( (j+1)!/j! )=j+1


E_{j}=a^{j}!


(1·a·...(m)...·a^{m})[W]...(n+(-m))...[W](1·a·...(n)...·a^{n})=(1·a·...(n)...·a^{n})
(1·a·...(m)...·a^{m})[M]...(n+(-m))...[M](1·a·...(n)...·a^{n})=(1·a·...(m)...·a^{m})
a^{m}![W]...(n+(-m))...[W]a^{n}!=a^{n}!
a^{m}![M]...(n+(-m))...[M]a^{n}!=a^{m}!


Ps[E_{j}] = (a^{j+1}!/a^{j}!)=a^{j+1}


E_{j} = {a_{1},...,a_{j}}


{a_{1},...,a_{m}}[W]...(n+(-m))...[W]{a_{1},...,a_{n}}={a_{1},...,a_{n}}
{a_{1},...,a_{m}}[M]...(n+(-m))...[M]{a_{1},...,a_{n}}={a_{1},...,a_{m}}
E_{m}[W]...(n+(-m))...[W]E_{n}=E_{n}
E_{m}[M]...(n+(-m))...[M]E_{n}=E_{m}




Ps[E_{j}] = {a_{1},...,a_{j+1}} --- {a_{1},...,a_{j}}={a_{j+1}}


Ad[ {x€R : d_{j}(x,c) < s_{j}} ] = {x€R : d_{j}(x,c) [< s_{j}} & s_{j+1} > s_{j} >] 0
Int[ {x€R : d_{j}(x,c) [< s_{j}} ] = {x€R : d_{j}(x,c) < s_{j}} & s_{j+1} > s_{j} >] 0
Fr[ {x€R : d_{j}(x,c) [< s_{j}} , {x€R : d_{j}(x,c) < s_{j}} ]={s_{j}}


{x€R : d_{m}(x,c) [< s_{m}}[W]...(n+(-m))...[W]{x€R : d_{n}(x,c) [< s_{n}} = {x€R : d_{n}(x,c) [< s_{n}}
{x€R : d_{m}(x,c) [< s_{m}}[M]...(n+(-m))...[M]{x€R : d_{n}(x,c) [< s_{n}} = {x€R : d_{n}(x,c) [< s_{m}}
{x€R : d_{m}(x,c) [< s_{m}} [<< ...(n+(-m))... [<< {x€R : d_{n}(x,c) [< s_{n}}


{x€R : d_{m}(x,c) < s_{m}}[W]...(n+(-m))...[W]{x€R : d_{n}(x,c) < s_{n}} = {x€R : d_{n}(x,c) < s_{n}}
{x€R : d_{m}(x,c) < s_{m}}[M]...(n+(-m))...[M]{x€R : d_{n}(x,c) < s_{n}} = {x€R : d_{m}(x,c) < s_{m}}
{x€R : d_{m}(x,c) < s_{m}} [<< ...(n+(-m))... [<< {x€R : d(x,c) < s_{n}}


Ps[ {x€R : d_{j}(x,c) [< s_{j}} ]={ x€R : d_{j+1}(x,c)+(-1)d_{j}(x,c) [< s_{j+1}+(-1)s_{j} }
Ps[ {x€R : d_{j}(x,c) < s_{j}} ]={ x€R : d_{j+1}(x,c)+(-1)d_{j}(x,c) < s_{j+1}+(-1)s_{j} }


Ad[ ((-1)a_{j},a_{j})_{R} ]=[(-1)a_{j},a_{j}]_{R} & a_{j} < a_{j+1}
Int[ [(-1)a_{j},a_{j}]_{R} ]=((-1)a_{j},a_{j})_{R} & a_{j} < a_{j+1}
Fr[ [(-1)a_{j},a_{j}]_{R} , ((-1)a_{j},a_{j})_{R} ]={ (-1)a_{j},a_{j} }


Ps[ [(-1)a_{j},a_{j}]_{R} ]={ x€R : (-1)a_{j+1} [< x [< (-1)a_{j} or a_{j} [< x [< a_{j+1} }
Ps[ ((-1)a_{j},a_{j})_{R} ]={ x€R : (-1)a_{j+1} < x < (-1)a_{j} or a_{j} < x < a_{j+1} }

mínim comú múltiple

ab=mcm{a,b}·mcd{a,b}
(mcd{a,b}·p)·(mcd{a,b}·q)=mcm{a,b}·mcd{a,b}


mcm{a,b+c}·mcd{a,b+c}=mcm{a,b}·mcd{a,b}+mcm{a,c}·mcd{a,c}


mcm{a,a+b}·mcd{a,a+b}=a^{2}+mcm{a,b}·mcd{a,b}
mcm{a,a+b}=(a^{2}/mcd{a,b})+mcm{a,b}


mcm{3,6}=mcm{3,3+3}=(9/3)+3=3+3=6
mcm{3,7}=mcm{3,3+4}=9+12=21
mcm{3,8}=mcm{3,3+5}=9+15=24
mcm{3,9}=mcm{3,3+6}=(9/3)+6=3+6=9


mcm{5,10}=mcm{5,5+5}=(25/5)+5=5+5=10
mcm{10,15}=mcm{10,10+5}=(100/5)+10=20+10=30
mcm{2k,2k+1}=4k^{2}+2k=2k(2k+1)
mcm{2k,3k}=mcm{2k,2k+k}=(4k^{2}/k)+2k=4k+2k=6k
mcm{nk,(n+1)k}=mcm{nk,nk+k}=(n^{2}k^{2}/k)+nk=n^{2}k+nk=(n^{2}+n)k=n(n+1)k


(na)·(nb)=mcm{na,nb}·mcd{na,nb}
n^{2}·(ab)=mcm{na,nb}·mcd{na,nb}
n^{2}·mcm{a,b}·mcd{a,b}=mcm{na,nb}·(n·mcd{a,b})
n·mcm{a,b}=mcm{na,nb}


mcm{3,12}=mcm{3,3·4}=3·mcm{1,4}=3·4=12


n=mcm{n,1}·mcd{n,1}
n=mcm{n,1}


mcm{a,a+1}=a^{2}+a=a(a+1)

màxim comú divisor

a=mcd{a,b}·p
b=mcd{a,b}·q
a+b=mcd{a,b}·(q+p)


a=mcd{a,a+b}·k
a+b=mcd{a,a+b}·s
b=mcd{a,a+b}·(s+(-k))


mcd{a,a+b}·(s+(-k))=mcd{a,b}·q
mcd{a,a+b}·s=mcd{a,b}·q+mcd{a,a+b}·k
mcd{a,b}·(q+p)=mcd{a,b}·q+mcd{a,a+b}·k
mcd{a,b}·p=mcd{a,a+b}·k & mcd{p,k}=1
mcd{a,b}=mcd{a,a+b}·(k/p) & ( p=(mk) or  k=(np) )
mcd{a,b}=mcd{a,a+b}·(p/p) & ( (1/m)=(k/p) or  (1/n)=(p/k) ) ==> (n=1  & m=1)


mcd{a,b}=mcd{a,a+b}


mcd{a,a+1}=mcd{a,1}=1








mcd{8,24}=mcd{8,8+16}=mcd{8,16}=mcd{8,8+8}=mcd{8,8}=8
mcd{10,15}=mcd{10,10+5}=mcd{10,5}=mcd{5+5,5}=mcd{5,5}=5


mcd{4k+1,14k+3}=mcd{4k+1,4k+1+10k+2}=mcd{4k+1,10k+2}=...
...mcd{4k+1,6k+1}=mcd{4k+1,2k}=mcd{2k+2k+1,2k}=mcd{2k+1,2k}=1




a=mcd{a,b}·p
b=mcd{a,b}·q
na=mcd{a,b}·np
nb=mcd{a,b}·nq
na=mcd{na,nb}·p
nb=mcd{na,nb}·q
n(a+b)=mcd{na,nb}·(p+q)
n·mcd{a,b}·(p+q)=mcd{na,nb}·(p+q)
n·mcd{a,b}=mcd{na,nb}


descens inductiu:
mcd{(n+1)a,(n+1)b}=mcd{na,nb}+mcd{a,b}
mcd{(n+1)a,(n+1)b}=n·mcd{a,b}+mcd{a,b}
mcd{(n+1)a,(n+1)b}=(n+1)·mcd{a,b}


mcd{3k,6k}=mcd{(2+1)k,(2+1)2k}=mcd{2k,4k}+mcd{k,2k}=2k+k=3k






a=mcd{a,na}·p
na=mcd{a,na}·np
(n+1)a=mcd{a,na}·(n+1)p
(n+1)a=mcd{a,a+na}·(n+1)p
(n+1)a=mcd{a,(n+1)a}·(n+1)p


mcd{6k+1,12k+2}=mcd{6k+1,2·(6k+1)}=6k+1






mcd{a,a^{n}}=a


n+1=2k+1 & n^{2}+1=4k^{2}+1
mcd{2k+1,2k(2k)+1}=1 & k=(1/2)
n+1=2k+2 & n^{2}+1=4k^{2}+4k+2
mcd{2k+2,4k^{2}+4k+2}=2·mcd{k+1,2k^{2}+k+k+1}=...
...2·mcd{k+1,k(2k+1)}=2 & 2k+1=m(1+(1/k)) & k=1



mcd{a,b}=1 & d=a+(-b)
nd=an+n(-b)
(n+1)d=a(n+1)+(n+1)(-b)
mcd{an,nb}=n


6=12+(-6)
6=2·6+6·(-1)
4=16+(-12)
4=4·4+4·(-3)
10=20+(-10)
10=2·10+10·(-1)
24=36+(-12)
12·2=3·12+12·(-1)
12=18+(-6)
6·2=3·6+6·(-1)
12=72+(-60)
12=6·12+12·(-5)
54=72+(-18)
18·3=4·18+18·(-1)
105=120+(-15)
15·7=8·15+15·(-1)




1=3·5+2·(-7)


mcd{2k,3k+1}=mcd{2k,k+1}
si k=2p+1 ==> mcd{2k,k+1}=mcd{4p+2,2p+2}=2·mcd{2p+1,p+1}=2·mcd{p,p+1}=2
si k=2p ==> mcd{2k,k+1}=mcd{4p,2p+1}=1 & 2p+1=4q & p=2q+(-1)(1/2)




a^{2}=(mcd{a,b}·p)^{2}
b^{2}=(mcd{a,b}·q)^{2}
2ab=(mcd{a,b})^{2}·(2pq)
mcd{a^{2},2ab,b^{2}}=(mcd{a,b})^{2}


a^{n} = (mcd{a,b}·p)^{n}
b^{n} = (mcd{a,b}·q)^{n}
[ n // k ]a^{n+(-k)}b^{k} = (mcd{a,b})^{n}·( [ n // k ]p^{n+(-k)}q^{k} )
[ n // n+(-k) ]a^{k}b^{n+(-k)} = (mcd{a,b})^{n}·( [ n // n+(-k) ]p^{k}q^{n+(-k)} )
mcd{a^{n},...,[ n // k ]a^{n+(-k)}b^{k},...,[ n // n+(-k) ]a^{k}b^{n+(-k)},...,b^{n}}=(mcd{a,b})^{n}





ax+by=au_{n}+bv_{n}
ax+by=a(x+(-1)nb)+b(y+na)
ax+by=a(x+(-1)nb)+b(y+na)+(-1)ab+ab
ax+by=a(x+(-1)(n+1)b)+b(y+(n+1)a)
ax+by=au_{n+1}+bv_{n+1}

funcions inverses hiperboliques

d_{x}[arcsinh[n](x)]=(n^{n+1}+x^{(n+1)})^{(-1)(n+1)}
d_{x}[arcsinh[n](x)]=(1/n)·(1+(x/n)^{(n+1)})^{(-1)(n+1)}
d_{x}[arcsinh[n](sinh[n](x))]=cosh[n](x)·(1/n)·(1+(sinh[n](x)/n)^{(n+1)})^{(-1)(n+1)}=1


d_{x}[arccosh[n](x)]=((-1)n^{n+1}+x^{(n+1)})^{(-1)(n+1)}
d_{x}[arccosh[n](x)]=(1/n)·((-1)+(x/n)^{(n+1)})^{(-1)(n+1)}
d_{x}[arccosh[n](cosh[n](x))]=sinh[n](x)·(1/n)·((-1)+(cosh[n](x)/n)^{(n+1)})^{(-1)(n+1)}=1


int[( sinh[n](x) )^{m}]d[x] =( 1/(m+1) )( sinh[n](x) )^{m+1} [o(x)o] ( sinh[n](x) )^{[o(x)o](-1)}
int[( cosh[n](x) )^{m}]d[x] =( 1/(m+1) )( cosh[n](x) )^{m+1} [o(x)o] ( cosh[n](x) )^{[o(x)o](-1)}

constructor dual-falacis

es defineish el constructor dual falacis:
com a demostracions per constrocter ponens de teoremes dual-falacis,
que només es compleish a 1


teorema dual-falacis
m+n=m+1
(-m)+(-n)=(-m)+(-1)


teorema dual-falacis
nx+...(m)...+nx=mx
(-n)x+...(m)...+(-n)x=(-m)x


teorema dual-falacis
x^{n}·...(m)...·x^{n}=x^{m}
x^{(-n)}·...(m)...·x^{(-n)}=x^{(-m)}


teorema dual-falacis
( n+...(m)...+n )^{p}=m^{p}
( (-n)+...(m)...+(-n) )^{p}=(-m)^{p}
demostració:
( n+...(m)...+n )^{p}=m^{p}
p·ln( ( n+...(m)...+n ) )=ln( ( n+...(m)...+n )^{p} )= ln(m^{p})=p·ln(m)
ln(mn)=ln(m)
ln(m)+ln(n)=ln(m)
n=1
demostració:
( (-n)+...(m)...+(-n) )^{p}=(-m)^{p}
p·ln( ( (-n)+...(m)...+(-n) ) )=ln( ( (-n)+...(m)...+(-n) )^{p} )=ln((-m)^{p})=p·ln(-m)
ln(m(-n))=ln(-m)
ln((-m)n)=ln(-m)
ln(-m)+ln(n)=ln(-m)
n=1


teorema dual-falacis
x^{n}=x
x^{-n}=(1/x)
demostració:
x^{n}=x
n·ln(x)=ln(x^{n})=ln(x)
n=1
demostració:
x^{(-n)}=(1/x)
(-n)·ln(x)=ln(x^{(-n)})=ln(1/x)=(-1)ln(x)
n=1


teorema dual-falacis
m < m+n < m+2
(-m) > (-m)+(-n) > (-m)+(-2)
demostració:
m < m+n < m+2
(-m)+m < (-m)+m+n < (-m)+m+2
0 < n < 2
n=1
demostració:
(-m) > (-m)+(-n) > (-m)+(-2)
m+(-m) > m+(-m)+(-n) > m+(-m)+(-2)
0 > (-n) > (-2)
n=1






teorema dual-falacis:
int[( sin[n](x) )^{n}]d[x] = (-1)·(1/n)·(cos[n](x))^{n}
int[( cos[n](x) )^{n}]d[x] = (1/n)·(sin[n](x))^{n}
demostració: 
...int[(1/n)·(y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(-1)·(1/(n+1))·n^{n}·int[( (-1)(n+1)(y/n)^{n}·(1/n) )/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
(-1)·n^{n}·(1/n)·(cos[n](x)/n)^{n}=(-1)·(1/n)·(cos[n](x))^{n}
demostració:
...int[(-1)(1/n)·(y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(1/(n+1))·n^{n}·int[( (-1)(n+1)(y/n)^{n}·(1/n) )/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
n^{n}·(1/n)·(sin[n](x)/n)^{n} = (1/n)·(sin[n](x))^{n}


teorema dual-falacis:
int[( sin[n](x) )^{2n+1}]d[x] = ...
...(-1)·(1/n)·(sin[n](x))^{n+1}·(cos[n](x))^{n}+(-1)·(1/n)( (n+1)/(2n+1) )·(cos[n](x))^{2n+1} )
int[( cos[n](x) )^{2n+1}]d[x] = ...
...(1/n)·(cos[n](x))^{n+1}·(sin[n](x))^{n}+(1/n)( (n+1)/(2n+1) )·(sin[n](x))^{2n+1} )
demostració:
...int[y^{n+1}·(1/n)·(y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(-1)·int[(y^{n+1}/(n+1))·n^{n}·((-1)(n+1)(y/n)^{n}·(1/n)/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(-1)n^{n}( (y^{n+1}/(n+1))·((n+1)/n)(1+(-1)(y/n)^{n+1})^{(n/(n+1))}+...
...(-1)int[y^{n}·((n+1)/n)·(1+(-1)(y/n)^{n+1})^{(n/(n+1))}]d[y] )=...
...(-1)n^{n}( (y^{n+1}/n)(1+(-1)(y/n)^{n+1})^{(n/(n+1))}+...
...n^{n}int[(-1)(n+1)(y/n)^{n}·(1/n)·(1+(-1)(y/n)^{n+1})^{(n/(n+1))}]d[y] )=...
...(-1)n^{n}(y^{n+1}/n)(1+(-1)(y/n)^{n+1})^{(n/(n+1))}+...
...(-1)n^{2n}((n+1)/(2n+1))·(1+(-1)(y/n)^{n+1})^{((2n+1)/(n+1))}=...
...(-1)·(1/n)·(sin[n](x))^{n+1}·(cos[n](x))^{n}+(-1)·(1/n)( (n+1)/(2n+1) )·(cos[n](x))^{2n+1} )


demostració:
...int[y^{n+1}·(1/n)·((-1)·y^{n}/(1+(-1)(y/n)^{n+1})^{(-1)(1/(n+1))})]d[y]=...
...(1/n)·(cos[n](x))^{n+1}·(sin[n](x))^{n}+(1/n)( (n+1)/(2n+1) )·(sin[n](x))^{2n+1} )

funcions eliptiques inverses

d_{x}[arcsin[n](x)]=(n^{n+1}+(-1)x^{(n+1)})^{(-1)(n+1)}
d_{x}[arcsin[n](x)]=(1/n)·(1+(-1)(x/n)^{(n+1)})^{(-1)(n+1)}
d_{x}[arcsin[n](sin[n](x))]=cos[n](x)·(1/n)·(1+(-1)(sin[n](x)/n)^{(n+1)})^{(-1)(n+1)}=1


d_{x}[arccos[n](x)]=(-1)·(n^{n+1}+(-1)x^{(n+1)})^{(-1)(n+1)}
d_{x}[arccos[n](x)]=(-1)·(1/n)·(1+(-1)(x/n)^{(n+1)})^{(-1)(n+1)}
d_{x}[arccos[n](cos[n](x))]=(-1)·sin[n](x)·(-1)·(1/n)·(1+(-1)(cos[n](x)/n)^{(n+1)})^{(-1)(n+1)}=1


int[( sin[n](x) )^{m}]d[x] =( 1/(m+1) )( sin[n](x) )^{m+1} [o(x)o] ( sin[n](x) )^{[o(x)o](-1)}
int[( cos[n](x) )^{m}]d[x] =( 1/(m+1) )( cos[n](x) )^{m+1} [o(x)o] ( cos[n](x) )^{[o(x)o](-1)}