cadenes de funcions - topología algebràica

f(x) = y+x

g(x) = (-y)+x

cadena per composició:

   0 ----> f(0)   ---->...(n)...---->   (fo...(n)...of)(0)

(-0) ----> g(-0) ---->...(n)...----> (go...(n)...og)(-0)

f(x) = yx

g(x) = (1/y)·x

   1 ---->    f(1)   ---->...(n)...---->   (fo...(n)...of)(1)

(1/1) ----> g(1/1) ---->...(n)...----> (go...(n)...og)(1/1)

f(x) = ln(x)

g(x) = e^{x} = exp(x)

1 ---->  f(1)   ---->...(n)...---->   (fo...(n)...of)(exp(...(n)...exp(0)...(n)...))

0 ----> g(0)   ---->...(n)...---->    (go...(n)...og)(ln(...(n)...ln(1)...(n)...))

f_{n}(x) = n^{k}+x

g_{n}(x) = (-1)·n^{k}+x

   0 ---->  f_{1}(0)   ---->...(n)...---->   (f_{n}o...(n)...of_{1})(0)

(-0) ----> g_{1}(-0) ---->...(n)...---->    (g_{n}o...(n)...og_{1})(-0)

f_{n}(x) = n^{k}·x

g_{n}(x) = (1/n^{k})·x

1      ---->  f_{1}(1)   ---->...(n)...---->   (f_{n}o...(n)...of_{1})(1)

(1/1) ----> g_{1}(1/1)   ---->...(n)...---->  (g_{n}o...(n)...og_{1})(1/1)

f(x^{n}) = d_{x}[ x^{n} ]

g(x^{(1/n)}) = d_{x}[ x^{(1/n)} ]

1 ---->  f(1)   ---->...(n)...---->   (fo...(n)...of)(1)

1 ----> g(1)   ---->...(n)...---->    (go...(n)...og)(1)

f(x^{(-n)}) = d_{x}[ x^{(-n)} ]

g(x^{(1/(-n))}) = d_{x}[ x^{(1/(-n))} ]

1 ---->  f(1)   ---->...(n)...---->   (fo...(n)...of)(1)

1 ----> g(1)   ---->...(n)...---->    (go...(n)...og)(1)