Tuesday, 21 September 2010

A rough answer about variations

Question 1: rather easy...
i)yes since x=+/- sqrt (k) y, no matter it's positive or negative it can be written in forms of x=cy.
ii)yes for the same reason.
iii)no except x=y (bonus for x=y) since log x = k log y, x=y^k.
iv, v)By contradiction, i.e., when x=cy, f(x)=kf(y), f(cy)=kf(y) is solvable (Bonus for solvable)  i.e., unique numbers of x,y is obtainable so x,y is no longer a variables.
b)So far I can only find three characteristics:
1)fixed point 0: f(0)=kf(0).
2)Strictly increasing: k^nf(y)=f(c^n)y
3)Everywhere differentiable: Proof left for readers.
For geometrical approach the obvious way is "parametic equations".

Question 2: hard
a)It must be multivariate functions.
b,c) They can be non-partial variations since the comoponent of the product has the same nature.
d)They can be non-partial variations as long as the degree of the function is unique.
e,f)Kept open.

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