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        riddles >> putnam exam (pure math) >> Scalable but not additive functions
        
(Message started by: mistaken_id on Feb 1^st, 2010, 6:14pm)

Title: Scalable but not additive functions
Post by mistaken_id on Feb 1^st, 2010, 6:14pm

Can anyone give some examples of scalable but not additive functions:

Scalable function: f(ax) = af(x)

Additive function: f(x+y)=f(x)+f(y)

Title: Re: Scalable but not additive functions
Post by Aryabhatta on Feb 1^st, 2010, 10:26pm

What is the Domain? Range? Do you have any restrictions on continuity/differentiability?

Assuming f:R->R and has a continuous derivative, then f(x) must be of the form f(x) = kx.

We can assume 0 < a < 1.

f(ax) = af(x) implies

f(a^nx) = a^n * f(x)

Taking limit as n->oo, f(0) = 0.

Now if g(x) = f'(x) then

g(ax) = g(x) (differentiating f(ax) = a f(x))

so
g(a^n * x ) = g(x)

Taking Limit as n-> oo, g(0) = g(x), hence g(x) is constant.

This f is linear, and since f(0) = 0, f(x) = kx.

Title: Re: Scalable but not additive functions
Post by towr on Feb 2^nd, 2010, 4:14am

on 02/01/10 at 18:14:45, mistaken_id wrote:

Can anyone give some examples of scalable but not additive functions:

Scalable function: f(ax) = af(x)

Additive function: f(x+y)=f(x)+f(y)

Take any scalable function f, then we have
f(x+y) = f((x+y) *1) = (x+y) * f(1) = x*f(1) + y*f(1) = f(x) + f(y)
So any scalable function is additive if 1 is in its domain.

Title: Re: Scalable but not additive functions
Post by Aryabhatta on Feb 2^nd, 2010, 7:32am

on 02/02/10 at 04:14:18, towr wrote:

Take any scalable function f, then we have
f(x+y) = f((x+y) *1) = (x+y) * f(1) = x*f(1) + y*f(1) = f(x) + f(y)
So any scalable function is additive if 1 is in its domain.

a is a constant (I think), so you just can't use any arbitrary scaling factor. In fact with an arbitrary scaling factor f(x) = 0 is the only function (in reals).

Title: Re: Scalable but not additive functions
Post by towr on Feb 2^nd, 2010, 7:44am

on 02/02/10 at 07:32:52, Aryabhatta wrote:

a is a constant (I think), so you just can't use any arbitrary scaling factor. In fact with an arbitrary scaling factor f(x) = 0 is the only function (in reals).

Are you sure? Why wouldn't f(x)=x work?

Title: Re: Scalable but not additive functions
Post by Aryabhatta on Feb 2^nd, 2010, 9:01am

on 02/02/10 at 07:44:30, towr wrote:

Are you sure? Why wouldn't f(x)=x work?

Duh! I must have made a mistake in the my 'proof' somewhere.

Anyway, I believe the intent is to have a as constant.

Title: Re: Scalable but not additive functions
Post by Grimbal on Feb 3^rd, 2010, 1:37am

on 02/01/10 at 18:14:45, mistaken_id wrote:

Can anyone give some examples of scalable but not additive functions:

Scalable function: f(ax) = af(x)

Additive function: f(x+y)=f(x)+f(y)

abs(x) is scalable for any positive constant a, but not additive.

In fact, there is little reason why a scalable function (with a constant a) should be additive. Any function of the form
f(x) = |x|·exp(h(ln(|x|)))
f(0) = 0
where h(x) is a periodic function with period ln(a) would be scalable but not additive.

Title: Re: Scalable but not additive functions
Post by towr on Feb 3^rd, 2010, 3:10am

on 02/03/10 at 01:37:26, Grimbal wrote:

abs(x) is scalable for any positive constant a, but not additive.

It's additive for x and y that have the same sign, though.

Title: Re: Scalable but not additive functions
Post by Grimbal on Feb 3^rd, 2010, 5:12am

Makes me think that all functions are additive on the range [c,2c[. ;)