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   Scalable but not additive functions
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   Author  Topic: Scalable but not additive functions  (Read 5584 times)
mistaken_id
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Scalable but not additive functions  
« on: Feb 1st, 2010, 6:14pm »
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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)
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Aryabhatta
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Re: Scalable but not additive functions  
« Reply #1 on: Feb 1st, 2010, 10:26pm »
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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.
 
 
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Re: Scalable but not additive functions  
« Reply #2 on: Feb 2nd, 2010, 4:14am »
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on Feb 1st, 2010, 6:14pm, 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.
« Last Edit: Feb 2nd, 2010, 4:24am by towr » IP Logged

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Aryabhatta
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Re: Scalable but not additive functions  
« Reply #3 on: Feb 2nd, 2010, 7:32am »
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on Feb 2nd, 2010, 4:14am, 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).
« Last Edit: Feb 2nd, 2010, 7:34am by Aryabhatta » IP Logged
towr
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Re: Scalable but not additive functions  
« Reply #4 on: Feb 2nd, 2010, 7:44am »
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on Feb 2nd, 2010, 7:32am, 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?
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Re: Scalable but not additive functions  
« Reply #5 on: Feb 2nd, 2010, 9:01am »
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on Feb 2nd, 2010, 7:44am, 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.
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Grimbal
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Re: Scalable but not additive functions  
« Reply #6 on: Feb 3rd, 2010, 1:37am »
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on Feb 1st, 2010, 6:14pm, 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.
« Last Edit: Feb 3rd, 2010, 3:08am by Grimbal » IP Logged
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Re: Scalable but not additive functions  
« Reply #7 on: Feb 3rd, 2010, 3:10am »
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on Feb 3rd, 2010, 1:37am, 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.
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Re: Scalable but not additive functions  
« Reply #8 on: Feb 3rd, 2010, 5:12am »
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Makes me think that all functions are additive on the range [c,2c[.  Wink
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