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Topic: Scalable but not additive functions (Read 5507 times) 

mistaken_id
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Scalable but not additive functions
« on: Feb 1^{st}, 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 1^{st}, 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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towr
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Re: Scalable but not additive functions
« Reply #2 on: Feb 2^{nd}, 2010, 4:14am » 
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on Feb 1^{st}, 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 2^{nd}, 2010, 4:24am by towr » 
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Aryabhatta
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Re: Scalable but not additive functions
« Reply #3 on: Feb 2^{nd}, 2010, 7:32am » 
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on Feb 2^{nd}, 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 2^{nd}, 2010, 7:34am by Aryabhatta » 
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towr
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Re: Scalable but not additive functions
« Reply #4 on: Feb 2^{nd}, 2010, 7:44am » 
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on Feb 2^{nd}, 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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Aryabhatta
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Re: Scalable but not additive functions
« Reply #5 on: Feb 2^{nd}, 2010, 9:01am » 
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on Feb 2^{nd}, 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 3^{rd}, 2010, 1:37am » 
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on Feb 1^{st}, 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 3^{rd}, 2010, 3:08am by Grimbal » 
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towr
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Re: Scalable but not additive functions
« Reply #7 on: Feb 3^{rd}, 2010, 3:10am » 
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on Feb 3^{rd}, 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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Grimbal
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Re: Scalable but not additive functions
« Reply #8 on: Feb 3^{rd}, 2010, 5:12am » 
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Makes me think that all functions are additive on the range [c,2c[.


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