1. ## weird scaling law...

Hey guys, can any of you tell me if there's a function (f(x): R->R) apart from linear, that satisfies the following:

f( (x - f(x))/2 ) = C * f(x)

where C is a real constant. I don't really know how to touch this problem... I'm a physics PhD student interested in fractal-like behaviour, and the answer to this question would help me a great deal. Even if it turns out, that no such function exists.
Thanks a lot for your help!

2. Oh, a solution exists, since f(x) = 0 satisfies the equation. I imagine you want a non-trivial solution, however. I'll have to think some more to figure out another solution, which may or may not exist.

3. Another special case: $C=1,$ and $f(x)=-x.$

4. Yep, these are solutions but I was wondering whether a non-linear one existed. It might not, but I'd have absolutely no idea how to prove that
Thanks for having a look, mate!

5. I read that you're in mathematical physics. What field are you working in?

6. I'll be thinking some more about this problem. It's very interesting. It looks a lot like Schröder's Equation, although in that equation, I'm not sure that $h$ is allowed to depend on $\Psi,$ like you have.

My background is mathematical physics. My dissertation was on the mathematics of solitons in optical fibers. Since then, I've done experimental work in fiber optics, and then mostly LabVIEW programming. Right now, I work in fuel cells, doing systems integration. I build fuel cell test stands, essentially. How about you?

7. Yeah it does look a bit like Schröder's equation, but like you said, the fact that in my case depends on makes it really difficult to handle...
Wow, fiber optics and fuel cells! Really interesting stuff!! I do computer simulations mainly for the fracture of grainy materials, to get to know about the physics of how cracks propagate, branch, etc... You're still at Virginia Tech, right? What's the physics scene there like? I'm asking cos I'll be hunting for post doc positions in about half a year hopefully

8. I am no longer at Virginia Tech. I am working in industry, for a small company. That's fun, because I get to wear multiple hats, like IT.

The physics scene at VT is something I have to admit I'm not really in the loop on. Yes, I took physics courses there, but I did mathematical physics through the math department. I think more like a mathematician than a physicist. I could have wished that Virginia Tech had quantum computing, which is what I would have loved to work in. But they don't have it. Virginia Tech has a very strong condensed matter group (would your field be in CM?), as well as a strong (of course) mathematical physics group. They also have some decent particle theorists. The very best lecturer I ever had at VT was Tatsu Takeuchi, who is a high-energy theorist there. Alas, all I ever was able to do with him was audit his classes. That guy, while brilliant himself, was able to make complex, difficult things clear. Not easy, but clear. The very worst lecturer I ever had was also there: Luke Mo, a high-energy experimentalist. Mo is no longer there. Mo was nominated for the Nobel prize, as I understand, but he didn't understand students who weren't as brilliant as him (like me).

I don't know what their chaos group is like, or if they even have one.