I have installed Mathematica 9 on Ubuntu 13.10. I was wondering how the program updates itself? There is no repository for Mathematica visible in the Software Center. I am just asking out of interest.
Mathematica is a proprietary application and not part of Ubuntu, and not available on the Ubuntu store. If it provides updates to itself, it does it completely outside the realm of the Ubuntu archive and packaging framework. You didn't install it using that framework, so there's no reason it would be updated through it.
If you want specific details about how it works, you should contact Wolfram for support on the product.