How to wake the Mac remotely so I can use screen sharing

The main thing to tackle is booting up the xServes. Luckily, Intel xServes (which yours should be if they're running 10.6) have Lights-out-Management (LOM) that should support this sort of thing (all Macs support Wake-on-LAN, but that only works on a local network).

Apple Remote Desktop (ARD) has support for this. It looks like you'll need to forward a few ports (looks like 623 for the LOM, and then a bunch for ARD - do a search), unless you have some sort of VPN access to your home network.

The basic method will be, use ARD to power up your xServes, then remotely connect via ARD or a VNC client, build your Mathematica app and run it. Then shut down the xServes.

If you have a Mac with you, all this can be done via ARD, but if you want to do it with an iPad, you'll need your desktop Mac running whenever you want to remotely access it. The process there would be to enable Screen Sharing (in the Sharing panel of System Prefs), forward the appropriate ports, then use an app like Screens to connect to your desktop Mac, run ARD and Mathematica remotely on it, and do the above.

If you don't have your desktop Mac running all the time, you could in theory use Wake-on-LAN to turn it on, then access it with Screens, but that requires some other device to be always on, remotely accessible and capable of sending the WoL packet to the Mac (some routers are capable of this, typically with 3rd party firmware).

Some more info would be helpful here. What app are you using on your iOS devices and Mac to connect to your Mini? What happens when you try to connect—do any errors appear, or does nothing appear to happen?

First off, make sure Screen Sharing is enabled in Sharing Preferences (Remote Management is only for using Apple Remote Desktop, so don't bother with that unless you're actually using that program). Double-check that Allow access is set to either "All Users", or "Only these users" with your user name included in the box.

Now try connecting from another Mac, using the Finder's Connect to Server dialog (under the Go menu, or ⌘K). Under Server Address, type in vnc://IP_OF_YOUR_MINI then hit connect. If you can connect that way, but not using the Finder's Shared panel, then the problem is likely in the Bonjour service that advertises the screen sharing option.

Another thing to check is what version of Apple Remote Desktop Client you have installed. If you have 3.6, there's a bug that can cause timeouts for Screen Sharing/VNC clients. The temporary fix is removing /System/Library/CoreServices/RemoteManagement and re-installing 3.5.3. This worked for me on Lion, I don't know if it will work on Mountain Lion. Make sure to keep a backup copy of the files you delete, in case the install doesn't work.