Where are the Trimble Positions Desktop Add-In 10.1 and ArcGIS Mobile Project Center 10.1 files stored on the PC? 

Trimble Positions and Mobile Project Center log files

Click on the above photo to enlarge it.

If you or your IT staff need to access the behind the scenes files of Trimble Positions Desktop Add-In 10.1 and ArcGIS Mobile Project Center 10.1 the above jpg shows you where to find them.