In some embodiments, an apparatus includes logical interrupt identification number creation logic to receive physical processor identification numbers and create logical processor identification numbers through using the physical processor identification numbers. The focus detecting apparatus may include a mark disposed on a primary image plane. In the preferred embodiment of the invention, a valid hash value is determined, preferably based on the state information and preferably by locating the valid hash value within a table of valid hash values indexed by the state information. If there is a mobile version available, the personal computer may generate a QR code that represents an encoded version of a link for the mobile version of the software application.