Dar-al-Islam has a major feature you're overlooking:
* ease of transport of information. (your mathematics expert in Persia could pass on information to fellows on the Hajj from Indonesia, Spain, Africa, and Mongolia; with the Academy of Gundishapur, how far could the knowledge travel within a single century?)
That really depends on the size of Sassanid Persia. When they had the Byzantines on the ropes around 620, they occupied Egypt, Syria, Anatolia, Yemen, Sindh, and Bactria in addition to their core lands. Could they hold this? If there is no Islam and Heraclius's counter attack fails, it seems possible.
In the case of mathematics, these lands give them access to all they need to duplicate the advances made by Islamic scholars.
Their merchants reached throughout India, Central Asia, and China. They also had trade with Axum. The empire was supported by the trade that flowed through it. There is no reason to suppose that they would suddenly shut this trade down. On the contrary, with access to the Mediterranean, they are likely to expand their trade there.
Further, there is no reason to assume that the knowledge would stay bottled up in Persia. For example, when Justinian shut down the Academy of Athens, the Neoplatonics moved to Gundishapur. When they decided they didn't like it there, they were allowed to return to Greece. Of course, they may try to control the spread of some knowledge, just as the Chinese did with gunpowder. However, we all know how well that worked.
let me be perfectly clear: I agree that, eventually, the knowledge will be learned. however, I disagree that there is no difference in how it will be distributed, nor in the timescales involved with either invention or distrobution.
I agree that there could very well be differences in the time it takes for distribution. It could take more or less time. For example, if Persia continues the religious tolerance shown under Khosrau I and does not continue to expand militarily into Europe, there would probably be no Crusades. Without the Crusades to sour relations, Christian Europe might encounter Persia science earlier. Then again, the Göktürks could rage across Persia around AD 700 and destroy all records of Persian science.