Thanks, Jerry, for pointing out Steven den Beste's excellent essay:
The great strength of Western (especially American) culture is that it is particularly good at adopting the best it sees from competing systems and improving itself in that way. Because of that it has gotten progressively stronger and more widespread.
The great weakness of extremist Arab Muslim culture is that it does not do so. It is the nature of its meme that it rejects any change. But because of that, it is becoming progressively less competitive. It's now backed into a corner: it can fight or die.
There is, of course, a third alternative: They could grow and adapt. They could take their rightful place as a valued contributor to world culture. (I am reminded of what could have been every time I see the word "algebra".) Nothing would please me more than to see them choose this option.
But they must choose it. And I don't think there's much chance of it.