Don and Rebecca did an excellent job of catching my (unintentionally expressed) meaning in the difference between "Rebecca's taking over" and "Rebecca taking over," although I disagree with Rebecca if she is implying that the two are almost identical in meaning. I agree with Jesperson that there is a difference in the two between "nexus" and "junction." The noun absolute, as direct object, expresses the nexal meaning -- "Rebecca" and "taking over" are equally important. On the other hand, the gerund, modified by a possessive, emphasizes the "taking over."
     Martha says that no grammarian considers the noun absolute as a noun, but Paul Roberts does, in Understanding Grammar (p. 355). He gives two examples, one of a subject -- "Linda in trouble was ample reason for my going." and one as an appositive -- "That's Pagahead for you, the right hand not keeping the left aware of what is going on."
    As for Quirk, we need to know what his philosophical bias is. By that I mean that the structuralist and early transformational grammars were developed with NO interest in what a sentence MEANS. Of course, my primary objection to Martha, Quirk, and now Richard Veit, is that they like grammar (and grammar books)  too much.   Their explanations are too long, too complex, and, ultimately, too sterile for the general public. Martha, for example, says that we should teach the noun absolute, but what does that mean? We have been "teaching" grammar for decades, but there is little evidence that students have learned much from it.
     As Richard says, "But for those who love playing with these things, there are further
complications to keep us busy."  My problem is that while "we" are busy "playing," most students, and teachers, cannot recognize basic subjects, verbs, clauses, etc. In the KISS approach, noun absolutes are in the last level -- the level at which students mop up any words that they cannot otherwise analyze. That means that they would be forced to see the noun absolute in "My hand behind my back, I couldn't catch the ball." Students would not need to see the absolute as direct object. They could analyze "They saw the soldiers leaving." as "soldiers," (DO) and "leaving" as a gerundive (participle). Doing so, of course, would make them miss the fine points of meaning that Don and Rebecca caught, but for most students, that would be a fine point.
      Paul Doniger raised a good question about my statement that most teachers are not prepared. There are, of course, no studies on this. My impression is based on things such as the many questions and comments that I received while editor of SIS, and, for example, on the fact that the intended standardized tests in England were canceled because the teachers noted that they themselves could not identify clauses. It might, by the way, be an interesting project for ATEG to see what the current state of knowledge about grammar actually is. Could the group come up with a test that would be administered to teachers in a number of school systems? (I'd bet that even if the group came up with such a test, the teachers would object -- because they do not feel comfortable with their knowledge of grammar.)
Ed V.