By Walther Burchard (auth.)
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This can easily be verified from Eq. 95). 44) These are equations which were already derived in Chap. 2. The recursive equation can now be solved step by step and yields the final result 00 00 U~(1) = ~ P " - I ~. = ~ ( p , ) , - 1 , . 42a) n=l and U6(1) =Pw(qi)z = •0 + [ ( N ( 1 ) ) ¢ . 42b) The advantage of the cascade process compared with the method applied in Chap. 2 is that the probability matrix P follows immediately from the generating function formalism as being the coefficient matrix to the components of the next higher generation, and this makes the role of a transition matrix immediately clear.
In fact, within the sphere or radius dr, one is always certain to find the selected unit. The other limit is fulfilled because at distances much larger than the radius of gyration there will be no other monomeric unit of the same molecule. Clearly, the space correlation function ? (r) is closely related to the segment-density distribution, and since the scattering intensity as a function of the scattering angle is the Fourier transform of ~, (r), there exists a close relationship between the density distribution and the angular distribution of the scattered light.
The transition probabilities refer here to the transition from the first to the second generation. For example, Pbbis the number of black units in the second generation divided by the number of black units in the first generation, etc. In individual rooted trees, the transition probabilities differ from generation to generation. However, if all rooted trees with a black unit are considered, the transition probabilitiesbecome on average independent of the generation, if n > 0. 110) with a similar expression for (N (1))B, and I is the unit column vector.