Asymptotic method, defined

There is, however, a simpler method, which consists in examining the asymptotic behaviour of the form function H(q) of the entire chain. We know indeed [see (7.3.30) and Appendix E] that in the asymptotic domain defined by the condition... 

The asymptotic method Here we choose the interarrival-time SCV of the renewal process so that its asymptotic variance [defined in (18)] agrees with that of N. This means we choose so that... 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

Using field theoretic methods this divergence has been analyzed to some extent. The analysis strongly suggests that the expansion is in fact an asymptotic expansion in the mathematical sense. Using the powerful method of Borel transformation one can resum the expansion to get a unique finite result. We can thus state that the theory is well defined and all the qualitative... 

The derivation given above of the stationary Kohn functional [ K] depends on logic that is not changed if the functions Fo and l< of Eq. (8.5) are replaced in each channel by any functions for which the Wronskian condition mm — m 0 = l is satisfied [245, 191]. The complex Kohn method [244, 237, 440] exploits this fact by defining continuum basis functions consistent with the canonical form cv() = I.a = T, where T is the complex-symmetric multichannel transition matrix. These continuum basis functions have the asymptotic forms... 

Other asymptotic forms consistent with unit Wronskian define different but equally valid Green functions, with different values of the asymptotic coefficient of u>i. In particular, if w k 2 exp i(kr — ln), this determines the outgoing-wave Green function, and the asymptotic coefficient of w is the single-channel F-matrix, F sin ij. This is the basis of the T-matrix method [342, 344], which has been used for electron-molecule scattering calculations [126], It is assumed that Avf is regular at the origin and that Ad vanishes more rapidly than r 2 for r — oo. 

In this paper, solutions to three important heterogeneous diffusion problems are presented, and their implications for transport in biological systems are discussed. While the detailed methods of solutions and subtleties are presented in other papers (2-6), the asymptotic solutions are easily described, and they define the important physics of diffusion for most of the ranges of interest. In particular, a) nonsteady-state diffusion through oil-water multilaminates (2,3) b) desorption from oil-water multilaminates (4) and... 

Caprani et al. [104], defining the cut-off frequency as the intersection of the low and high-frequency asymptotes, as indicated on Fig. 10.18, have given an approximate method to deduce the size of active sites on a partially blocked electrode from the ratio of the two cut-off frequencies ... 

The problem that has been defined here possesses the cold-boundary difficulty discussed in Section 5.3.2 and can be approached by the same variety of methods presented in Section 5.3. The asymptotic approach of Section 5.3.6 has been seen to be most attractive and will be adopted here. Thus we shall treat the Zel dovich number... 

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