Propagator representation

Karger, J., and Heink, W. (1983). The propagator representation of molecular transport in microporous crystallites. J. Magn. Reson. 51, 1-7. 

The usual way to analyze the PFG NMR data for NMR tracer desorption measurements is illustrated in Fig. 21 [48] the intensity of the slowly decaying component (corresponding to the broader distribution within the propagator representation) coincides with the relative amount of molecules [l-7(t)j which, for the given observation time A, have remained in their crystallites (Fig. 21a,b). 

As an example. Fig. 1 shows the propagator representation of molecular self-diffusion of ethane in beds of zeolite Na,Ca-A with two different crystalhte sizes. Being symmetric in z, for simphcity the propagator is only represented... 

Fig.l Propagator representation of the self-diffusion of ethane in zeolite Na,Ca-A with mean crystallite radii of R = p.m (a) and 0.4 p.m (b). From [57] with permission... 

Now, looking on Eq. (3.48) as compared with the previously used quantum mechanical form, see (2.10) with (2.21), and Volume I/Chapter 4 of the present five-volume work, the present propagator representation would be resumed as ... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

Rather than making this statement, one should consider first whether the representation of the Y-variablc is appropriate. What wc did here was to take categorical information as a quantitative value. So if wc have, for instance, a vector of class 1 and one of c lass 9 falling into the same neuron, the weights of the output layer will be adapted to a value between 1 and 9, which docs not make much sense. Thus, it is necessary to choose another representation with one layer for each biological activity. The architecture of such a counter-propagation network is shown in Figure 10.1 -11. Each of the nine layers in the output block corresponds to a different MOA. 

We assume that in (4.38) and (4.39), all velocities are measured with respect to the same coordinate system (at rest in the laboratory) and the particle velocity is normal to the shock front. When a plane shock wave propagates from one material into another the pressure (stress) and particle velocity across the interface are continuous. Therefore, the pressure-particle velocity plane representation proves a convenient framework from which to describe the plane Impact of a gun- or explosive-accelerated flyer plate with a sample target. Also of importance (and discussed below) is the interaction of plane shock waves with a free surface or higher- or lower-impedance media. 

Figure lb gives a graphical representation of the steps involved in the leap-frog propagation. The current velocity v , which is necessary for calculating the kinetic energy, can be calculated as... 

Rumelhart, D.E., Hinton, G.E. and Williams, R.J. (1986) Learning internal representations by error propagation. In Parallel Distributed Processing, Rumelhart, D.E. and McClelland, J.L. (eds.), M.I.T. Press, Cambridge, Mass. 

Fig. 7. Schematic representation of hydration causing crack propagation in a wedge test specimen. The increase in volume upon hydration induces stresses at the crack tip that promote crack growth 19,391.

Although eq. (10.103) for the propagator appears to involve the same effort as the perturbation approach (sum over all excited states, eq. (10.18)), the actual calculation of the propagator is somewhat different. Returning to the time representation of the polarization propagator, it may be written in terms of a commutator. 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

---