Time averaging approximation

Mixed Quantum/Classical Approach Coupled Chromophores and the Time Averaging Approximation... 

For the case of coupled chromophores the approximation is to average the matrix k(i) [98]. Thus in the time-averaging approximation (TAA) we replace ic(f) in Eq. (7) with... 

We have described our most recent efforts to calculate vibrational line shapes for liquid water and its isotopic variants under ambient conditions, as well as to calculate ultrafast observables capable of shedding light on spectral diffusion dynamics, and we have endeavored to interpret line shapes and spectral diffusion in terms of hydrogen bonding in the liquid. Our approach uses conventional classical effective two-body simulation potentials, coupled with more sophisticated quantum chemistry-based techniques for obtaining transition frequencies, transition dipoles and polarizabilities, and intramolecular and intermolecular couplings. In addition, we have used the recently developed time-averaging approximation to calculate Raman and IR line shapes for H20 (which involves... 

One problem yet to be solved theoretically involves ultrafast echo and pump-probe experiments on H20. Jansen has extended the time-averaging approximation to nonlinear ultrafast spectroscopy [164], meaning that one is now in the position of calculating 2DIR spectra for liquid water, which would allow for direct comparison with results from the exciting new experiments [73, 74]. 

In applications, one is often interested in approximating time averages over a time interval [0, T] via associated mean values of a , k = 1. ..Tfr. For T (or r) small enough, the above backward analysis may lead to much better error estimates than the worst case estimates of forward analysis. 

Its operation ean be seen as analogous to a meehanieal flywheel and a one-piston engine. The L-C Alter, like the flywheel, stores energy between the power pulses of the driver. The input to the L-C Alter choke input filter) is the ehopped input voltage. The L-C Alter volt-time averages this duty-cyele modulated input voltage waveform. The L-C Altering funetion ean be approximated by... 

In the next step, ethylene is polymerized by the action of triethylaluminum at approximately 120°C and 130 atmospheres to trialkylalu-minum. Typical reaction time is approximately 140 minutes for an average C12 alcohol production ... 

Figure 1 shows the hierarchical tree that corresponds to Table I. The scale at the top of the tree corresponds to the predicted time to failure in months. The conditions at the top of the tree are given by 71 C, 72%, and 8.3 suns and correspond to an expected failure time of approximately 16 months. The first split in the tree is associated with a temperature of 95 C on the left branch and 50 C on the right branch, which correspond to average predicted lifetimes of approximately four and 27 months, respectively. The next two splits are seen to be based on relative humidity and ultraviolet radiation. 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

An approximate equation for the profile of the time-averaged velocity for steady turbulent flow of a Newtonian fluid through a pipe of circular... 

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