Correlation function equal-time cross

For short times, the correlation function (7(f) depends on the microscopic details of the dynamics as the system crosses from to 38. These motions take place on a molecular time scale rmoi essentially equal to the time required to move through the transition region. For times f larger than rmoi but still very small compared to the reaction time rrxn (if the crossing event is rare rrxn L> rmoi such that such an intermediate time regime exists), C(f) can be replaced by an approximation linear in time. Using the detailed balance condition k jk = h )/ h ) [33] one then obtains... 

Time domain electromagnetic (EM) migration is based on downward extrapolation of the residual field in reverse time. In this section I will show that electromagnetic migration, as the solution of the boundary value problem for the adjoint Maxwell s equation, can be clearly associated with solution of the inverse problem in the time domain. In particular, I will demonstrate that the gradient of the residual field energy flow functional with respect to the perturbation of the model conductivity is equal to the vector cross-correlation function between the predicted field for the given... 

According to the correlation theorem (Press et ah, 1987, p. 383), the inverse Fourier transform of the product of spectrum of one function and the complex conjugate spectrum of another function is equal to correlation of these functions. Therefore, we can write the numerator in formula (15.38) as a cross correlation of the time derivatives of the back-propagated scattered field and the incident wavefield ... 

Here, as in the case of the one-photon spectrum, 1 ) is the bright state on the upper electronic potential energy surface which corresponds to the final state / on the ground electronic state. Equation (49) is a half Fourier transform in that it is limited to positive values of the time. One can regard it as an ordinary Fourier transform by defining the cross-correlation function to equal zero for negative times. Such a function is called causal in the theory of Fourier transform (51) and this puts conditions on the analytic properties of the Raman amplitude. These will be further discussed in Sec. IV. 

Therefore we mention only two examples. The nonrelativistic 3 parameter for s-electrons is equal to 2, while the inclusion of both relativistic and correlational corrections leads in the case of 5s-electrons in Xe to a very complicated curve >, given in Fig. 2 together with recent experimental data. The action of both 5p and 4d excitations upon 5s-electrons is very essential. The inclusion of many-electron correlations leads to a very complex behaviour of D(a)) as a function of O), D((jo) even changing its sign several times. The detailed information on D(a)) may be obtained in a so called complete experiment For photoionization it includes measurement of partial subshell cross section and angular distribution, and of electron spin direction, i.e. its polarization. The polarization follows all variations of D(o)), Recently, extensive calculations of polarization parameters for noble gases were performedand the RPAE predictions are now confirmed expe-... 

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