Exponentially correlated

The influence of the electronic coupling on the electron transfer rate was determined by changing the length of the (A T)n bridge. As expected, the rate decreased as the number n of the A T base pairs between electron donor and electron acceptor increased [4, 7]. But, surprisingly, the exponential correlation of Eq. (1) between the rate kEr and the distance is not valid for short distances. The plots in Fig. 3 and Fig. 4 show that at 6 A the electron transfer rate /cEt is much faster than expected [4, 7]. 

In order to be able to evaluate data with a reasonable number of parameters, the mode analysis assumes, as a first approximation, that the exponential correlation of the correlations [Eq. (18)] is maintained, and only the relaxation rates 1/tp are allowed to depend on a general form on the mode index... 

The water vapor available in the air plays a decisive part in the formation of precipitation. The amount of water that can be retained in the air in the form of vapor is primarily dependent on the temperature Warm air masses can absorb more humidity than cold air masses, whereby there is an exponential correlation between temperature and saturation humidity (Fig. 1, [4]). 

The previously mentioned release of aluminium at low pH is particularly important because of its toxic effects on organisms. As expected from theory [9], drawing the average aluminium concentrations against the mean pH a negative exponential correlation is obtained (Fig. 6) A1 increases with decreasing pH and lakes with the lowest pH have the highest aluminium concentrations. 

A possible step in this direction can be made through use of earlier relaxation studies on other systems. Hunt and Powles,— when studying the proton relaxation in liquids and glasses, found the relaxation best described by a "defect-diffusion" model, in which a non-exponential correlation function corresponding to diffusion is Included together with the usual exponential function corresponding to rotational motion. The correlation function is taken as the product of the two independent reorientation pro-... 

The simplest motional description is isotropic tumbling characterized by a single exponential correlation time ( ). This model has been successfully employed to interpret carbon-13 relaxation in a few cases, notably the methylene carbons in polyisobutylene among the well studied systems ( ). However, this model is unable to account for relaxation in many macromolecular systems, for instance polystyrene (6) and poly(phenylene oxide)(7,... 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

To compare the time scales of the dynamics characterization produced by each model, the spectral density or correlation function can be written as a distribution of exponential correlation times. For a correlation function, (t), the general expression is CO... 

Exponentially correlated gaussian type wave function... 

The non-Markovian heat bath presents an exponential correlation function characterized by rc. 

In order to compare the results from PD with the simulations, the input properties, and silo geometry have to be the same. As the results of PD in the previous section are based on a first order negative exponential correlation function, this function was used as well for the simulations. Three different input properties were used with different characteristic volumes (Vci) 40 and 400 m3. The silo volume was approximately 40,000 m3, with a height of 50 m and a diameter of 32 m. This relatively high silo was chosen because constant angles over the silo height are required (as indicated in Fig. 1) for a thorough comparison with PD. As the input properties are realizations of a stochastic process 400 repetitions were done per simulation. 

A natural extension of a Hylleraas-type basis might include correlation in the exponential function. Hylleraas[24] introduced this type of exponentially correlated (EC) wave function, using a single term of the form,... 

In this Table we also find a second form of exponentially correlated function s]. Such functions have been referred to as explicitly correlated Gaussians (ECG). They take the form,... 

It is in this context that exponentially correlated basis sets (EC or ECG) become particularly interesting, since such bases behave as if they contained high metric terms, but in a very compact form. With multiple nonlinear parameters, a trial wave function may be able to represent the wave function at the cusps accurately while retaining sufficient flexibility to correctly reproduce its asymptotic behavior. 

The correlation function R(-, ) in (2) is central to this statistical model. The power-exponential class of correlation functions is a popular choice, for its computational simplicity and because it has been successful in many applications. The power-exponential correlation function is... 

The power-exponential correlation function (3), for example, is of this product form. To computere x), the integral on the right-hand side of (22) is evaluated as... 

In this group of subjects there was no relationship between ferritin and Fe-PAE-9> but serum iron was correlated with PAE-9 with an r of 0 508 at p<0 02 (Figure 6) That is subjects with higher serum iron tended to excrete more iron 9 days or more after the dose Possibly subjects with better iron status have some hold-up of iron in the gut and excrete it later Likewise Fe-PAE-9 was exponentially correlated with the percentage of iron absorbed from the dose with r 0 35 and a p<0 005 (Figure 7) This means that the more iron which was absorbed the less which was excreted at day 10 or later ... 

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