Double-exponential method

Access to a fourth-generation Frechet dendron can thus be shortened by one stage, since no first-generation dendron can be formed by this method - i.e. by doubling of the numbers of generations. 

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Fig. 2.5 Double-exponential method (C=coupling group, F=functional group P=protected group)...

I 33 Accelerated Approaches to Dendrimers Double exponential methode... 

Recently, new methods such as orthogonal synthesis, double-stage convergent method, double-exponential method, hypermonomer method, and click chemistry have been disclosed for synthesis of dendrimers [5]. 

Recent reports 54 seem to indicate that the resolution of the notoriously difficult solid-state spectra of coals may be enhanced by such techniques as double exponential multiplication and convolution difference. Differential relaxation behaviour as discussed in connection with intermolecular effects in carbohydrates and low temperature methods may further improve identification. 

Once the background is subtracted, the component of the spectrum due to the annihilation of ortho-positronium is usually visible (see Figure 6.5(a), curve (ii) and the fitted line (iv)). The analysis of the spectrum can now proceed, and a number of different methods have been applied to derive annihilation rates and the amplitudes of the various components. One method, introduced by Orth, Falk and Jones (1968), applies a maximum-likelihood technique to fit a double exponential function to the free-positron and ortho-positronium components (where applicable). Alternatively, the fits to the components can be made individually, if their decay rates are sufficiently well separated, by fitting to the longest component (usually ortho-positronium) first and then subtracting this from the... 

By fitting the depolarized time correlation function data with a double exponential model, we obtained Djj values inconsistent with the other results we obtained. However, by plotting the values obtained for the faster component of the double exponential fit against K, and by plotting the resulting intercepts and slopes against concentration, we obtained = 10.9 x 10" cm2/sec and ifg = 8.24 sec . The results are consistent with those obtained from other methods. Thus, there may be a low-frequency component of the depolarized time correlation function arising possibly from dust. 

These accelerated growth procedures have been used by a number of groups to synthesize dendritic poly(amides),222-224 poly(esters),225 poly(ether urethanes),226 and chiral poly(ethers).227 The orthogonality of the focal and peripheral protecting groups required for double exponential growth also enables facile modification at either the periphery or focal point. Although this approach is one of the most rapid methods yet reported for the synthesis of well-... 

The same method works with other decay functions. With a double exponential, the dimensions of V and A are increased accordingly. 

Deconvolution of the observed decay pulse was necessary because of the finite width (approx. 1.9ns FWHM) of the excitation pulse. This was carried out using a Fourier transform method (4). Despite the non-trlvlal computational requirements and the need to filter out the high frequency components of the Fourier division before taking the Inverse transform (5) It was felt that this method was the most direct way of generating the true Impulse response function of the system. The resulting fluorescence decay curves were fit to a single (or double) exponential. The phase plane plot method of Demas (6) and others (2), was also used for deconvolution but with less success. 

Regular exponential smoothing model estimates the constant level (L) to forecast future demands. Holt s model improves it by estimating both the level (L) and trend factor (T). It adjusts both the level and trend factor using exponential smoothing. Holt s method is also known as double exponential smoothing or trend adjusted exponential smoothing method. 

Like the traditional mass bias correction approaches, the double-spike method also relies on the choice of the mass bias model. The original formalism of the double spikes employed the linear mass bias law and, although double-spike calibration equations adapted for the exponential mass bias discrimination are available, linear models are still often used owing to their simplicity (see, for example, [50-52]). The caveat here is that erroneous results can be obtained when a linear correction is applied to data that do not follow such behavior. This is illustrated below. 

In order to illustrate the method, we can refer to the measurements of the isothermal decay of nonlinear d33 proportional to the square root of the SHG intensity, of DR19 in a polyimide films [71]. Polyimides are good candidates for NLO devices since they offer high thermal and good chemical stability. The second order nonlinear coefficient d33 have been fitted with a double exponential function ... 

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