Logarithmic double

Each of the three approaches will be applied in this section to the transformed retention times of the 23 chalcones with eight chromatographic elution methods in Table 31.2. The transformation is defined by the successive operations of logarithms, double-centering and global normalization which is typical for the method of spectral map analysis (SMA) ... 

Because of the double sound path involved in PE measurements of the back wall echo, we approximate the corresponding attenuation at a certain frequency to be twice as large as the attenuation that would be obtained by an ordinary TT measurement. We propose to use the logarithm of the absolute value of the Fourier transform of the back wall echo as input data, i.e... 

Noise. Technical differences exist between personal noise dosimeters and high accuracy sound level meters and these may alter the usual preference for personal monitors. But it is exposure to noise rather than general room noise that must be estimated for comparison with noise exposure criteria, the logarithmic expression and alternative means of summation (3 vs 5 db doubling) compHcate statistics. Exposure criteria for both dose and peak exposure must be evaluated, and space and time variabiUty of noise intensity can be immense. 

Fig. 11. Logarithm of heating time compared with temperature to attain various degrees of degradation of wood (35). No parentheses indicates weight loss on oven heating a single parenthesis indicates modulus of mpture on oven heating and double parentheses indicates weight loss on heating beneath...

The general function giving straight lines on double logarithmic plots is ... 

Both sets of experiments seem to support the proportionality of crack opening displacement 5C = 2w and molecular mass Mc between crosslinks as indicated by the slope 1 in the double logarithmic plot (Fig. 7.5). Even if Mc had to be adjusted due to doubts about the front factor in Eq. (4.3), the proportionality would stay unaffected. Consequently, the size of the deformation zone ahead of the crack is determined by the length of the molecular strands in the chemical network. 

As an example, consider the polymerization of methyl methacrylate, initiated by a,a -azobisisobutyronitrile.4 The dependence of the initial rate on the concentration of the initiator is displayed in Fig. 1-1, which shows them on a double logarithmic scale. The points define a straight line with a least-squares slope of0.496. Clearly the reaction is half-order with respect to the concentration of the initiator. 

Figure 9.72. Correction for logarithmic mean temperature difference for double shell pass exchanger...

Most normal animal cells in culture grow at a logarithmic rate until they achieve a particular density (i.e., their saturation density) (Figure 11). A typical growth rate for animal cells in culture is one doubling per day, and a typical saturation density... 

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

RIA PREC.dat Two hundred thirty-eight calibration data sets were collected and analyzed for repeatability (within group CV) and plotted against the mean concentration. In a double-logarithmic plot the pattern seen in Fig. 4.6 appears. 

Most pectin solutions behave like Newtonian liquids below a pectin concentration of about 1 % (w/w). Onogi (1966) derived the critical concentration of polymer solutions from plotting the double logarithmic curves of viscosity (ii) against concentration at constant shear rates. Each curve consists of two straight lines intersecting at the critical concentration. At higher... 

Fig. 141.—Double logarithmic plot of [77]0 against M for several polymer series polyisobutylene in benzene at 24°C, O polystyrene in cyclohexane at 34°C, cellulose tricaprylate in T-phenylpropyl alcohol at 48 C, Q and...

Preprocessing by log double-centering consists of first taking logarithms, and then to center the data both by rows and by columns ... 

The transformation by log double-centering has received various names among which spectral mapping [13], logarithmic analysis [14], saturated RC association model [15], log-bilinear model [16] and spectral map analysis or SMA for short [17]. 

The logarithmic transformation prior to column- or double-centered PCA (Section 31.3) can be considered as a special case of non-linear PCA. The procedure tends to make the row- and column-variances more homogeneous, and allows us to interpret the resulting biplots in terms of log ratios. 

The log-linear model (LLM) is closely related to correspondence factor analysis (CFA). Both methods pursue the same objective, i.e. the analysis of the association (or correspondence) between the rows and columns of a contingency table. In CFA this can be obtained by means of double-closure of the data in LLM this is achieved by means of double-centring of the logarithmic data. 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

Fig. 5. Double logarithmic plot of zero-shear rate viscosity against concentration and molar mass... 

Now consider the effect of temperature on the rate of reaction. A qualitative observation is that most reactions go faster as the temperature increases. An increase in temperature of 10°C from room temperature typically doubles the rate of reaction for organic species in solution. It is found in practice that if the logarithm of the reaction rate constant is plotted against the inverse of absolute temperature, it tends to follow a straight line. Thus, at the same concentration, but at two different temperatures ... 

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