Inflexion points

Thus, the points of inflexion occur after n - Vn and n + Vn plate volumes of mobile phase has passed through the column. It follows, the volume of mobile phase that has passed through the column between the inflexion points will be... 

Now, it has already been shown that the retention volume of a solute is given by n(vm + Kvs), and twice the standard deviation of the peak at the inflexion points is given by 2 /11 (v + Kvg)... 

The various relationships concerning the interconversion between un-ionised and ionised or different resonant forms of indicators referred to in Section 10.7 apply equally well to those indicators used for non-aqueous titrations. However, in this type of titration the colour change exhibited by an indicator at the end point is not always the same for different titrations as it depends upon the nature of the titrand to which it has been added. The colour corresponding to the correct end point may be established by carrying out a potentiometric titration while simultaneously observing the colour change of the indicator. The appropriate colour corresponds to the inflexion point of the titration curve (see Section 15.18). 

In Fig. 15.7 are presented (a) the part of the experimental titration curve in the vicinity of the equivalence point (b) the first derivative curve, i.e. the slope of the titration curve as a function of V (the equivalence point is indicated by the maximum, which corresponds to the inflexion in the titration curve) and (c) the second derivative curve, i.e. the slope of curve (b) as a function of V (the second derivative becomes zero at the inflexion point and provides a more exact measurement of the equivalence point). 

The a—time curves for the vacuum decomposition at 593—693 K of lanthanum oxalate [1098] are sigmoid. Following a short induction period (E = 164 kJ mole-1), the inflexion point occurred at a 0.15 and the Prout—Tompkins equation [eqn. (9)] was applied (E = 133 kJ mole-1). Young [29] has suggested, however, that a more appropriate analysis is that exponential behaviour [eqn. (8)] is followed by obedience to the contracting volume equation [eqn. (7), n = 3]. Similar kinetic characteristics were found [1098] for several other lanthanide oxalates and the sequence of relative stabilities established was Gd > Sm > Nd > La > Pr > Ce. The behaviour of europium(III) oxalate [1100] is exceptional in that Eu3+ is readily reduced... 

As (T increases, pairs of inflexion points disappear and give rise to a smoother representation of the signal. 

The position of the inflexion points at higher values of [Pg.224]

If we coimect the positions of the same inflexion points over various values of straight lines, we create an interval tree of scales, as shown in Fig. 7 for the signals of Fig. 5. The interval tree allows us to generate two very important pieces of information about the trends of a measured variable ... 

Scale-space filtering provides a multiscale description of a signal s trends in terms of its inflexion points (second-order zero crossings). The only legal sequences of triangles between two adjacent inflexion points are (in terms of triangular episodes) ... 

Inflexion points never disappear as we move from more abstract to more detailed descriptions. 

Fictitious inflexion points are not generated as the scale of filtering increases. 

In Section II we defined the trend of a measured variable as a strictly ordered sequence of scaling episodes. Since each scaling episode is defined by its bounding inflexion points, it is clear that the extraction of trends necessitates the localization of inflexion points of the measured variable at various scales of the scale-space image. Finally, the interval tree of scale (see Section II) indicates that there is a finite number of distinct sequences of inflexion points, implying a finite number of distinct trends. The question that we will try to answer in this section is, How can you use the wavelet-based decomposition of signals in order to identify the distinct sequences of inflexion points and thus of the signal s trends ... 

A wavelet defined as above is called a first-order wavelet. From Eq. (21) we conclude that the extrema points of the first-order wavelet transform provide the position of the inflexion points of the scaled signal at any level of scale. Similarly, if i/ (f) = d it)/dt, then the zero crossings of the wavelet transform correspond to the inflexion points of the original signal smoothed (i.e., scaled) by the scaling function, tj/it) (Mallat, 1991). 

The evolution of the inflexion points, as given by the local extrema of the detail signals, also characterize the regularity of the original signal s inflexion points. 

For the energy calibration the spectrum from the Au(0) foil measured at the same time as the alfalfa spectra was used. The value of 11.918 keV was assigned to the first inflexion point of the Au(0) foil. The XANES energy... 

The comparison of measurements for oxidation of polyisoprene indicates that the increase of chemiluminescence precedes that of oxygen uptake (Figure 12) and that a maximum of chemiluminescence is situated around an inflexion point of the oxygen absorption curve. 

The rate constant k, which involves the rate constant of bimolecular decomposition of hydroperoxides khi from the Scheme 2 may thus be determined, so that we find the value of [d(f/Imax)/df], which is the slope of chemiluminescence-time record at its inflexion point. This ratio is function of k as follows ... 

Figure 20 An increase of induction time of oxidation of polypropylene stabilized by Irganox 1010 (points 1) due to prior sample annealing at 130°C (points 2) in oxygen. The induction time corresponds to the time of cross-section of the straight line passing the CL inflexion point and time axis. It was determined for an oxygen atmosphere and temperature 150°C.

Full details of the kinetics will be published in due course, but for the present it suffices to say that for 1 under the conditions which concern us here the reaction curves beyond the inflexion point are of first order, and that for 5 this applies to the whole reaction curves. As is shown in Figures 2a and b, the reactions obey the kinetic equation... 

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