Minimum negativity

Substances formed at a given potential give rise to a diminution of the reflected intensity as compared with that at the reference potential. As a consequence, the relative reflection band presents a minimum ( negative going band ). On the contrary, the reflected intensity for substances consumed at a given potential present a maximum ( positive going band ). 

Minimum negative partial atomic charge (HYBOT) [23] also Qnegmin... 

Minimum-Negativity-Constrained Fourier Spectrum Continuation... 

The constraint of minimum negativity (Howard, 1981) applies to data for which it is known that the correctly restored function should be all positive. For our formulation, we want to find the coefficients of v(k) that best satisfy this constraint. These coefficients will be those that minimize the negative deviations in the total function u(k) + v(k). The sum of the squared values of the negative deviations is given by... 

Although this procedure was developed from the constraint of minimum negativity, it will easily accommodate other constraints also, with only slight modification. Note that if the summation is not over a different set of data points for each iteration, but over a fixed set of points, the summation need be computed only once, because u(k) and the sinusoids are constant for each value of the variable k. This is true for the finite-extent constraint, in which... 

Fig. 1 Restoration by inverse filtering of the low-frequency band followed by spectral restoration of the high-frequency band with the constraint of minimum negativity, (a) Inverse-filtered result shown in Fig. 4(a) of Chapter 9. Six coefficients are retained, (b) Restored function produced by restoring 16 (complex) coefficients to the Fourier spectrum by applying the constraint of minimum negativity.

We have found it to be generally true that the constraint of minimum negativity produces results much superior to those obtained with the constraint of finite extent. The minimum-negativity procedure has also been found to be extraordinarily insensitive to noise and other error. This is contrasted with the equations resulting from the constraint of finite extent, for which usually only a narrow band of coefficients may be permitted restoration to achieve a stable solution. The best overall results, however, are obtained with a combination of the two constraints. 

Figure 2 shows the constraint of minimum negativity applied to the same deconvolution as that shown in Fig. 1 but with different truncation points for the Fourier spectrum. Figure 2(a) shows restoration to the inverse-filtered estimate with seven (complex) coefficients retained, and illustrates the distortion occurring when too many noise-laden coefficients are retained in the Fourier spectrum. From Fig. 3(b) of Chapter 9 it is evident that the seventh coefficient contains a large amount of noise error. Figure 2(b) shows restoration to the inverse-filtered result with only five complex coefficients retained in the Fourier spectrum. It differs little from the restoration with only six coefficients retained in the inverse-filtered estimate shown in Fig. 1. For both cases shown in Fig. 2, 16 complex coefficients were restored.

Fig. 3 Restoration of Fourier spectrum to the inverse filtering of two noisy infrared peaks using the constraint of minimum negativity, (a) Two merged infrared peaks, (b) Inverse filtering of the infrared peaks with the spectrum truncated after the 10th coefficient, (c) Spectrum restored by minimizing the sum of the squares of the negative regions of the inverse-filtered result. Sixteen (unique complex) coefficients were restored.

Fig. 10 Result of restoring 62 (31 complex) coefficients to the inverse-filtered spectrum of the function shown in Fig. 9 with the constraint of minimum negativity. The broadened spectrum of these narrower peaks necessitates recovering more spectral components to obtain reasonable results.

To further reduce the computational burden, an attempt was made to separate the variables. To see how this may be implemented, let us consider Eq. (A.l), which enforces the minimum negativity constraint. Note that it may also be written ... 

In the case of surface-bound molecules, due to the characteristics of the current obtained when a sequence of potential pulses is applied (see Sect. 6.4.1.2), the use of DSCVC is only recommended for the analysis of non-reversible electrochemical reactions, since for very fast electrochemical reactions (i.e., for values of the dimensionless rate constant which fulfill log( 0r) >0.5), the current becomes negligible, in accordance with Eq. (6.132). The response obtained in DSCVC when non-reversible electrochemical reactions are considered presents two peaks, one maximum positive fV dscvc) an(J one minimum negative (v Sdscvc) which appear for values of the applied potentials EMax and Emin, respectively (with i// >scvc = / >scvc/The cross potential value, at which dscvc = 0,... 

In practice, the key question will be the standard (i.e. the intensity) of review employed in asking what would amount to an excessive negative effect i.e. does it have to be the minimum negative effect possible, while still achieving the goal, or is a less strict standard appropriate ... 

Finally, we plot the wave function for the second excited state 2i(x, y) (see Fig. 4.27e). It has two maxima (positive) and two minima (negative) located at the values 0.25 and 0.75 for x and y. There are two nodal lines, along x = 0.5 and y = 0.5. They divide the x-y plane into quadrants, each of which contains a single maximum (positive) or minimum (negative) value. Make sure that you see how these characteristics trace back to the one-dimensional solutions in Figure 4.24. Figure 4.27f shows the contour plots for y). As the magnitude (absolute... 

The trajectory of the magnetization after the r.f. pulse is shown in Figure 7c. As the magnetization turns around the z axis the projection on the y axis evolves between a maximum positive and a minimum negative value (Figure 7d). The reason why the maxima and minims have not always the same... 

---